VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.6% → 99.7%

Time: 8.3s

Alternatives: 21

Speedup: 2.1×

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: 76.6% 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: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -2 \cdot 10^{+26}:\\ \;\;\;\;\frac{-1 - t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 60000:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot \sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -2e+26)
     (/ (- -1.0 t_0) (sin B))
     (if (<= F 60000.0)
       (+ (/ (- x) (tan B)) (/ F (* (sqrt (fma 2.0 x (fma F F 2.0))) (sin B))))
       (/ (- 1.0 t_0) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -2e+26) {
		tmp = (-1.0 - t_0) / sin(B);
	} else if (F <= 60000.0) {
		tmp = (-x / tan(B)) + (F / (sqrt(fma(2.0, x, fma(F, F, 2.0))) * sin(B)));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -2e+26)
		tmp = Float64(Float64(-1.0 - t_0) / sin(B));
	elseif (F <= 60000.0)
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(F / Float64(sqrt(fma(2.0, x, fma(F, F, 2.0))) * 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]}, If[LessEqual[F, -2e+26], N[(N[(-1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 60000.0], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(F / N[(N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.0000000000000001e26

   1. Initial program 45.8%

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

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. 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 99.8

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

   5. Applied rewrites 99.8%

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

   if -2.0000000000000001e26 < F < 6e4

   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. Add Preprocessing
   3. 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

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

      6. lift-+.f64 N/A

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

      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

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      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

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

   4. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 99.5

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

   6. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. associate-+r+ N/A

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

      9. pow2 N/A

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

      10. metadata-eval N/A

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

      11. sqrt-pow1 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. inv-pow N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

   8. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. *-rgt-identity N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. associate-/l/ N/A

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

      10. lower-/.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-sqrt.f64 N/A

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

      15. lift-sin.f64 99.6

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

   10. Applied rewrites 99.6%

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

   if 6e4 < F

   1. Initial program 55.7%

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

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   4. 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 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+26}:\\ \;\;\;\;\frac{-1 - \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 60000:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot \sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 92.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ t_1 := \cos B \cdot x\\ \mathbf{if}\;F \leq -7000000:\\ \;\;\;\;\frac{-1 - t\_1}{\sin B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{t\_0}{B}\\ \mathbf{elif}\;F \leq 39000:\\ \;\;\;\;\frac{-x}{B} + \frac{t\_0}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ F (sqrt (fma 2.0 x (fma F F 2.0))))) (t_1 (* (cos B) x)))
   (if (<= F -7000000.0)
     (/ (- -1.0 t_1) (sin B))
     (if (<= F 7.5e-58)
       (+ (/ (- x) (tan B)) (/ t_0 B))
       (if (<= F 39000.0)
         (+ (/ (- x) B) (/ t_0 (sin B)))
         (/ (- 1.0 t_1) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = F / sqrt(fma(2.0, x, fma(F, F, 2.0)));
	double t_1 = cos(B) * x;
	double tmp;
	if (F <= -7000000.0) {
		tmp = (-1.0 - t_1) / sin(B);
	} else if (F <= 7.5e-58) {
		tmp = (-x / tan(B)) + (t_0 / B);
	} else if (F <= 39000.0) {
		tmp = (-x / B) + (t_0 / sin(B));
	} else {
		tmp = (1.0 - t_1) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0))))
	t_1 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -7000000.0)
		tmp = Float64(Float64(-1.0 - t_1) / sin(B));
	elseif (F <= 7.5e-58)
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(t_0 / B));
	elseif (F <= 39000.0)
		tmp = Float64(Float64(Float64(-x) / B) + Float64(t_0 / sin(B)));
	else
		tmp = Float64(Float64(1.0 - t_1) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -7000000.0], N[(N[(-1.0 - t$95$1), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.5e-58], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 39000.0], N[(N[((-x) / B), $MachinePrecision] + N[(t$95$0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$1), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7e6

   1. Initial program 49.7%

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

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. 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 99.8

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

   5. Applied rewrites 99.8%

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

   if -7e6 < F < 7.50000000000000002e-58

   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. Add Preprocessing
   3. 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

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

      6. lift-+.f64 N/A

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

      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

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      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

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

   4. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 99.6

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

   6. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. associate-+r+ N/A

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

      9. pow2 N/A

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

      10. metadata-eval N/A

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

      11. sqrt-pow1 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. inv-pow N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in B around 0

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

   11. Applied rewrites 85.4%

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

   if 7.50000000000000002e-58 < F < 39000

   1. Initial program 99.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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

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

      6. lift-+.f64 N/A

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

      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

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      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

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

   4. Applied rewrites 98.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 98.7

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

   6. Applied rewrites 98.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. associate-+r+ N/A

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

      9. pow2 N/A

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

      10. metadata-eval N/A

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

      11. sqrt-pow1 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. inv-pow N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in B around 0

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

       1. associate-*r/ N/A

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

       2. associate-*r/ N/A

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

       3. mul-1-neg N/A

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

       4. lower-/.f64 N/A

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

       5. lower-neg.f64 86.2

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

   11. Applied rewrites 86.2%

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

   if 39000 < F

   1. Initial program 55.7%

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

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   4. 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 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7000000:\\ \;\;\;\;\frac{-1 - \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{B}\\ \mathbf{elif}\;F \leq 39000:\\ \;\;\;\;\frac{-x}{B} + \frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{if}\;F \leq -6 \cdot 10^{+150}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{t\_0}{B}\\ \mathbf{elif}\;F \leq 39000:\\ \;\;\;\;\frac{-x}{B} + \frac{t\_0}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ F (sqrt (fma 2.0 x (fma F F 2.0))))))
   (if (<= F -6e+150)
     (+ (- (/ x B)) (/ -1.0 (sin B)))
     (if (<= F 7.5e-58)
       (+ (/ (- x) (tan B)) (/ t_0 B))
       (if (<= F 39000.0)
         (+ (/ (- x) B) (/ t_0 (sin B)))
         (/ (- 1.0 (* (cos B) x)) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = F / sqrt(fma(2.0, x, fma(F, F, 2.0)));
	double tmp;
	if (F <= -6e+150) {
		tmp = -(x / B) + (-1.0 / sin(B));
	} else if (F <= 7.5e-58) {
		tmp = (-x / tan(B)) + (t_0 / B);
	} else if (F <= 39000.0) {
		tmp = (-x / B) + (t_0 / sin(B));
	} else {
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0))))
	tmp = 0.0
	if (F <= -6e+150)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(-1.0 / sin(B)));
	elseif (F <= 7.5e-58)
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(t_0 / B));
	elseif (F <= 39000.0)
		tmp = Float64(Float64(Float64(-x) / B) + Float64(t_0 / 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[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -6e+150], N[((-N[(x / B), $MachinePrecision]) + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.5e-58], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 39000.0], N[(N[((-x) / B), $MachinePrecision] + N[(t$95$0 / 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 < -6.00000000000000025e150

