VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.6% → 99.7%

Time: 6.5s

Alternatives: 19

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -4 \cdot 10^{+20}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 10^{+43}:\\ \;\;\;\;\mathsf{fma}\left(-F, \frac{-1}{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{0.5}}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, 1, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -4e+20)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F 1e+43)
       (fma (- F) (/ -1.0 (* (sin B) (pow (fma 2.0 x (fma F F 2.0)) 0.5))) t_0)
       (fma (/ 1.0 (sin B)) 1.0 t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -4e+20) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1e+43) {
		tmp = fma(-F, (-1.0 / (sin(B) * pow(fma(2.0, x, fma(F, F, 2.0)), 0.5))), t_0);
	} else {
		tmp = fma((1.0 / sin(B)), 1.0, t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -4e+20)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1e+43)
		tmp = fma(Float64(-F), Float64(-1.0 / Float64(sin(B) * (fma(2.0, x, fma(F, F, 2.0)) ^ 0.5))), t_0);
	else
		tmp = fma(Float64(1.0 / sin(B)), 1.0, t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4e+20], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e+43], N[((-F) * N[(-1.0 / N[(N[Sin[B], $MachinePrecision] * N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * 1.0 + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4e20

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.5%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-lft-identity N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 56.5

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

   8. Applied rewrites 56.5%

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

   if -4e20 < F < 1.00000000000000001e43

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. mult-flip N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-flip N/A

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

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

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

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

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lift-/.f64 N/A

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

      17. lift-neg.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 84.9

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

   5. Applied rewrites 84.9%

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

   7. Applied rewrites 84.9%

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

   if 1.00000000000000001e43 < F

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 56.7%

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

Alternative 2: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5 \cdot 10^{+26}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 95000000:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, 1, \frac{-x}{\tan B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -5e+26)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 95000000.0)
       (- (/ F (* (sin B) (sqrt (fma x 2.0 (fma F F 2.0))))) t_0)
       (fma (/ 1.0 (sin B)) 1.0 (/ (- x) (tan B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -5e+26) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 95000000.0) {
		tmp = (F / (sin(B) * sqrt(fma(x, 2.0, fma(F, F, 2.0))))) - t_0;
	} else {
		tmp = fma((1.0 / sin(B)), 1.0, (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -5e+26)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 95000000.0)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(x, 2.0, fma(F, F, 2.0))))) - t_0);
	else
		tmp = fma(Float64(1.0 / sin(B)), 1.0, Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5e+26], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 95000000.0], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * 1.0 + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.0000000000000001e26

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.5%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-lft-identity N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 56.5

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

   8. Applied rewrites 56.5%

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

   if -5.0000000000000001e26 < F < 9.5e7

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. mult-flip N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-flip N/A

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

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

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

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

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lift-/.f64 N/A

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

      17. lift-neg.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 84.9

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

   5. Applied rewrites 84.9%

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

   7. Applied rewrites 84.9%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

   9. Applied rewrites 85.0%

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

   if 9.5e7 < F

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 56.7%

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

Alternative 3: 91.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.7 \cdot 10^{+20}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.000175:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, 1, \frac{-x}{\tan B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2.7e+20)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.000175)
       (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) (/ F B)) t_0)
       (fma (/ 1.0 (sin B)) 1.0 (/ (- x) (tan B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2.7e+20) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.000175) {
		tmp = (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * (F / B)) - t_0;
	} else {
		tmp = fma((1.0 / sin(B)), 1.0, (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.7e+20)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.000175)
		tmp = Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - t_0);
	else
		tmp = fma(Float64(1.0 / sin(B)), 1.0, Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.7e+20], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.000175], N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * 1.0 + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.7e20

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.5%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-lft-identity N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 56.5

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

   8. Applied rewrites 56.5%

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

   if -2.7e20 < F < 1.74999999999999998e-4

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.2

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

   4. Applied rewrites 62.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. distribute-frac-neg N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower--.f64 N/A

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

   6. Applied rewrites 62.3%

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

   if 1.74999999999999998e-4 < F

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 56.7%

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

Alternative 4: 85.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.7 \cdot 10^{+20}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+30}:\\ \;\;\;\;t\_0 \cdot \frac{F}{B} - t\_1\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-t\_0}}, -1 \cdot \frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-F, \frac{-1}{B \cdot F}, \frac{-x}{\tan B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (fma x 2.0 (fma F F 2.0)) -0.5)) (t_1 (/ x (tan B))))
   (if (<= F -2.7e+20)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F 2e+30)
       (- (* t_0 (/ F B)) t_1)
       (if (<= F 2.1e+97)
         (fma (- F) (/ 1.0 (/ (sin B) (- t_0))) (* -1.0 (/ x B)))
         (fma (- F) (/ -1.0 (* B F)) (/ (- x) (tan B))))))))

C

double code(double F, double B, double x) {
	double t_0 = pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -2.7e+20) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= 2e+30) {
		tmp = (t_0 * (F / B)) - t_1;
	} else if (F <= 2.1e+97) {
		tmp = fma(-F, (1.0 / (sin(B) / -t_0)), (-1.0 * (x / B)));
	} else {
		tmp = fma(-F, (-1.0 / (B * F)), (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.7e+20)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= 2e+30)
		tmp = Float64(Float64(t_0 * Float64(F / B)) - t_1);
	elseif (F <= 2.1e+97)
		tmp = fma(Float64(-F), Float64(1.0 / Float64(sin(B) / Float64(-t_0))), Float64(-1.0 * Float64(x / B)));
	else
		tmp = fma(Float64(-F), Float64(-1.0 / Float64(B * F)), Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.7e+20], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 2e+30], N[(N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 2.1e+97], N[((-F) * N[(1.0 / N[(N[Sin[B], $MachinePrecision] / (-t$95$0)), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-F) * N[(-1.0 / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.7e20

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.5%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-lft-identity N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 56.5

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

   8. Applied rewrites 56.5%

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

   if -2.7e20 < F < 2e30

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.2

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

   4. Applied rewrites 62.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. distribute-frac-neg N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower--.f64 N/A

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

   6. Applied rewrites 62.3%

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

   if 2e30 < F < 2.10000000000000012e97

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. mult-flip N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-flip N/A

