VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.3% → 99.5%

Time: 7.3s

Alternatives: 32

Speedup: 1.4×

Specification

Math

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

FPCore

(FPCore (F B x)
 :precision binary64
 (+
  (- (* x (/ 1.0 (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: 76.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (F B x)
 :precision binary64
 (+
  (- (* x (/ 1.0 (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: 99.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := F \cdot \sin B\\ t_1 := \frac{1}{t\_0}\\ t_2 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -1.2 \cdot 10^{+154}:\\ \;\;\;\;-1 \cdot \left(F \cdot \left(t\_1 + \frac{x \cdot \cos B}{t\_0}\right)\right)\\ \mathbf{elif}\;F \leq 7000:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, t\_1, t\_2\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* F (sin B))) (t_1 (/ 1.0 t_0)) (t_2 (/ (- x) (tan B))))
   (if (<= F -1.2e+154)
     (* -1.0 (* F (+ t_1 (/ (* x (cos B)) t_0))))
     (if (<= F 7000.0)
       (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B)) t_2)
       (fma F t_1 t_2)))))

C

double code(double F, double B, double x) {
	double t_0 = F * sin(B);
	double t_1 = 1.0 / t_0;
	double t_2 = -x / tan(B);
	double tmp;
	if (F <= -1.2e+154) {
		tmp = -1.0 * (F * (t_1 + ((x * cos(B)) / t_0)));
	} else if (F <= 7000.0) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), t_2);
	} else {
		tmp = fma(F, t_1, t_2);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(F * sin(B))
	t_1 = Float64(1.0 / t_0)
	t_2 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -1.2e+154)
		tmp = Float64(-1.0 * Float64(F * Float64(t_1 + Float64(Float64(x * cos(B)) / t_0))));
	elseif (F <= 7000.0)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), t_2);
	else
		tmp = fma(F, t_1, t_2);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.2e+154], N[(-1.0 * N[(F * N[(t$95$1 + N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7000.0], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], N[(F * t$95$1 + t$95$2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.20000000000000007e154

   1. Initial program 76.3%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 50.3

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

   4. Applied rewrites 50.3%

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

   if -1.20000000000000007e154 < F < 7e3

   1. Initial program 76.3%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   if 7e3 < F

   1. Initial program 76.3%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 51.9

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

   6. Applied rewrites 51.9%

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

Alternative 2: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ t_1 := F \cdot \sin B\\ \mathbf{if}\;F \leq -2.9 \cdot 10^{+157}:\\ \;\;\;\;F \cdot \frac{-1}{t\_1} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7000:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{t\_1}, t\_0\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))) (t_1 (* F (sin B))))
   (if (<= F -2.9e+157)
     (- (* F (/ -1.0 t_1)) (/ x B))
     (if (<= F 7000.0)
       (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B)) t_0)
       (fma F (/ 1.0 t_1) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double t_1 = F * sin(B);
	double tmp;
	if (F <= -2.9e+157) {
		tmp = (F * (-1.0 / t_1)) - (x / B);
	} else if (F <= 7000.0) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), t_0);
	} else {
		tmp = fma(F, (1.0 / t_1), t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	t_1 = Float64(F * sin(B))
	tmp = 0.0
	if (F <= -2.9e+157)
		tmp = Float64(Float64(F * Float64(-1.0 / t_1)) - Float64(x / B));
	elseif (F <= 7000.0)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), t_0);
	else
		tmp = fma(F, Float64(1.0 / t_1), t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.9e+157], N[(N[(F * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7000.0], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(F * N[(1.0 / t$95$1), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.89999999999999988e157

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   8. Applied rewrites 57.7%

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

   9. Taylor expanded in F around -inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 35.4

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

   11. Applied rewrites 35.4%

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

   if -2.89999999999999988e157 < F < 7e3

   1. Initial program 76.3%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   if 7e3 < F

   1. Initial program 76.3%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 51.9

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

   6. Applied rewrites 51.9%

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

Alternative 3: 95.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := F \cdot \sin B\\ \mathbf{if}\;F \leq -2.9 \cdot 10^{+157}:\\ \;\;\;\;F \cdot \frac{-1}{t\_0} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{+20}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\cos B} - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{t\_0}, \frac{-x}{\tan B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* F (sin B))))
   (if (<= F -2.9e+157)
     (- (* F (/ -1.0 t_0)) (/ x B))
     (if (<= F 1.2e+20)
       (/ (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) (/ F (cos B))) x) (tan B))
       (fma F (/ 1.0 t_0) (/ (- x) (tan B)))))))