   1. Initial program 13.0%

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

   3. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   4. 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

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 92.8

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

   8. Applied rewrites 92.8%

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

   if -6.00000000000000025e150 < F < 7.50000000000000002e-58

   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. Add Preprocessing
   3. 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

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

      6. lift-+.f64 N/A

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

      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

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      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

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

   4. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 99.6

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

   6. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. associate-+r+ N/A

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

      9. pow2 N/A

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

      10. metadata-eval N/A

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

      11. sqrt-pow1 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. inv-pow N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in B around 0

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

   11. Applied rewrites 84.0%

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

   if 7.50000000000000002e-58 < F < 39000

   1. Initial program 99.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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

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

      6. lift-+.f64 N/A

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

      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

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      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

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

   4. Applied rewrites 98.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 98.7

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

   6. Applied rewrites 98.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. associate-+r+ N/A

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

      9. pow2 N/A

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

      10. metadata-eval N/A

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

      11. sqrt-pow1 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. inv-pow N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in B around 0

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

       1. associate-*r/ N/A

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

       2. associate-*r/ N/A

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

       3. mul-1-neg N/A

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

       4. lower-/.f64 N/A

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

       5. lower-neg.f64 86.2

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

   11. Applied rewrites 86.2%

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

   if 39000 < F

   1. Initial program 55.7%

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

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   4. 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 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6 \cdot 10^{+150}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{B}\\ \mathbf{elif}\;F \leq 39000:\\ \;\;\;\;\frac{-x}{B} + \frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;F \leq -6 \cdot 10^{+150}:\\ \;\;\;\;t\_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{+120}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{F \cdot \frac{\mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(2, x, 2\right)}{F}}{F}, -0.5, 1\right)}{F}}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x B))))
   (if (<= F -6e+150)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 4.2e+120)
       (+ (/ (- x) (tan B)) (/ (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) B))
       (+
        t_0
        (/ (* F (/ (fma (/ (/ (fma 2.0 x 2.0) F) F) -0.5 1.0) F)) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (F <= -6e+150) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 4.2e+120) {
		tmp = (-x / tan(B)) + ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / B);
	} else {
		tmp = t_0 + ((F * (fma(((fma(2.0, x, 2.0) / F) / F), -0.5, 1.0) / F)) / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	tmp = 0.0
	if (F <= -6e+150)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 4.2e+120)
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / B));
	else
		tmp = Float64(t_0 + Float64(Float64(F * Float64(fma(Float64(Float64(fma(2.0, x, 2.0) / F) / F), -0.5, 1.0) / F)) / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[F, -6e+150], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.2e+120], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(F * N[(N[(N[(N[(N[(2.0 * x + 2.0), $MachinePrecision] / F), $MachinePrecision] / F), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.00000000000000025e150

   1. Initial program 13.0%

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

   3. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   4. 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

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 92.8

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

   8. Applied rewrites 92.8%

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

   if -6.00000000000000025e150 < F < 4.2000000000000001e120

   1. Initial program 98.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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

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

      6. lift-+.f64 N/A

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

      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

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      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

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

   4. Applied rewrites 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 99.5

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

   6. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. associate-+r+ N/A

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

      9. pow2 N/A

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

      10. metadata-eval N/A

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

      11. sqrt-pow1 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. inv-pow N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in B around 0

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

   11. Applied rewrites 80.0%

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

   if 4.2000000000000001e120 < F

   1. Initial program 39.0%

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

   3. Taylor expanded in F around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 67.9%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 47.4

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

   8. Applied rewrites 47.4%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -0.5, 1\right)}{F} \]
   9. Step-by-step derivation

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. associate-*l/ N/A

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

      7. lower-/.f64 N/A

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

   10. Applied rewrites 79.0%

       \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{F \cdot \frac{\mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(2, x, 2\right)}{F}}{F}, -0.5, 1\right)}{F}}{\sin B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6 \cdot 10^{+150}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{+120}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F \cdot \frac{\mathsf{fma}\left(\frac{\frac{\mathsf{fma}\left(2, x, 2\right)}{F}}{F}, -0.5, 1\right)}{F}}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-9} \lor \neg \left(x \leq 2.75 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{-x}{\tan B} + \frac{-1}{\mathsf{fma}\left(B \cdot B, -0.16666666666666666, 1\right) \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B} + \frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -2.9e-9) (not (<= x 2.75e-14)))
   (+ (/ (- x) (tan B)) (/ -1.0 (* (fma (* B B) -0.16666666666666666 1.0) B)))
   (+ (/ (- x) B) (/ (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -2.9e-9) || !(x <= 2.75e-14)) {
		tmp = (-x / tan(B)) + (-1.0 / (fma((B * B), -0.16666666666666666, 1.0) * B));
	} else {
		tmp = (-x / B) + ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -2.9e-9) || !(x <= 2.75e-14))
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(-1.0 / Float64(fma(Float64(B * B), -0.16666666666666666, 1.0) * B)));
	else
		tmp = Float64(Float64(Float64(-x) / B) + Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -2.9e-9], N[Not[LessEqual[x, 2.75e-14]], $MachinePrecision]], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(N[(N[(B * B), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / B), $MachinePrecision] + N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.89999999999999991e-9 or 2.74999999999999996e-14 < x

   1. Initial program 83.3%

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

   3. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   4. 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 94.0

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

   5. Applied rewrites 94.0%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {B}^{2}}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{B}^{2}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{\color{blue}{2}}\right)} \]

      4. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \left(B \cdot B\right)\right)} \]

      5. lower-*.f64 95.3

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)} \]

   8. Applied rewrites 95.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)}} \]

   10. Applied rewrites 95.5%

       \[\leadsto \color{blue}{\frac{-x}{\tan B} + \frac{-1}{\mathsf{fma}\left(B \cdot B, -0.16666666666666666, 1\right) \cdot B}} \]

   if -2.89999999999999991e-9 < x < 2.74999999999999996e-14

   1. Initial program 70.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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