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

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

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

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

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lift-/.f64 N/A

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

      17. lift-neg.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.1

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

   8. Applied rewrites 58.1%

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

   if 2.10000000000000012e97 < F

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. mult-flip N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-flip N/A

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

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

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

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

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lift-/.f64 N/A

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

      17. lift-neg.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval 70.5

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

   8. Applied rewrites 70.5%

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

   9. Taylor expanded in F around inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 50.6

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

   11. Applied rewrites 50.6%

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

Alternative 5: 85.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.7 \cdot 10^{+20}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+30}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - t\_0\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(-F, \frac{-1}{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{0.5}}, -1 \cdot \frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-F, \frac{-1}{B \cdot F}, \frac{-x}{\tan B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2.7e+20)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 2e+30)
       (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) (/ F B)) t_0)
       (if (<= F 2.1e+97)
         (fma
          (- F)
          (/ -1.0 (* (sin B) (pow (fma 2.0 x (fma F F 2.0)) 0.5)))
          (* -1.0 (/ x B)))
         (fma (- F) (/ -1.0 (* B F)) (/ (- x) (tan B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2.7e+20) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 2e+30) {
		tmp = (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * (F / B)) - t_0;
	} else if (F <= 2.1e+97) {
		tmp = fma(-F, (-1.0 / (sin(B) * pow(fma(2.0, x, fma(F, F, 2.0)), 0.5))), (-1.0 * (x / B)));
	} else {
		tmp = fma(-F, (-1.0 / (B * F)), (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.7e+20)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 2e+30)
		tmp = Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - t_0);
	elseif (F <= 2.1e+97)
		tmp = fma(Float64(-F), Float64(-1.0 / Float64(sin(B) * (fma(2.0, x, fma(F, F, 2.0)) ^ 0.5))), Float64(-1.0 * Float64(x / B)));
	else
		tmp = fma(Float64(-F), Float64(-1.0 / Float64(B * F)), Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.7e+20], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2e+30], N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2.1e+97], N[((-F) * N[(-1.0 / N[(N[Sin[B], $MachinePrecision] * N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-F) * N[(-1.0 / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.7e20

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.5%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-lft-identity N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 56.5

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

   8. Applied rewrites 56.5%

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

   if -2.7e20 < F < 2e30

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.2

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

   4. Applied rewrites 62.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. distribute-frac-neg N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower--.f64 N/A

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

   6. Applied rewrites 62.3%

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

   if 2e30 < F < 2.10000000000000012e97

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. mult-flip N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-flip N/A

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

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

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

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

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lift-/.f64 N/A

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

      17. lift-neg.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 84.9

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

   5. Applied rewrites 84.9%

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

   7. Applied rewrites 84.9%

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

   8. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.1

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

   10. Applied rewrites 58.1%

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

   if 2.10000000000000012e97 < F

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. mult-flip N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-flip N/A

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

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

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

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

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lift-/.f64 N/A

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

      17. lift-neg.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval 70.5

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

   8. Applied rewrites 70.5%

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

   9. Taylor expanded in F around inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 50.6

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

   11. Applied rewrites 50.6%

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

Alternative 6: 77.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{B}, -1, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;x \leq 2.55:\\ \;\;\;\;\mathsf{fma}\left(-F, \frac{-1}{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{0.5}}, -1 \cdot \frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -1.2e-15)
   (fma (/ 1.0 B) -1.0 (/ (- x) (tan B)))
   (if (<= x 2.55)
     (fma
      (- F)
      (/ -1.0 (* (sin B) (pow (fma 2.0 x (fma F F 2.0)) 0.5)))
      (* -1.0 (/ x B)))
     (* -1.0 (/ (* x (cos B)) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -1.2e-15) {
		tmp = fma((1.0 / B), -1.0, (-x / tan(B)));
	} else if (x <= 2.55) {
		tmp = fma(-F, (-1.0 / (sin(B) * pow(fma(2.0, x, fma(F, F, 2.0)), 0.5))), (-1.0 * (x / B)));
	} else {
		tmp = -1.0 * ((x * cos(B)) / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -1.2e-15)
		tmp = fma(Float64(1.0 / B), -1.0, Float64(Float64(-x) / tan(B)));
	elseif (x <= 2.55)
		tmp = fma(Float64(-F), Float64(-1.0 / Float64(sin(B) * (fma(2.0, x, fma(F, F, 2.0)) ^ 0.5))), Float64(-1.0 * Float64(x / B)));
	else
		tmp = Float64(-1.0 * Float64(Float64(x * cos(B)) / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -1.2e-15], N[(N[(1.0 / B), $MachinePrecision] * -1.0 + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.55], N[((-F) * N[(-1.0 / N[(N[Sin[B], $MachinePrecision] * N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.19999999999999997e-15

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.5%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 54.0

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

   9. Applied rewrites 54.0%

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

   if -1.19999999999999997e-15 < x < 2.5499999999999998

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. mult-flip N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-flip N/A

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

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

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

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

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lift-/.f64 N/A

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

      17. lift-neg.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 84.9

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

   5. Applied rewrites 84.9%

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

   7. Applied rewrites 84.9%

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

   8. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.1

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

   10. Applied rewrites 58.1%

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

   if 2.5499999999999998 < x

   1. Initial program 76.6%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 56.3

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

   4. Applied rewrites 56.3%

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

Alternative 7: 77.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{B}, -1, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;x \leq 2.55:\\ \;\;\;\;\mathsf{fma}\left(-F, \frac{-1}{\sin B} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, -1 \cdot \frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -1.2e-15)
   (fma (/ 1.0 B) -1.0 (/ (- x) (tan B)))
   (if (<= x 2.55)
     (fma
      (- F)
      (* (/ -1.0 (sin B)) (pow (fma 2.0 x (fma F F 2.0)) -0.5))
      (* -1.0 (/ x B)))
     (* -1.0 (/ (* x (cos B)) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -1.2e-15) {
		tmp = fma((1.0 / B), -1.0, (-x / tan(B)));
	} else if (x <= 2.55) {
		tmp = fma(-F, ((-1.0 / sin(B)) * pow(fma(2.0, x, fma(F, F, 2.0)), -0.5)), (-1.0 * (x / B)));
	} else {
		tmp = -1.0 * ((x * cos(B)) / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -1.2e-15)
		tmp = fma(Float64(1.0 / B), -1.0, Float64(Float64(-x) / tan(B)));
	elseif (x <= 2.55)
		tmp = fma(Float64(-F), Float64(Float64(-1.0 / sin(B)) * (fma(2.0, x, fma(F, F, 2.0)) ^ -0.5)), Float64(-1.0 * Float64(x / B)));
	else
		tmp = Float64(-1.0 * Float64(Float64(x * cos(B)) / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -1.2e-15], N[(N[(1.0 / B), $MachinePrecision] * -1.0 + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.55], N[((-F) * N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.19999999999999997e-15