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

      6. lift-+.f64 N/A

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

      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

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      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

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

   4. Applied rewrites 71.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 71.7

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

   6. Applied rewrites 71.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. associate-+r+ N/A

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

      9. pow2 N/A

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

      10. metadata-eval N/A

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

      11. sqrt-pow1 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. inv-pow N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

   8. Applied rewrites 71.8%

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

   9. Taylor expanded in B around 0

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

       1. associate-*r/ N/A

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

       2. associate-*r/ N/A

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

       3. mul-1-neg N/A

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

       4. lower-/.f64 N/A

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

       5. lower-neg.f64 61.1

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

   11. Applied rewrites 61.1%

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

4. Final simplification 76.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-9} \lor \neg \left(x \leq 2.75 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{-x}{\tan B} + \frac{-1}{\mathsf{fma}\left(B \cdot B, -0.16666666666666666, 1\right) \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B} + \frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;F \leq -6 \cdot 10^{+150}:\\ \;\;\;\;t\_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{+238}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{+255}:\\ \;\;\;\;t\_0 + \frac{F}{\sin B} \cdot \frac{1}{F}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x B))))
   (if (<= F -6e+150)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 4.8e+238)
       (+ (/ (- x) (tan B)) (/ (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) B))
       (if (<= F 4.6e+255)
         (+ t_0 (* (/ F (sin B)) (/ 1.0 F)))
         (/ (- 1.0 x) B))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (F <= -6e+150) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 4.8e+238) {
		tmp = (-x / tan(B)) + ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / B);
	} else if (F <= 4.6e+255) {
		tmp = t_0 + ((F / sin(B)) * (1.0 / F));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	tmp = 0.0
	if (F <= -6e+150)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 4.8e+238)
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / B));
	elseif (F <= 4.6e+255)
		tmp = Float64(t_0 + Float64(Float64(F / sin(B)) * Float64(1.0 / F)));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[F, -6e+150], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.8e+238], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.6e+255], N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -6.00000000000000025e150

   1. Initial program 13.0%

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

   3. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   4. 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

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 92.8

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

   8. Applied rewrites 92.8%

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

   if -6.00000000000000025e150 < F < 4.8e238

   1. Initial program 90.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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

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

      6. lift-+.f64 N/A

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

      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

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      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

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

   4. Applied rewrites 94.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 94.7

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

   6. Applied rewrites 94.7%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. associate-+r+ N/A

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

      9. pow2 N/A

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

      10. metadata-eval N/A

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

      11. sqrt-pow1 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. inv-pow N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

   8. Applied rewrites 94.8%

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

   9. Taylor expanded in B around 0

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

   11. Applied rewrites 76.5%

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

   if 4.8e238 < F < 4.6000000000000001e255

   1. Initial program 2.2%

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

   3. Taylor expanded in F around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   5. Applied rewrites 85.9%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 85.9

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

   8. Applied rewrites 85.9%

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

   9. Taylor expanded in F around inf

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

   11. Applied rewrites 85.9%

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

   if 4.6000000000000001e255 < F

   1. Initial program 20.0%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 27.2%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 79.4%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6 \cdot 10^{+150}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 4.8 \cdot 10^{+238}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{+255}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \frac{1}{F}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 56.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{if}\;B \leq 4800:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666 \cdot t\_0\right), t\_0\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B \cdot \left(1 + \left(B \cdot B\right) \cdot \left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ F (sqrt (fma 2.0 x (fma F F 2.0))))))
   (if (<= B 4800.0)
     (/
      (-
       (fma (* B B) (fma 0.3333333333333333 x (* 0.16666666666666666 t_0)) t_0)
       x)
      B)
     (+
      (* x (/ -1.0 (tan B)))
      (/
       -1.0
       (*
        B
        (+
         1.0
         (*
          (* B B)
          (- (* 0.008333333333333333 (* B B)) 0.16666666666666666)))))))))

C

double code(double F, double B, double x) {
	double t_0 = F / sqrt(fma(2.0, x, fma(F, F, 2.0)));
	double tmp;
	if (B <= 4800.0) {
		tmp = (fma((B * B), fma(0.3333333333333333, x, (0.16666666666666666 * t_0)), t_0) - x) / B;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / (B * (1.0 + ((B * B) * ((0.008333333333333333 * (B * B)) - 0.16666666666666666)))));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0))))
	tmp = 0.0
	if (B <= 4800.0)
		tmp = Float64(Float64(fma(Float64(B * B), fma(0.3333333333333333, x, Float64(0.16666666666666666 * t_0)), t_0) - x) / B);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / Float64(B * Float64(1.0 + Float64(Float64(B * B) * Float64(Float64(0.008333333333333333 * Float64(B * B)) - 0.16666666666666666))))));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 4800.0], N[(N[(N[(N[(B * B), $MachinePrecision] * N[(0.3333333333333333 * x + N[(0.16666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(B * N[(1.0 + N[(N[(B * B), $MachinePrecision] * N[(N[(0.008333333333333333 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 4800

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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

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

      6. lift-+.f64 N/A

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

      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

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      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

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

   4. Applied rewrites 84.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 84.8

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

   6. Applied rewrites 84.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. associate-+r+ N/A

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

      9. pow2 N/A

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

      10. metadata-eval N/A

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

      11. sqrt-pow1 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. inv-pow N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

   8. Applied rewrites 84.9%

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

   9. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]

   11. Applied rewrites 54.4%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666 \cdot \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right), \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right) - x}{B}} \]

   if 4800 < B

   1. Initial program 76.4%

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

   3. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   4. 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 53.6

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \color{blue}{\left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \color{blue}{{B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + {B}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \color{blue}{\frac{1}{6}}\right)\right)} \]

      4. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \left(B \cdot B\right) \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \left(B \cdot B\right) \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \left(B \cdot B\right) \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \left(B \cdot B\right) \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right)} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \left(B \cdot B\right) \cdot \left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}\right)\right)} \]

      9. lower-*.f64 47.0

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \left(B \cdot B\right) \cdot \left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666\right)\right)} \]