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.5%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 54.0

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

   9. Applied rewrites 54.0%

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

   if -1.19999999999999997e-15 < x < 2.5499999999999998

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. mult-flip N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 84.9%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.1

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

   6. Applied rewrites 58.1%

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

   if 2.5499999999999998 < x

   1. Initial program 76.6%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 56.3

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

   4. Applied rewrites 56.3%

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

Alternative 8: 77.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{B}, -1, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;x \leq 2.2:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F, -1 \cdot \frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -1.2e-15)
   (fma (/ 1.0 B) -1.0 (/ (- x) (tan B)))
   (if (<= x 2.2)
     (fma
      (/ 1.0 (sin B))
      (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F)
      (* -1.0 (/ x B)))
     (* -1.0 (/ (* x (cos B)) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -1.2e-15) {
		tmp = fma((1.0 / B), -1.0, (-x / tan(B)));
	} else if (x <= 2.2) {
		tmp = fma((1.0 / sin(B)), (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F), (-1.0 * (x / B)));
	} else {
		tmp = -1.0 * ((x * cos(B)) / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -1.2e-15)
		tmp = fma(Float64(1.0 / B), -1.0, Float64(Float64(-x) / tan(B)));
	elseif (x <= 2.2)
		tmp = fma(Float64(1.0 / sin(B)), Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F), Float64(-1.0 * Float64(x / B)));
	else
		tmp = Float64(-1.0 * Float64(Float64(x * cos(B)) / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -1.2e-15], N[(N[(1.0 / B), $MachinePrecision] * -1.0 + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.2], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.19999999999999997e-15

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.5%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 54.0

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

   9. Applied rewrites 54.0%

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

   if -1.19999999999999997e-15 < x < 2.2000000000000002

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.1

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

   6. Applied rewrites 58.1%

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

   if 2.2000000000000002 < x

   1. Initial program 76.6%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 56.3

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

   4. Applied rewrites 56.3%

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

Alternative 9: 73.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ t_1 := {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;t\_0 \leq -50000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2 + {F}^{2}}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+225}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-F, \frac{-1}{B \cdot F}, \frac{-x}{\tan B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (+
          (- (* x (/ 1.0 (tan B))))
          (*
           (/ F (sin B))
           (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))
        (t_1
         (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) (/ F B)) (/ x (tan B)))))
   (if (<= t_0 -50000.0)
     t_1
     (if (<= t_0 10.0)
       (/ F (* (sin B) (sqrt (+ 2.0 (pow F 2.0)))))
       (if (<= t_0 5e+225)
         t_1
         (fma (- F) (/ -1.0 (* B F)) (/ (- x) (tan B))))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	double t_1 = (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * (F / B)) - (x / tan(B));
	double tmp;
	if (t_0 <= -50000.0) {
		tmp = t_1;
	} else if (t_0 <= 10.0) {
		tmp = F / (sin(B) * sqrt((2.0 + pow(F, 2.0))));
	} else if (t_0 <= 5e+225) {
		tmp = t_1;
	} else {
		tmp = fma(-F, (-1.0 / (B * F)), (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = 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)))))
	t_1 = Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / tan(B)))
	tmp = 0.0
	if (t_0 <= -50000.0)
		tmp = t_1;
	elseif (t_0 <= 10.0)
		tmp = Float64(F / Float64(sin(B) * sqrt(Float64(2.0 + (F ^ 2.0)))));
	elseif (t_0 <= 5e+225)
		tmp = t_1;
	else
		tmp = fma(Float64(-F), Float64(-1.0 / Float64(B * F)), Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(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]}, Block[{t$95$1 = N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -50000.0], t$95$1, If[LessEqual[t$95$0, 10.0], N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+225], t$95$1, N[((-F) * N[(-1.0 / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < -5e4 or 10 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 4.99999999999999981e225

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.2

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

   4. Applied rewrites 62.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. distribute-frac-neg N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower--.f64 N/A

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

   6. Applied rewrites 62.3%

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

   if -5e4 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 10

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. mult-flip N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-flip N/A

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

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

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

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

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lift-/.f64 N/A

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

      17. lift-neg.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 84.9

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

   5. Applied rewrites 84.9%

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

   7. Applied rewrites 84.9%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-pow.f64 30.2

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

   10. Applied rewrites 30.2%

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

   if 4.99999999999999981e225 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))))))

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. mult-flip N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-flip N/A