   8. Applied rewrites 47.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \color{blue}{\left(1 + \left(B \cdot B\right) \cdot \left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666\right)\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4800:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666 \cdot \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right), \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B \cdot \left(1 + \left(B \cdot B\right) \cdot \left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 56.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{if}\;B \leq 4800:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666 \cdot t\_0\right), t\_0\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{-1}{\mathsf{fma}\left(B \cdot B, -0.16666666666666666, 1\right) \cdot B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ F (sqrt (fma 2.0 x (fma F F 2.0))))))
   (if (<= B 4800.0)
     (/
      (-
       (fma (* B B) (fma 0.3333333333333333 x (* 0.16666666666666666 t_0)) t_0)
       x)
      B)
     (+
      (/ (- x) (tan B))
      (/ -1.0 (* (fma (* B B) -0.16666666666666666 1.0) B))))))

C

double code(double F, double B, double x) {
	double t_0 = F / sqrt(fma(2.0, x, fma(F, F, 2.0)));
	double tmp;
	if (B <= 4800.0) {
		tmp = (fma((B * B), fma(0.3333333333333333, x, (0.16666666666666666 * t_0)), t_0) - x) / B;
	} else {
		tmp = (-x / tan(B)) + (-1.0 / (fma((B * B), -0.16666666666666666, 1.0) * B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0))))
	tmp = 0.0
	if (B <= 4800.0)
		tmp = Float64(Float64(fma(Float64(B * B), fma(0.3333333333333333, x, Float64(0.16666666666666666 * t_0)), t_0) - x) / B);
	else
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(-1.0 / Float64(fma(Float64(B * B), -0.16666666666666666, 1.0) * B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 4800.0], N[(N[(N[(N[(B * B), $MachinePrecision] * N[(0.3333333333333333 * x + N[(0.16666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(N[(N[(B * B), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 4800

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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

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

      6. lift-+.f64 N/A

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

      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

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      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

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

   4. Applied rewrites 84.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 84.8

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

   6. Applied rewrites 84.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. associate-+r+ N/A

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

      9. pow2 N/A

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

      10. metadata-eval N/A

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

      11. sqrt-pow1 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. inv-pow N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

   8. Applied rewrites 84.9%

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

   9. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]

   11. Applied rewrites 54.4%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666 \cdot \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right), \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right) - x}{B}} \]

   if 4800 < B

   1. Initial program 76.4%

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

   3. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   4. 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 53.6

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {B}^{2}}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{B}^{2}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{\color{blue}{2}}\right)} \]

      4. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \left(B \cdot B\right)\right)} \]

      5. lower-*.f64 46.4

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)} \]

   8. Applied rewrites 46.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)}} \]

   10. Applied rewrites 46.4%

       \[\leadsto \color{blue}{\frac{-x}{\tan B} + \frac{-1}{\mathsf{fma}\left(B \cdot B, -0.16666666666666666, 1\right) \cdot B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 56.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{if}\;B \leq 4800:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666 \cdot t\_0\right), t\_0\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ F (sqrt (fma 2.0 x (fma F F 2.0))))))
   (if (<= B 4800.0)
     (/
      (-
       (fma (* B B) (fma 0.3333333333333333 x (* 0.16666666666666666 t_0)) t_0)
       x)
      B)
     (+ (* x (/ -1.0 (tan B))) (/ -1.0 B)))))

C

double code(double F, double B, double x) {
	double t_0 = F / sqrt(fma(2.0, x, fma(F, F, 2.0)));
	double tmp;
	if (B <= 4800.0) {
		tmp = (fma((B * B), fma(0.3333333333333333, x, (0.16666666666666666 * t_0)), t_0) - x) / B;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0))))
	tmp = 0.0
	if (B <= 4800.0)
		tmp = Float64(Float64(fma(Float64(B * B), fma(0.3333333333333333, x, Float64(0.16666666666666666 * t_0)), t_0) - x) / B);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 4800.0], N[(N[(N[(N[(B * B), $MachinePrecision] * N[(0.3333333333333333 * x + N[(0.16666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 4800

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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

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

      6. lift-+.f64 N/A

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

      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

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      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

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

   4. Applied rewrites 84.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 84.8

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

   6. Applied rewrites 84.8%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. associate-+r+ N/A

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

      9. pow2 N/A

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

      10. metadata-eval N/A

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

      11. sqrt-pow1 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. inv-pow N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

   8. Applied rewrites 84.9%

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

   9. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]

   11. Applied rewrites 54.4%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666 \cdot \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right), \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right) - x}{B}} \]

   if 4800 < B

   1. Initial program 76.4%

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

   3. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   4. 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 53.6

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

   5. Applied rewrites 53.6%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 44.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4800:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666 \cdot \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right), \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{if}\;F \leq -7000000:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 280000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666 \cdot t\_0\right), t\_0\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ F (sqrt (fma 2.0 x (fma F F 2.0))))))
   (if (<= F -7000000.0)
     (+ (- (/ x B)) (/ -1.0 (sin B)))
     (if (<= F 280000000000.0)
       (/
        (-
         (fma
          (* B B)
          (fma 0.3333333333333333 x (* 0.16666666666666666 t_0))
          t_0)
         x)
        B)
       (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double t_0 = F / sqrt(fma(2.0, x, fma(F, F, 2.0)));
	double tmp;
	if (F <= -7000000.0) {
		tmp = -(x / B) + (-1.0 / sin(B));
	} else if (F <= 280000000000.0) {
		tmp = (fma((B * B), fma(0.3333333333333333, x, (0.16666666666666666 * t_0)), t_0) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0))))
	tmp = 0.0
	if (F <= -7000000.0)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(-1.0 / sin(B)));
	elseif (F <= 280000000000.0)
		tmp = Float64(Float64(fma(Float64(B * B), fma(0.3333333333333333, x, Float64(0.16666666666666666 * t_0)), t_0) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -7000000.0], N[((-N[(x / B), $MachinePrecision]) + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 280000000000.0], N[(N[(N[(N[(B * B), $MachinePrecision] * N[(0.3333333333333333 * x + N[(0.16666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -7e6

   1. Initial program 49.7%

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

   3. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   4. 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 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 85.8

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

   8. Applied rewrites 85.8%

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

   if -7e6 < F < 2.8e11

   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. Add Preprocessing
   3. 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

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

      6. lift-+.f64 N/A

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

      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

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      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

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

   4. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 99.5

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

   6. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. associate-+r+ N/A

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

      9. pow2 N/A

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

      10. metadata-eval N/A

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

      11. sqrt-pow1 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. inv-pow N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]