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

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

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

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

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lift-/.f64 N/A

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

      17. lift-neg.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval 70.5

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

   8. Applied rewrites 70.5%

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

   9. Taylor expanded in F around inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 50.6

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

   11. Applied rewrites 50.6%

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

Alternative 10: 73.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ t_1 := \frac{F}{B} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{\tan B}\\ \mathbf{if}\;t\_0 \leq -50000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2 + {F}^{2}}}\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+225}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-F, \frac{-1}{B \cdot F}, \frac{-x}{\tan B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (+
          (- (* x (/ 1.0 (tan B))))
          (*
           (/ F (sin B))
           (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))
        (t_1
         (-
          (* (/ F B) (/ 1.0 (sqrt (fma x 2.0 (fma F F 2.0)))))
          (/ x (tan B)))))
   (if (<= t_0 -50000.0)
     t_1
     (if (<= t_0 10.0)
       (/ F (* (sin B) (sqrt (+ 2.0 (pow F 2.0)))))
       (if (<= t_0 5e+225)
         t_1
         (fma (- F) (/ -1.0 (* B F)) (/ (- x) (tan B))))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	double t_1 = ((F / B) * (1.0 / sqrt(fma(x, 2.0, fma(F, F, 2.0))))) - (x / tan(B));
	double tmp;
	if (t_0 <= -50000.0) {
		tmp = t_1;
	} else if (t_0 <= 10.0) {
		tmp = F / (sin(B) * sqrt((2.0 + pow(F, 2.0))));
	} else if (t_0 <= 5e+225) {
		tmp = t_1;
	} else {
		tmp = fma(-F, (-1.0 / (B * F)), (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = 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)))))
	t_1 = Float64(Float64(Float64(F / B) * Float64(1.0 / sqrt(fma(x, 2.0, fma(F, F, 2.0))))) - Float64(x / tan(B)))
	tmp = 0.0
	if (t_0 <= -50000.0)
		tmp = t_1;
	elseif (t_0 <= 10.0)
		tmp = Float64(F / Float64(sin(B) * sqrt(Float64(2.0 + (F ^ 2.0)))));
	elseif (t_0 <= 5e+225)
		tmp = t_1;
	else
		tmp = fma(Float64(-F), Float64(-1.0 / Float64(B * F)), Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(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]}, Block[{t$95$1 = N[(N[(N[(F / B), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -50000.0], t$95$1, If[LessEqual[t$95$0, 10.0], N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+225], t$95$1, N[((-F) * N[(-1.0 / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < -5e4 or 10 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 4.99999999999999981e225

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.2

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

   4. Applied rewrites 62.2%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 62.2

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 62.2

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

   6. Applied rewrites 62.2%

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

   8. Applied rewrites 62.3%

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

   if -5e4 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 10

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. mult-flip N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-flip N/A

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

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

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

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

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lift-/.f64 N/A

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

      17. lift-neg.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 84.9

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

   5. Applied rewrites 84.9%

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

   7. Applied rewrites 84.9%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-pow.f64 30.2

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

   10. Applied rewrites 30.2%

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

   if 4.99999999999999981e225 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))))))

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. mult-flip N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 84.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-flip N/A

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

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

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

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

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. metadata-eval N/A

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

      15. metadata-eval N/A

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

      16. lift-/.f64 N/A

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

      17. lift-neg.f64 N/A

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

      18. mul-1-neg N/A

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

      19. lower-neg.f64 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval 70.5

          \[\leadsto \mathsf{fma}\left(-F, -1 \cdot \frac{{\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{B}, \frac{-x}{\tan B}\right) \]

   8. Applied rewrites 70.5%

      \[\leadsto \mathsf{fma}\left(-F, \color{blue}{-1 \cdot \frac{{\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{B}}, \frac{-x}{\tan B}\right) \]

   9. Taylor expanded in F around inf

      \[\leadsto \mathsf{fma}\left(-F, \frac{-1}{\color{blue}{B \cdot F}}, \frac{-x}{\tan B}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{-1}{B \cdot \color{blue}{F}}, \frac{-x}{\tan B}\right) \]

       2. lower-*.f64 50.6

          \[\leadsto \mathsf{fma}\left(-F, \frac{-1}{B \cdot F}, \frac{-x}{\tan B}\right) \]

   11. Applied rewrites 50.6%

       \[\leadsto \mathsf{fma}\left(-F, \frac{-1}{\color{blue}{B \cdot F}}, \frac{-x}{\tan B}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 70.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{1}{B}, -1, \frac{-x}{\tan B}\right)\\ \mathbf{if}\;x \leq -4.3 \cdot 10^{-35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-84}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2 + {F}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (fma (/ 1.0 B) -1.0 (/ (- x) (tan B)))))
   (if (<= x -4.3e-35)
     t_0
     (if (<= x 2.05e-84) (/ F (* (sin B) (sqrt (+ 2.0 (pow F 2.0))))) t_0))))

C

double code(double F, double B, double x) {
	double t_0 = fma((1.0 / B), -1.0, (-x / tan(B)));
	double tmp;
	if (x <= -4.3e-35) {
		tmp = t_0;
	} else if (x <= 2.05e-84) {
		tmp = F / (sin(B) * sqrt((2.0 + pow(F, 2.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(Float64(1.0 / B), -1.0, Float64(Float64(-x) / tan(B)))
	tmp = 0.0
	if (x <= -4.3e-35)
		tmp = t_0;
	elseif (x <= 2.05e-84)
		tmp = Float64(F / Float64(sin(B) * sqrt(Float64(2.0 + (F ^ 2.0)))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(1.0 / B), $MachinePrecision] * -1.0 + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.3e-35], t$95$0, If[LessEqual[x, 2.05e-84], N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.3000000000000002e-35 or 2.05000000000000003e-84 < x

   1. Initial program 76.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. mult-flip N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{1}{\sin B} \cdot \left(F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right)} \]

   3. Applied rewrites 85.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in F around -inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, \color{blue}{-1}, \frac{-x}{\tan B}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 56.5%

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, \color{blue}{-1}, \frac{-x}{\tan B}\right) \]

   7. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{B}}, -1, \frac{-x}{\tan B}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 54.0

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{B}}, -1, \frac{-x}{\tan B}\right) \]

   9. Applied rewrites 54.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{B}}, -1, \frac{-x}{\tan B}\right) \]

   if -4.3000000000000002e-35 < x < 2.05000000000000003e-84

   1. Initial program 76.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(F\right)}{\mathsf{neg}\left(\sin B\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. mult-flip N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(F\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sin B\right)}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(F\right)\right) \cdot \left(\frac{1}{\mathsf{neg}\left(\sin B\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(F\right), \frac{1}{\mathsf{neg}\left(\sin B\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right)} \]

   3. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-F, \frac{-1}{\sin B} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{-1}{\sin B} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}, \frac{-x}{\tan B}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{-1}{\sin B}} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}, \frac{-x}{\tan B}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{-1 \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}}, \frac{-x}{\tan B}\right) \]

      4. div-flip N/A

         \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{1}{\frac{\sin B}{-1 \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}}, \frac{-x}{\tan B}\right) \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{1}{\frac{\sin B}{-1 \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}}, \frac{-x}{\tan B}\right) \]