   11. Applied rewrites 49.8%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666 \cdot \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right), \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right) - x}{B}} \]

   if 2.8e11 < F

   1. Initial program 53.9%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 30.6%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 45.4%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7000000:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 280000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666 \cdot \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right), \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.9% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{if}\;F \leq -115000000:\\ \;\;\;\;\left(-x\right) \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right)}{B} + \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)}\\ \mathbf{elif}\;F \leq 280000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666 \cdot t\_0\right), t\_0\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ F (sqrt (fma 2.0 x (fma F F 2.0))))))
   (if (<= F -115000000.0)
     (+
      (* (- x) (/ (fma -0.3333333333333333 (* B B) 1.0) B))
      (/ -1.0 (* B (+ 1.0 (* -0.16666666666666666 (* B B))))))
     (if (<= F 280000000000.0)
       (/
        (-
         (fma
          (* B B)
          (fma 0.3333333333333333 x (* 0.16666666666666666 t_0))
          t_0)
         x)
        B)
       (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double t_0 = F / sqrt(fma(2.0, x, fma(F, F, 2.0)));
	double tmp;
	if (F <= -115000000.0) {
		tmp = (-x * (fma(-0.3333333333333333, (B * B), 1.0) / B)) + (-1.0 / (B * (1.0 + (-0.16666666666666666 * (B * B)))));
	} else if (F <= 280000000000.0) {
		tmp = (fma((B * B), fma(0.3333333333333333, x, (0.16666666666666666 * t_0)), t_0) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0))))
	tmp = 0.0
	if (F <= -115000000.0)
		tmp = Float64(Float64(Float64(-x) * Float64(fma(-0.3333333333333333, Float64(B * B), 1.0) / B)) + Float64(-1.0 / Float64(B * Float64(1.0 + Float64(-0.16666666666666666 * Float64(B * B))))));
	elseif (F <= 280000000000.0)
		tmp = Float64(Float64(fma(Float64(B * B), fma(0.3333333333333333, x, Float64(0.16666666666666666 * t_0)), t_0) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -115000000.0], N[(N[((-x) * N[(N[(-0.3333333333333333 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(B * N[(1.0 + N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 280000000000.0], N[(N[(N[(N[(B * B), $MachinePrecision] * N[(0.3333333333333333 * x + N[(0.16666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.15e8

   1. Initial program 49.7%

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

   3. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   4. 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 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {B}^{2}}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{B}^{2}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{\color{blue}{2}}\right)} \]

      4. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \left(B \cdot B\right)\right)} \]

      5. lower-*.f64 73.8

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)} \]

   8. Applied rewrites 73.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)}} \]

   9. Taylor expanded in B around 0

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

       1. lower-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1 + \frac{-1}{3} \cdot {B}^{2}}{\color{blue}{B}}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \left(B \cdot B\right)\right)} \]

       2. +-commutative N/A

          \[\leadsto \left(-x \cdot \frac{\frac{-1}{3} \cdot {B}^{2} + 1}{B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \left(B \cdot B\right)\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{3}, {B}^{2}, 1\right)}{B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \left(B \cdot B\right)\right)} \]

       4. pow2 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{3}, B \cdot B, 1\right)}{B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \left(B \cdot B\right)\right)} \]

       5. lift-*.f64 62.5

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right)}{B}\right) + \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)} \]

   11. Applied rewrites 62.5%

       \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right)}{B}}\right) + \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)} \]

   if -1.15e8 < F < 2.8e11

   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. Add Preprocessing
   3. 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

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

      6. lift-+.f64 N/A

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

      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

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      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

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

   4. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 99.5

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

   6. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. associate-+r+ N/A

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

      9. pow2 N/A

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

      10. metadata-eval N/A

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

      11. sqrt-pow1 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. inv-pow N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]

   11. Applied rewrites 49.8%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666 \cdot \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right), \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right) - x}{B}} \]

   if 2.8e11 < F

   1. Initial program 53.9%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 30.6%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 45.4%

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

4. Final simplification 51.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -115000000:\\ \;\;\;\;\left(-x\right) \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right)}{B} + \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)}\\ \mathbf{elif}\;F \leq 280000000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666 \cdot \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right), \frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.8% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.8 \cdot 10^{+118}:\\ \;\;\;\;\left(-x\right) \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right)}{B} + \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)}\\ \mathbf{elif}\;F \leq 3.25 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -0.5, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.8e+118)
   (+
    (* (- x) (/ (fma -0.3333333333333333 (* B B) 1.0) B))
    (/ -1.0 (* B (+ 1.0 (* -0.16666666666666666 (* B B))))))
   (if (<= F 3.25e-6)
     (/ (- (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) 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 <= -5.8e+118) {
		tmp = (-x * (fma(-0.3333333333333333, (B * B), 1.0) / B)) + (-1.0 / (B * (1.0 + (-0.16666666666666666 * (B * B)))));
	} else if (F <= 3.25e-6) {
		tmp = ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - 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 <= -5.8e+118)
		tmp = Float64(Float64(Float64(-x) * Float64(fma(-0.3333333333333333, Float64(B * B), 1.0) / B)) + Float64(-1.0 / Float64(B * Float64(1.0 + Float64(-0.16666666666666666 * Float64(B * B))))));
	elseif (F <= 3.25e-6)
		tmp = Float64(Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - 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, -5.8e+118], N[(N[((-x) * N[(N[(-0.3333333333333333 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(B * N[(1.0 + N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.25e-6], N[(N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $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 < -5.80000000000000032e118

   1. Initial program 34.0%

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

   3. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   4. 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

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {B}^{2}}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{B}^{2}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{\color{blue}{2}}\right)} \]

      4. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \left(B \cdot B\right)\right)} \]

      5. lower-*.f64 67.8

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)} \]

   8. Applied rewrites 67.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)}} \]

   9. Taylor expanded in B around 0

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

       1. lower-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1 + \frac{-1}{3} \cdot {B}^{2}}{\color{blue}{B}}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \left(B \cdot B\right)\right)} \]

       2. +-commutative N/A

          \[\leadsto \left(-x \cdot \frac{\frac{-1}{3} \cdot {B}^{2} + 1}{B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \left(B \cdot B\right)\right)} \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{3}, {B}^{2}, 1\right)}{B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \left(B \cdot B\right)\right)} \]

       4. pow2 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{3}, B \cdot B, 1\right)}{B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \left(B \cdot B\right)\right)} \]

       5. lift-*.f64 55.4

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right)}{B}\right) + \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)} \]