      6. lower-unsound-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\color{blue}{\frac{\sin B}{-1 \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}}, \frac{-x}{\tan B}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\color{blue}{\left(2 \cdot x + \mathsf{fma}\left(F, F, 2\right)\right)}}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\color{blue}{2 \cdot x} + \mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\color{blue}{\left(\mathsf{fma}\left(F, F, 2\right) + 2 \cdot x\right)}}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\color{blue}{\left(\mathsf{fma}\left(F, F, 2\right) + 2 \cdot x\right)}}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      11. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\color{blue}{\left(F \cdot F + 2\right)} + 2 \cdot x\right)}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\left(\color{blue}{F \cdot F} + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\color{blue}{\left(F \cdot F + 2\right)} + 2 \cdot x\right)}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}}}, \frac{-x}{\tan B}\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)}}}, \frac{-x}{\tan B}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)}}}, \frac{-x}{\tan B}\right) \]

      17. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\color{blue}{\left(-\frac{1}{2}\right)}}}}, \frac{-x}{\tan B}\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{\color{blue}{\mathsf{neg}\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)}}}, \frac{-x}{\tan B}\right) \]

      19. lower-neg.f64 84.9

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{\color{blue}{-{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}}}, \frac{-x}{\tan B}\right) \]

   5. Applied rewrites 84.9%

      \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{1}{\frac{\sin B}{-{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}, \frac{-x}{\tan B}\right) \]

   7. Applied rewrites 84.9%

      \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{-1}{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{0.5}}}, \frac{-x}{\tan B}\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B \cdot \sqrt{2 + {F}^{2}}}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F}{\color{blue}{\sin B \cdot \sqrt{2 + {F}^{2}}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{F}{\sin B \cdot \color{blue}{\sqrt{2 + {F}^{2}}}} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{F}{\sin B \cdot \sqrt{\color{blue}{2 + {F}^{2}}}} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{F}{\sin B \cdot \sqrt{2 + {F}^{2}}} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{F}{\sin B \cdot \sqrt{2 + {F}^{2}}} \]

      6. lower-pow.f64 30.2

         \[\leadsto \frac{F}{\sin B \cdot \sqrt{2 + {F}^{2}}} \]

   10. Applied rewrites 30.2%

       \[\leadsto \color{blue}{\frac{F}{\sin B \cdot \sqrt{2 + {F}^{2}}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 58.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 2.65:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \mathsf{fma}\left(2, x, {F}^{2}\right)}}\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{B}, -1, \frac{-x}{\tan B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 2.65)
   (/ (fma -1.0 x (/ F (sqrt (+ 2.0 (fma 2.0 x (pow F 2.0)))))) B)
   (fma (/ 1.0 B) -1.0 (/ (- x) (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 2.65) {
		tmp = fma(-1.0, x, (F / sqrt((2.0 + fma(2.0, x, pow(F, 2.0)))))) / B;
	} else {
		tmp = fma((1.0 / B), -1.0, (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 2.65)
		tmp = Float64(fma(-1.0, x, Float64(F / sqrt(Float64(2.0 + fma(2.0, x, (F ^ 2.0)))))) / B);
	else
		tmp = fma(Float64(1.0 / B), -1.0, Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 2.65], N[(N[(-1.0 * x + N[(F / N[Sqrt[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] * -1.0 + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.64999999999999991

   1. Initial program 76.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(F\right)}{\mathsf{neg}\left(\sin B\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. mult-flip N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(F\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sin B\right)}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(F\right)\right) \cdot \left(\frac{1}{\mathsf{neg}\left(\sin B\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(F\right), \frac{1}{\mathsf{neg}\left(\sin B\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right)} \]

   3. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-F, \frac{-1}{\sin B} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{-1}{\sin B} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}, \frac{-x}{\tan B}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{-1}{\sin B}} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}, \frac{-x}{\tan B}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{-1 \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}}, \frac{-x}{\tan B}\right) \]

      4. div-flip N/A

         \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{1}{\frac{\sin B}{-1 \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}}, \frac{-x}{\tan B}\right) \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{1}{\frac{\sin B}{-1 \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}}, \frac{-x}{\tan B}\right) \]

      6. lower-unsound-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\color{blue}{\frac{\sin B}{-1 \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}}, \frac{-x}{\tan B}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\color{blue}{\left(2 \cdot x + \mathsf{fma}\left(F, F, 2\right)\right)}}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\color{blue}{2 \cdot x} + \mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\color{blue}{\left(\mathsf{fma}\left(F, F, 2\right) + 2 \cdot x\right)}}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\color{blue}{\left(\mathsf{fma}\left(F, F, 2\right) + 2 \cdot x\right)}}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      11. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\color{blue}{\left(F \cdot F + 2\right)} + 2 \cdot x\right)}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\left(\color{blue}{F \cdot F} + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\color{blue}{\left(F \cdot F + 2\right)} + 2 \cdot x\right)}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}}}, \frac{-x}{\tan B}\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)}}}, \frac{-x}{\tan B}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)}}}, \frac{-x}{\tan B}\right) \]

      17. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\color{blue}{\left(-\frac{1}{2}\right)}}}}, \frac{-x}{\tan B}\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{\color{blue}{\mathsf{neg}\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)}}}, \frac{-x}{\tan B}\right) \]

      19. lower-neg.f64 84.9

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{\color{blue}{-{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}}}, \frac{-x}{\tan B}\right) \]

   5. Applied rewrites 84.9%

      \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{1}{\frac{\sin B}{-{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}, \frac{-x}{\tan B}\right) \]

   7. Applied rewrites 84.9%

      \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{-1}{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{0.5}}}, \frac{-x}{\tan B}\right) \]

   8. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{-1 \cdot x + \frac{F}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot x + \frac{F}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}}{\color{blue}{B}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}{B} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \mathsf{fma}\left(2, x, {F}^{2}\right)}}\right)}{B} \]

      7. lower-pow.f64 44.4

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \mathsf{fma}\left(2, x, {F}^{2}\right)}}\right)}{B} \]

   10. Applied rewrites 44.4%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \mathsf{fma}\left(2, x, {F}^{2}\right)}}\right)}{B}} \]

   if 2.64999999999999991 < B

   1. Initial program 76.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. mult-flip N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{1}{\sin B} \cdot \left(F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right)} \]

   3. Applied rewrites 85.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in F around -inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, \color{blue}{-1}, \frac{-x}{\tan B}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 56.5%

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, \color{blue}{-1}, \frac{-x}{\tan B}\right) \]

   7. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{B}}, -1, \frac{-x}{\tan B}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 54.0

         \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{B}}, -1, \frac{-x}{\tan B}\right) \]

   9. Applied rewrites 54.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{B}}, -1, \frac{-x}{\tan B}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 56.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -2.7 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, -1 \cdot \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{+31}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \mathsf{fma}\left(2, x, {F}^{2}\right)}}\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))))
   (if (<= F -2.7e+20)
     (fma t_0 -1.0 (* -1.0 (/ x B)))
     (if (<= F 2.2e+31)
       (/ (fma -1.0 x (/ F (sqrt (+ 2.0 (fma 2.0 x (pow F 2.0)))))) B)
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double tmp;
	if (F <= -2.7e+20) {
		tmp = fma(t_0, -1.0, (-1.0 * (x / B)));
	} else if (F <= 2.2e+31) {
		tmp = fma(-1.0, x, (F / sqrt((2.0 + fma(2.0, x, pow(F, 2.0)))))) / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -2.7e+20)
		tmp = fma(t_0, -1.0, Float64(-1.0 * Float64(x / B)));
	elseif (F <= 2.2e+31)
		tmp = Float64(fma(-1.0, x, Float64(F / sqrt(Float64(2.0 + fma(2.0, x, (F ^ 2.0)))))) / B);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.7e+20], N[(t$95$0 * -1.0 + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.2e+31], N[(N[(-1.0 * x + N[(F / N[Sqrt[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.7e20

   1. Initial program 76.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. mult-flip N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{\sin B} \cdot F\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\frac{1}{\sin B} \cdot \left(F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right)} \]

   3. Applied rewrites 85.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in F around -inf

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, \color{blue}{-1}, \frac{-x}{\tan B}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 56.5%

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, \color{blue}{-1}, \frac{-x}{\tan B}\right) \]

   7. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, -1, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, -1, -1 \cdot \color{blue}{\frac{x}{B}}\right) \]

      2. lower-/.f64 37.0

         \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, -1, -1 \cdot \frac{x}{\color{blue}{B}}\right) \]

   9. Applied rewrites 37.0%

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, -1, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]

   if -2.7e20 < F < 2.2000000000000001e31

   1. Initial program 76.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(F\right)}{\mathsf{neg}\left(\sin B\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. mult-flip N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(F\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sin B\right)}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(F\right)\right) \cdot \left(\frac{1}{\mathsf{neg}\left(\sin B\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(F\right), \frac{1}{\mathsf{neg}\left(\sin B\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right)} \]

   3. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-F, \frac{-1}{\sin B} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{-1}{\sin B} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}, \frac{-x}{\tan B}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{-1}{\sin B}} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}, \frac{-x}{\tan B}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{-1 \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}}, \frac{-x}{\tan B}\right) \]

      4. div-flip N/A

         \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{1}{\frac{\sin B}{-1 \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}}, \frac{-x}{\tan B}\right) \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{1}{\frac{\sin B}{-1 \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}}, \frac{-x}{\tan B}\right) \]

      6. lower-unsound-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\color{blue}{\frac{\sin B}{-1 \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}}, \frac{-x}{\tan B}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\color{blue}{\left(2 \cdot x + \mathsf{fma}\left(F, F, 2\right)\right)}}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\color{blue}{2 \cdot x} + \mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\color{blue}{\left(\mathsf{fma}\left(F, F, 2\right) + 2 \cdot x\right)}}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\color{blue}{\left(\mathsf{fma}\left(F, F, 2\right) + 2 \cdot x\right)}}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      11. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\color{blue}{\left(F \cdot F + 2\right)} + 2 \cdot x\right)}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\left(\color{blue}{F \cdot F} + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\color{blue}{\left(F \cdot F + 2\right)} + 2 \cdot x\right)}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}}}, \frac{-x}{\tan B}\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)}}}, \frac{-x}{\tan B}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)}}}, \frac{-x}{\tan B}\right) \]

      17. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\color{blue}{\left(-\frac{1}{2}\right)}}}}, \frac{-x}{\tan B}\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{\color{blue}{\mathsf{neg}\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)}}}, \frac{-x}{\tan B}\right) \]

      19. lower-neg.f64 84.9

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{\color{blue}{-{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}}}, \frac{-x}{\tan B}\right) \]

   5. Applied rewrites 84.9%

      \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{1}{\frac{\sin B}{-{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}, \frac{-x}{\tan B}\right) \]

   7. Applied rewrites 84.9%

      \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{-1}{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{0.5}}}, \frac{-x}{\tan B}\right) \]

   8. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{-1 \cdot x + \frac{F}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot x + \frac{F}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}}{\color{blue}{B}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}{B} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \mathsf{fma}\left(2, x, {F}^{2}\right)}}\right)}{B} \]

      7. lower-pow.f64 44.4

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \mathsf{fma}\left(2, x, {F}^{2}\right)}}\right)}{B} \]

   10. Applied rewrites 44.4%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \mathsf{fma}\left(2, x, {F}^{2}\right)}}\right)}{B}} \]

   if 2.2000000000000001e31 < F

   1. Initial program 76.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 16.9

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 16.9%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 47.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.98:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \mathsf{fma}\left(2, x, {F}^{2}\right)}}\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.98)
   (/ (fma -1.0 x (/ F (sqrt (+ 2.0 (fma 2.0 x (pow F 2.0)))))) B)
   (/ 1.0 (sin B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.98) {
		tmp = fma(-1.0, x, (F / sqrt((2.0 + fma(2.0, x, pow(F, 2.0)))))) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.98)
		tmp = Float64(fma(-1.0, x, Float64(F / sqrt(Float64(2.0 + fma(2.0, x, (F ^ 2.0)))))) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.98], N[(N[(-1.0 * x + N[(F / N[Sqrt[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.97999999999999998

   1. Initial program 76.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(F\right)}{\mathsf{neg}\left(\sin B\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. mult-flip N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(F\right)\right) \cdot \frac{1}{\mathsf{neg}\left(\sin B\right)}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(F\right)\right) \cdot \left(\frac{1}{\mathsf{neg}\left(\sin B\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(F\right), \frac{1}{\mathsf{neg}\left(\sin B\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right)} \]