   11. Applied rewrites 55.4%

       \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right)}{B}}\right) + \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)} \]

   if -5.80000000000000032e118 < F < 3.2499999999999998e-6

   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. Add Preprocessing
   3. 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

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

      6. lift-+.f64 N/A

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

      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

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      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

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

   4. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 99.5

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

   6. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. associate-+r+ N/A

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

      9. pow2 N/A

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

      10. metadata-eval N/A

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

      11. sqrt-pow1 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. inv-pow N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in B around 0

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

   11. Applied rewrites 51.8%

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

   if 3.2499999999999998e-6 < F

   1. Initial program 56.3%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 31.7%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 28.0

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

   8. Applied rewrites 28.0%

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

   9. Taylor expanded in F around inf

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

       1. lower--.f64 N/A

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

   11. Applied rewrites 45.7%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -0.5, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.8 \cdot 10^{+118}:\\ \;\;\;\;\left(-x\right) \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right)}{B} + \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)}\\ \mathbf{elif}\;F \leq 3.25 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -0.5, 1\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.4 \cdot 10^{+127}:\\ \;\;\;\;x \cdot \frac{-1}{B} + \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)}\\ \mathbf{elif}\;F \leq 3.25 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -0.5, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.4e+127)
   (+ (* x (/ -1.0 B)) (/ -1.0 (* B (+ 1.0 (* -0.16666666666666666 (* B B))))))
   (if (<= F 3.25e-6)
     (/ (- (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) 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+127) {
		tmp = (x * (-1.0 / B)) + (-1.0 / (B * (1.0 + (-0.16666666666666666 * (B * B)))));
	} else if (F <= 3.25e-6) {
		tmp = ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - 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+127)
		tmp = Float64(Float64(x * Float64(-1.0 / B)) + Float64(-1.0 / Float64(B * Float64(1.0 + Float64(-0.16666666666666666 * Float64(B * B))))));
	elseif (F <= 3.25e-6)
		tmp = Float64(Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - 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+127], N[(N[(x * N[(-1.0 / B), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(B * N[(1.0 + N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.25e-6], N[(N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $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 < -7.39999999999999959e127

   1. Initial program 30.7%

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

   3. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   4. 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

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {B}^{2}}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{B}^{2}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{\color{blue}{2}}\right)} \]

      4. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \left(B \cdot B\right)\right)} \]

      5. lower-*.f64 68.8

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)} \]

   8. Applied rewrites 68.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)}} \]

   9. Taylor expanded in B around 0

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

   11. Applied rewrites 54.6%

       \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)} \]

   if -7.39999999999999959e127 < F < 3.2499999999999998e-6

   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. Add Preprocessing
   3. 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

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

      6. lift-+.f64 N/A

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

      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

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      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

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

   4. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 99.5

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

   6. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. associate-+r+ N/A

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

      9. pow2 N/A

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

      10. metadata-eval N/A

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

      11. sqrt-pow1 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. inv-pow N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in B around 0

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

   11. Applied rewrites 51.7%

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

   if 3.2499999999999998e-6 < F

   1. Initial program 56.3%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 31.7%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 28.0

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

   8. Applied rewrites 28.0%

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

   9. Taylor expanded in F around inf

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

       1. lower--.f64 N/A

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

   11. Applied rewrites 45.7%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -0.5, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.4 \cdot 10^{+127}:\\ \;\;\;\;x \cdot \frac{-1}{B} + \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)}\\ \mathbf{elif}\;F \leq 3.25 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -0.5, 1\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.25 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -0.5, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1e+154)
   (/ (- -1.0 x) B)
   (if (<= F 3.25e-6)
     (/ (- (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) 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 <= -1e+154) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.25e-6) {
		tmp = ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - 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 <= -1e+154)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 3.25e-6)
		tmp = Float64(Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) - 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, -1e+154], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3.25e-6], N[(N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $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 < -1.00000000000000004e154

   1. Initial program 13.0%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 25.4%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 64.7%

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

   if -1.00000000000000004e154 < F < 3.2499999999999998e-6

   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. Add Preprocessing
   3. 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

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

      6. lift-+.f64 N/A

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

      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

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      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

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

   4. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 99.5

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

   6. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. associate-+r+ N/A

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

      9. pow2 N/A

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

      10. metadata-eval N/A

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

      11. sqrt-pow1 N/A

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

      12. pow2 N/A

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

      13. +-commutative N/A

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

      14. inv-pow N/A

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

      15. sqrt-div N/A

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

      16. metadata-eval N/A

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in B around 0

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

   11. Applied rewrites 49.8%

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

   if 3.2499999999999998e-6 < F

   1. Initial program 56.3%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 31.7%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 28.0

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

   8. Applied rewrites 28.0%

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

   9. Taylor expanded in F around inf

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

       1. lower--.f64 N/A

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

   11. Applied rewrites 45.7%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -0.5, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.25 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -0.5, 1\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 51.7% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+64}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.25 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -0.5, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e+64)
   (/ (- -1.0 x) B)
   (if (<= F 3.25e-6)
     (/ (- (* F (/ 1.0 (sqrt (fma F F 2.0)))) 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 <= -2e+64) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.25e-6) {
		tmp = ((F * (1.0 / sqrt(fma(F, F, 2.0)))) - 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 <= -2e+64)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 3.25e-6)
		tmp = Float64(Float64(Float64(F * Float64(1.0 / sqrt(fma(F, F, 2.0)))) - 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, -2e+64], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3.25e-6], N[(N[(N[(F * N[(1.0 / N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $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 < -2.00000000000000004e64

   1. Initial program 43.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 33.5%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 59.0%

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

   if -2.00000000000000004e64 < F < 3.2499999999999998e-6

   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. Add Preprocessing
   3. 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

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

      6. lift-+.f64 N/A

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

      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

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      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

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 49.9%

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

   8. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. pow2 N/A

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

      4. lift-fma.f64 49.9

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

   10. Applied rewrites 49.9%

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

   if 3.2499999999999998e-6 < F

   1. Initial program 56.3%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 31.7%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 28.0