   3. Applied rewrites 84.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-F, \frac{-1}{\sin B} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{-x}{\tan B}\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{-1}{\sin B} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}, \frac{-x}{\tan B}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{-1}{\sin B}} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}, \frac{-x}{\tan B}\right) \]

      3. associate-*l/ N/A

         \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{-1 \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}}, \frac{-x}{\tan B}\right) \]

      4. div-flip N/A

         \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{1}{\frac{\sin B}{-1 \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}}, \frac{-x}{\tan B}\right) \]

      5. lower-unsound-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{1}{\frac{\sin B}{-1 \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}}, \frac{-x}{\tan B}\right) \]

      6. lower-unsound-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\color{blue}{\frac{\sin B}{-1 \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}}, \frac{-x}{\tan B}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\color{blue}{\left(2 \cdot x + \mathsf{fma}\left(F, F, 2\right)\right)}}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\color{blue}{2 \cdot x} + \mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\color{blue}{\left(\mathsf{fma}\left(F, F, 2\right) + 2 \cdot x\right)}}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      10. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\color{blue}{\left(\mathsf{fma}\left(F, F, 2\right) + 2 \cdot x\right)}}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      11. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\color{blue}{\left(F \cdot F + 2\right)} + 2 \cdot x\right)}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\left(\color{blue}{F \cdot F} + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      13. lift-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\color{blue}{\left(F \cdot F + 2\right)} + 2 \cdot x\right)}^{\frac{-1}{2}}}}, \frac{-x}{\tan B}\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}}}, \frac{-x}{\tan B}\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)}}}, \frac{-x}{\tan B}\right) \]

      16. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)}}}, \frac{-x}{\tan B}\right) \]

      17. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{-1 \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\color{blue}{\left(-\frac{1}{2}\right)}}}}, \frac{-x}{\tan B}\right) \]

      18. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{\color{blue}{\mathsf{neg}\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)}}}, \frac{-x}{\tan B}\right) \]

      19. lower-neg.f64 84.9

          \[\leadsto \mathsf{fma}\left(-F, \frac{1}{\frac{\sin B}{\color{blue}{-{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}}}, \frac{-x}{\tan B}\right) \]

   5. Applied rewrites 84.9%

      \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{1}{\frac{\sin B}{-{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}}, \frac{-x}{\tan B}\right) \]

   7. Applied rewrites 84.9%

      \[\leadsto \mathsf{fma}\left(-F, \color{blue}{\frac{-1}{\sin B \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{0.5}}}, \frac{-x}{\tan B}\right) \]

   8. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{-1 \cdot x + \frac{F}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot x + \frac{F}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}}{\color{blue}{B}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}{B} \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}{B} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \mathsf{fma}\left(2, x, {F}^{2}\right)}}\right)}{B} \]

      7. lower-pow.f64 44.4

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \mathsf{fma}\left(2, x, {F}^{2}\right)}}\right)}{B} \]

   10. Applied rewrites 44.4%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \mathsf{fma}\left(2, x, {F}^{2}\right)}}\right)}{B}} \]

   if 0.97999999999999998 < B

   1. Initial program 76.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 16.9

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 16.9%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 37.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.95 \cdot 10^{+20}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 3300000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{F}, \frac{F}{B}, \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot 0.3333333333333333 - x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.95e+20)
   (/ -1.0 (sin B))
   (if (<= F 3300000000.0)
     (fma (/ -1.0 F) (/ F B) (/ (- (* (* (* B B) x) 0.3333333333333333) x) B))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.95e+20) {
		tmp = -1.0 / sin(B);
	} else if (F <= 3300000000.0) {
		tmp = fma((-1.0 / F), (F / B), (((((B * B) * x) * 0.3333333333333333) - x) / B));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.95e+20)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 3300000000.0)
		tmp = fma(Float64(-1.0 / F), Float64(F / B), Float64(Float64(Float64(Float64(Float64(B * B) * x) * 0.3333333333333333) - x) / B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.95e+20], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3300000000.0], N[(N[(-1.0 / F), $MachinePrecision] * N[(F / B), $MachinePrecision] + N[(N[(N[(N[(N[(B * B), $MachinePrecision] * x), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.95e20

   1. Initial program 76.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.2

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -2.95e20 < F < 3.3e9

   1. Initial program 76.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 62.2

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\color{blue}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 62.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{\color{blue}{B}} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      5. lower-pow.f64 36.0

         \[\leadsto \frac{0.3333333333333333 \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   7. Applied rewrites 36.0%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   9. Applied rewrites 36.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{F}{B}, \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot 0.3333333333333333 - x}{B}\right)} \]

   10. Taylor expanded in F around -inf

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{F}}, \frac{F}{B}, \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot \frac{1}{3} - x}{B}\right) \]
   11. Step-by-step derivation

       1. lower-/.f64 22.4

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{F}}, \frac{F}{B}, \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot 0.3333333333333333 - x}{B}\right) \]

   12. Applied rewrites 22.4%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{F}}, \frac{F}{B}, \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot 0.3333333333333333 - x}{B}\right) \]

   if 3.3e9 < F

   1. Initial program 76.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 16.9

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 16.9%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 31.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.7 \cdot 10^{+20}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{F}, \frac{F}{B}, \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot 0.3333333333333333 - x}{B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.7e+20)
   (/ -1.0 (sin B))
   (fma (/ 1.0 F) (/ F B) (/ (- (* (* (* B B) x) 0.3333333333333333) x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.7e+20) {
		tmp = -1.0 / sin(B);
	} else {
		tmp = fma((1.0 / F), (F / B), (((((B * B) * x) * 0.3333333333333333) - x) / B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.7e+20)
		tmp = Float64(-1.0 / sin(B));
	else
		tmp = fma(Float64(1.0 / F), Float64(F / B), Float64(Float64(Float64(Float64(Float64(B * B) * x) * 0.3333333333333333) - x) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.7e+20], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / F), $MachinePrecision] * N[(F / B), $MachinePrecision] + N[(N[(N[(N[(N[(B * B), $MachinePrecision] * x), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -2.7e20