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

   8. Applied rewrites 28.0%

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

   9. Taylor expanded in F around inf

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

       1. lower--.f64 N/A

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

   11. Applied rewrites 45.7%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -0.5, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+64}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.25 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -0.5, 1\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 51.7% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}\\ \mathbf{if}\;F \leq -38:\\ \;\;\;\;\frac{\left(t\_0 \cdot 0.5 - 1\right) - x}{B}\\ \mathbf{elif}\;F \leq 3.25 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \frac{1}{\sqrt{\mathsf{fma}\left(2, x, 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, -0.5, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (fma 2.0 x 2.0) (* F F))))
   (if (<= F -38.0)
     (/ (- (- (* t_0 0.5) 1.0) x) B)
     (if (<= F 3.25e-6)
       (/ (- (* F (/ 1.0 (sqrt (fma 2.0 x 2.0)))) x) B)
       (/ (- (fma t_0 -0.5 1.0) x) B)))))

C

double code(double F, double B, double x) {
	double t_0 = fma(2.0, x, 2.0) / (F * F);
	double tmp;
	if (F <= -38.0) {
		tmp = (((t_0 * 0.5) - 1.0) - x) / B;
	} else if (F <= 3.25e-6) {
		tmp = ((F * (1.0 / sqrt(fma(2.0, x, 2.0)))) - x) / B;
	} else {
		tmp = (fma(t_0, -0.5, 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(fma(2.0, x, 2.0) / Float64(F * F))
	tmp = 0.0
	if (F <= -38.0)
		tmp = Float64(Float64(Float64(Float64(t_0 * 0.5) - 1.0) - x) / B);
	elseif (F <= 3.25e-6)
		tmp = Float64(Float64(Float64(F * Float64(1.0 / sqrt(fma(2.0, x, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(fma(t_0, -0.5, 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -38.0], N[(N[(N[(N[(t$95$0 * 0.5), $MachinePrecision] - 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3.25e-6], N[(N[(N[(F * N[(1.0 / N[Sqrt[N[(2.0 * x + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(t$95$0 * -0.5 + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -38

   1. Initial program 51.5%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 41.3%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 25.8

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

   8. Applied rewrites 25.8%

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

   9. Taylor expanded in F around -inf

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

       1. metadata-eval N/A

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

       2. associate--r+ N/A

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

       3. lower--.f64 N/A

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

   11. Applied rewrites 62.5%

       \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F} \cdot 0.5 - 1\right) - x}{B} \]

   if -38 < F < 3.2499999999999998e-6

   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. Add Preprocessing
   3. 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

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

      6. lift-+.f64 N/A

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

      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

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      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

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 47.5%

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

   8. Taylor expanded in F around 0

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

      1. lower-sqrt.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 47.5

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

   10. Applied rewrites 47.5%

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

   if 3.2499999999999998e-6 < F

   1. Initial program 56.3%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 31.7%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 28.0

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

   8. Applied rewrites 28.0%

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

   9. Taylor expanded in F around inf

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

       1. lower--.f64 N/A

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

   11. Applied rewrites 45.7%

       \[\leadsto \frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -0.5, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -38:\\ \;\;\;\;\frac{\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F} \cdot 0.5 - 1\right) - x}{B}\\ \mathbf{elif}\;F \leq 3.25 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot \frac{1}{\sqrt{\mathsf{fma}\left(2, x, 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -0.5, 1\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 44.5% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.7:\\ \;\;\;\;\frac{\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F} \cdot 0.5 - 1\right) - x}{B}\\ \mathbf{elif}\;F \leq 10^{-66}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(x \cdot -0.3333333333333333, B \cdot B, x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.7)
   (/ (- (- (* (/ (fma 2.0 x 2.0) (* F F)) 0.5) 1.0) x) B)
   (if (<= F 1e-66)
     (- (/ (fma (* x -0.3333333333333333) (* B B) x) B))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.7) {
		tmp = ((((fma(2.0, x, 2.0) / (F * F)) * 0.5) - 1.0) - x) / B;
	} else if (F <= 1e-66) {
		tmp = -(fma((x * -0.3333333333333333), (B * B), x) / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.7)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(2.0, x, 2.0) / Float64(F * F)) * 0.5) - 1.0) - x) / B);
	elseif (F <= 1e-66)
		tmp = Float64(-Float64(fma(Float64(x * -0.3333333333333333), Float64(B * B), x) / B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.7], N[(N[(N[(N[(N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] - 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1e-66], (-N[(N[(N[(x * -0.3333333333333333), $MachinePrecision] * N[(B * B), $MachinePrecision] + x), $MachinePrecision] / B), $MachinePrecision]), N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.70000000000000018

   1. Initial program 51.5%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 41.3%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 25.8

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

   8. Applied rewrites 25.8%

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

   9. Taylor expanded in F around -inf

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

       1. metadata-eval N/A

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

       2. associate--r+ N/A

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

       3. lower--.f64 N/A

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

   11. Applied rewrites 62.5%

       \[\leadsto \frac{\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F} \cdot 0.5 - 1\right) - x}{B} \]

   if -4.70000000000000018 < F < 9.9999999999999998e-67

   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. Add Preprocessing
   3. 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

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

      6. lift-+.f64 N/A

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

      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

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      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

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{x \cdot \cos B}{\sin B} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{x \cdot \cos B}{\sin B} \]

      4. lower-*.f64 N/A

         \[\leadsto -\frac{x \cdot \cos B}{\sin B} \]

      5. lower-cos.f64 N/A

         \[\leadsto -\frac{x \cdot \cos B}{\sin B} \]

      6. lift-sin.f64 70.2

         \[\leadsto -\frac{x \cdot \cos B}{\sin B} \]

   7. Applied rewrites 70.2%

      \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]

   8. Taylor expanded in B around 0

      \[\leadsto -\frac{x + {B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -\frac{x + {B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]

      2. +-commutative N/A

         \[\leadsto -\frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) + x}{B} \]

      3. *-commutative N/A

         \[\leadsto -\frac{\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot {B}^{2} + x}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x, {B}^{2}, x\right)}{B} \]

      5. distribute-rgt-out-- N/A

         \[\leadsto -\frac{\mathsf{fma}\left(x \cdot \left(\frac{-1}{2} - \frac{-1}{6}\right), {B}^{2}, x\right)}{B} \]

      6. metadata-eval N/A

         \[\leadsto -\frac{\mathsf{fma}\left(x \cdot \frac{-1}{3}, {B}^{2}, x\right)}{B} \]

      7. lower-*.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(x \cdot \frac{-1}{3}, {B}^{2}, x\right)}{B} \]

      8. pow2 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(x \cdot \frac{-1}{3}, B \cdot B, x\right)}{B} \]

      9. lift-*.f64 36.9

         \[\leadsto -\frac{\mathsf{fma}\left(x \cdot -0.3333333333333333, B \cdot B, x\right)}{B} \]

   10. Applied rewrites 36.9%

       \[\leadsto -\frac{\mathsf{fma}\left(x \cdot -0.3333333333333333, B \cdot B, x\right)}{B} \]

   if 9.9999999999999998e-67 < F

   1. Initial program 63.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 31.7%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.5%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.7:\\ \;\;\;\;\frac{\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F} \cdot 0.5 - 1\right) - x}{B}\\ \mathbf{elif}\;F \leq 10^{-66}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(x \cdot -0.3333333333333333, B \cdot B, x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 44.5% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 10^{-66}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(x \cdot -0.3333333333333333, B \cdot B, x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.4e-12)
   (/ (- -1.0 x) B)
   (if (<= F 1e-66)
     (- (/ (fma (* x -0.3333333333333333) (* B B) x) B))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.4e-12) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1e-66) {
		tmp = -(fma((x * -0.3333333333333333), (B * B), x) / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.4e-12)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1e-66)
		tmp = Float64(-Float64(fma(Float64(x * -0.3333333333333333), Float64(B * B), x) / B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.4e-12], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1e-66], (-N[(N[(N[(x * -0.3333333333333333), $MachinePrecision] * N[(B * B), $MachinePrecision] + x), $MachinePrecision] / B), $MachinePrecision]), N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.39999999999999987e-12

   1. Initial program 54.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 42.7%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.3%

      \[\leadsto \frac{-1 - x}{B} \]

   if -2.39999999999999987e-12 < F < 9.9999999999999998e-67

   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. Add Preprocessing
   3. 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

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\left(-\frac{1}{2}\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\color{blue}{\left(F \cdot F + 2\right)} + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      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

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + \color{blue}{2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)} \]

      9. lift-neg.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      10. lift-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \]

      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

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}} \]

   4. Applied rewrites 99.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{x \cdot \cos B}{\sin B} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{x \cdot \cos B}{\sin B} \]

      4. lower-*.f64 N/A

         \[\leadsto -\frac{x \cdot \cos B}{\sin B} \]

      5. lower-cos.f64 N/A

         \[\leadsto -\frac{x \cdot \cos B}{\sin B} \]

      6. lift-sin.f64 69.7

         \[\leadsto -\frac{x \cdot \cos B}{\sin B} \]

   7. Applied rewrites 69.7%

      \[\leadsto \color{blue}{-\frac{x \cdot \cos B}{\sin B}} \]

   8. Taylor expanded in B around 0

      \[\leadsto -\frac{x + {B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -\frac{x + {B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]

      2. +-commutative N/A

         \[\leadsto -\frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) + x}{B} \]

      3. *-commutative N/A

         \[\leadsto -\frac{\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot {B}^{2} + x}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x, {B}^{2}, x\right)}{B} \]

      5. distribute-rgt-out-- N/A

         \[\leadsto -\frac{\mathsf{fma}\left(x \cdot \left(\frac{-1}{2} - \frac{-1}{6}\right), {B}^{2}, x\right)}{B} \]

      6. metadata-eval N/A

         \[\leadsto -\frac{\mathsf{fma}\left(x \cdot \frac{-1}{3}, {B}^{2}, x\right)}{B} \]

      7. lower-*.f64 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(x \cdot \frac{-1}{3}, {B}^{2}, x\right)}{B} \]

      8. pow2 N/A

         \[\leadsto -\frac{\mathsf{fma}\left(x \cdot \frac{-1}{3}, B \cdot B, x\right)}{B} \]

      9. lift-*.f64 36.1

         \[\leadsto -\frac{\mathsf{fma}\left(x \cdot -0.3333333333333333, B \cdot B, x\right)}{B} \]

   10. Applied rewrites 36.1%

       \[\leadsto -\frac{\mathsf{fma}\left(x \cdot -0.3333333333333333, B \cdot B, x\right)}{B} \]

   if 9.9999999999999998e-67 < F

   1. Initial program 63.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 31.7%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 39.5%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 10^{-66}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(x \cdot -0.3333333333333333, B \cdot B, x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 44.5% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.9 \cdot 10^{-77}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.9e-77)
   (/ (- -1.0 x) B)
   (if (<= F 3.2e-56) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.9e-77) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.2e-56) {
		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 <= (-3.9d-77)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 3.2d-56) 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 <= -3.9e-77) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.2e-56) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.9e-77:
		tmp = (-1.0 - x) / B
	elif F <= 3.2e-56:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.9e-77)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 3.2e-56)
		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 <= -3.9e-77)
		tmp = (-1.0 - x) / B;
	elseif (F <= 3.2e-56)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.9e-77], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3.2e-56], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.89999999999999979e-77

   1. Initial program 65.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 52.9%

      \[\leadsto \frac{-1 - x}{B} \]

   if -3.89999999999999979e-77 < F < 3.19999999999999986e-56

   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. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 48.6%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

      2. lower-neg.f64 34.6

         \[\leadsto \frac{-x}{B} \]

   8. Applied rewrites 34.6%

      \[\leadsto \frac{-x}{B} \]

   if 3.19999999999999986e-56 < F

   1. Initial program 60.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 32.2%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 41.5%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 42.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.9 \cdot 10^{-77}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-56}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 37.2% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.9 \cdot 10^{-77}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.9e-77) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.9e-77) {
		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 <= (-3.9d-77)) 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 <= -3.9e-77) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.9e-77:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.9e-77)
		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 <= -3.9e-77)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.9e-77], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -3.89999999999999979e-77

   1. Initial program 65.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 43.1%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 52.9%

      \[\leadsto \frac{-1 - x}{B} \]

   if -3.89999999999999979e-77 < F

   1. Initial program 81.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 40.8%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

      2. lower-neg.f64 30.2

         \[\leadsto \frac{-x}{B} \]

   8. Applied rewrites 30.2%

      \[\leadsto \frac{-x}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.9 \cdot 10^{-77}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 29.9% accurate, 26.3× 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 76.3%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

5. Applied rewrites 41.5%

   \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

6. Taylor expanded in F around 0

   \[\leadsto \frac{-1 \cdot x}{B} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

   2. lower-neg.f64 29.3

      \[\leadsto \frac{-x}{B} \]

8. Applied rewrites 29.3%

   \[\leadsto \frac{-x}{B} \]

9. Final simplification 29.3%

   \[\leadsto \frac{-x}{B} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025061 
(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))))))