   1. Initial program 76.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.2

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -2.7e20 < F

   1. Initial program 76.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 62.2

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\color{blue}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 62.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{\color{blue}{B}} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      5. lower-pow.f64 36.0

         \[\leadsto \frac{0.3333333333333333 \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   7. Applied rewrites 36.0%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   9. Applied rewrites 36.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{F}{B}, \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot 0.3333333333333333 - x}{B}\right)} \]

   10. Taylor expanded in F around inf

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F}}, \frac{F}{B}, \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot \frac{1}{3} - x}{B}\right) \]
   11. Step-by-step derivation

       1. lower-/.f64 21.5

          \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{F}}, \frac{F}{B}, \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot 0.3333333333333333 - x}{B}\right) \]

   12. Applied rewrites 21.5%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F}}, \frac{F}{B}, \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot 0.3333333333333333 - x}{B}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 26.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot 0.3333333333333333 - x}{B}\\ \mathbf{if}\;F \leq -8.5 \cdot 10^{-263}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{F}, \frac{F}{B}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{F}, \frac{F}{B}, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- (* (* (* B B) x) 0.3333333333333333) x) B)))
   (if (<= F -8.5e-263)
     (fma (/ -1.0 F) (/ F B) t_0)
     (fma (/ 1.0 F) (/ F B) t_0))))

C

double code(double F, double B, double x) {
	double t_0 = ((((B * B) * x) * 0.3333333333333333) - x) / B;
	double tmp;
	if (F <= -8.5e-263) {
		tmp = fma((-1.0 / F), (F / B), t_0);
	} else {
		tmp = fma((1.0 / F), (F / B), t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(Float64(Float64(B * B) * x) * 0.3333333333333333) - x) / B)
	tmp = 0.0
	if (F <= -8.5e-263)
		tmp = fma(Float64(-1.0 / F), Float64(F / B), t_0);
	else
		tmp = fma(Float64(1.0 / F), Float64(F / B), t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(N[(N[(B * B), $MachinePrecision] * x), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[F, -8.5e-263], N[(N[(-1.0 / F), $MachinePrecision] * N[(F / B), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(1.0 / F), $MachinePrecision] * N[(F / B), $MachinePrecision] + t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if F < -8.49999999999999975e-263

   1. Initial program 76.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 62.2

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\color{blue}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 62.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{\color{blue}{B}} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      5. lower-pow.f64 36.0

         \[\leadsto \frac{0.3333333333333333 \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   7. Applied rewrites 36.0%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   9. Applied rewrites 36.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{F}{B}, \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot 0.3333333333333333 - x}{B}\right)} \]

   10. Taylor expanded in F around -inf

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{F}}, \frac{F}{B}, \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot \frac{1}{3} - x}{B}\right) \]
   11. Step-by-step derivation

       1. lower-/.f64 22.4

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{F}}, \frac{F}{B}, \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot 0.3333333333333333 - x}{B}\right) \]

   12. Applied rewrites 22.4%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{F}}, \frac{F}{B}, \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot 0.3333333333333333 - x}{B}\right) \]

   if -8.49999999999999975e-263 < F

   1. Initial program 76.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 62.2

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\color{blue}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 62.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{\color{blue}{B}} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      5. lower-pow.f64 36.0

         \[\leadsto \frac{0.3333333333333333 \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   7. Applied rewrites 36.0%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   9. Applied rewrites 36.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{F}{B}, \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot 0.3333333333333333 - x}{B}\right)} \]

   10. Taylor expanded in F around inf

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F}}, \frac{F}{B}, \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot \frac{1}{3} - x}{B}\right) \]
   11. Step-by-step derivation

       1. lower-/.f64 21.5

          \[\leadsto \mathsf{fma}\left(\frac{1}{\color{blue}{F}}, \frac{F}{B}, \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot 0.3333333333333333 - x}{B}\right) \]

   12. Applied rewrites 21.5%

       \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{F}}, \frac{F}{B}, \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot 0.3333333333333333 - x}{B}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 22.4% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{-1}{F}, \frac{F}{B}, \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot 0.3333333333333333 - x}{B}\right) \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (fma (/ -1.0 F) (/ F B) (/ (- (* (* (* B B) x) 0.3333333333333333) x) B)))

C

double code(double F, double B, double x) {
	return fma((-1.0 / F), (F / B), (((((B * B) * x) * 0.3333333333333333) - x) / B));
}

Julia

function code(F, B, x)
	return fma(Float64(-1.0 / F), Float64(F / B), Float64(Float64(Float64(Float64(Float64(B * B) * x) * 0.3333333333333333) - x) / B))
end

Wolfram

code[F_, B_, x_] := N[(N[(-1.0 / F), $MachinePrecision] * N[(F / B), $MachinePrecision] + N[(N[(N[(N[(N[(B * B), $MachinePrecision] * x), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.6%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in B around 0

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
3. Step-by-step derivation

   1. lower-/.f64 62.2

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\color{blue}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

4. Applied rewrites 62.2%

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

5. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{\color{blue}{B}} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. lower-pow.f64 36.0

      \[\leadsto \frac{0.3333333333333333 \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

7. Applied rewrites 36.0%

   \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

9. Applied rewrites 36.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{F}{B}, \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot 0.3333333333333333 - x}{B}\right)} \]

10. Taylor expanded in F around -inf

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{F}}, \frac{F}{B}, \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot \frac{1}{3} - x}{B}\right) \]
11. Step-by-step derivation

    1. lower-/.f64 22.4

       \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{F}}, \frac{F}{B}, \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot 0.3333333333333333 - x}{B}\right) \]

12. Applied rewrites 22.4%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{F}}, \frac{F}{B}, \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot 0.3333333333333333 - x}{B}\right) \]
13. Add Preprocessing

Alternative 19: 10.3% accurate, 26.5× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / 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 = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 76.6%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in F around -inf

   \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

   2. lower-sin.f64 17.2

      \[\leadsto \frac{-1}{\sin B} \]

4. Applied rewrites 17.2%

   \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

5. Taylor expanded in B around 0

   \[\leadsto \frac{-1}{B} \]
6. Step-by-step derivation

7. Applied rewrites 10.3%

   \[\leadsto \frac{-1}{B} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025154 
(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))))))