C

double code(double F, double B, double x) {
	double t_0 = F * sin(B);
	double tmp;
	if (F <= -2.9e+157) {
		tmp = (F * (-1.0 / t_0)) - (x / B);
	} else if (F <= 1.2e+20) {
		tmp = ((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * (F / cos(B))) - x) / tan(B);
	} else {
		tmp = fma(F, (1.0 / t_0), (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(F * sin(B))
	tmp = 0.0
	if (F <= -2.9e+157)
		tmp = Float64(Float64(F * Float64(-1.0 / t_0)) - Float64(x / B));
	elseif (F <= 1.2e+20)
		tmp = Float64(Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * Float64(F / cos(B))) - x) / tan(B));
	else
		tmp = fma(F, Float64(1.0 / t_0), Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.9e+157], N[(N[(F * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.2e+20], N[(N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(F * N[(1.0 / t$95$0), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.89999999999999988e157

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   8. Applied rewrites 57.7%

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

   9. Taylor expanded in F around -inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 35.4

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

   11. Applied rewrites 35.4%

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

   if -2.89999999999999988e157 < F < 1.2e20

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. sub-to-fraction N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 76.4%

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

   4. Taylor expanded in B around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. lower-cos.f64 84.7

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

   6. Applied rewrites 84.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-+l+ N/A

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

      9. +-commutative N/A

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

      10. lift-pow.f64 N/A

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

      11. pow2 N/A

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

      12. associate-/l* N/A

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

      13. lower-*.f64 N/A

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

   8. Applied rewrites 84.7%

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

   if 1.2e20 < F

   1. Initial program 76.3%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 51.9

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

   6. Applied rewrites 51.9%

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

Alternative 4: 94.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.2 \cdot 10^{+129}:\\ \;\;\;\;F \cdot \frac{\frac{-1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7000:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{F \cdot \sin B}, \frac{-x}{\tan B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.2e+129)
   (- (* F (/ (/ -1.0 F) (sin B))) (/ x B))
   (if (<= F 7000.0)
     (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F (sin B))) (/ x (tan B)))
     (fma F (/ 1.0 (* F (sin B))) (/ (- x) (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.2e+129) {
		tmp = (F * ((-1.0 / F) / sin(B))) - (x / B);
	} else if (F <= 7000.0) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / sin(B))) - (x / tan(B));
	} else {
		tmp = fma(F, (1.0 / (F * sin(B))), (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.2e+129)
		tmp = Float64(Float64(F * Float64(Float64(-1.0 / F) / sin(B))) - Float64(x / B));
	elseif (F <= 7000.0)
		tmp = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / sin(B))) - Float64(x / tan(B)));
	else
		tmp = fma(F, Float64(1.0 / Float64(F * sin(B))), Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.2e+129], N[(N[(F * N[(N[(-1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7000.0], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.1999999999999999e129

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   8. Applied rewrites 57.7%

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

   9. Taylor expanded in F around -inf

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

       1. lower-/.f64 35.4

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

   11. Applied rewrites 35.4%

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

   if -1.1999999999999999e129 < F < 7e3

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.3

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

   3. Applied rewrites 76.4%

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

   if 7e3 < F

   1. Initial program 76.3%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 51.9

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

   6. Applied rewrites 51.9%

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

Alternative 5: 92.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -4.1:\\ \;\;\;\;F \cdot \frac{\frac{-1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5}}{\sin B}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{F \cdot \sin B}, t\_0\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -4.1)
     (- (* F (/ (/ -1.0 F) (sin B))) (/ x B))
     (if (<= F 1.8e-7)
       (fma F (/ (pow (fma 2.0 x 2.0) -0.5) (sin B)) t_0)
       (fma F (/ 1.0 (* F (sin B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -4.1) {
		tmp = (F * ((-1.0 / F) / sin(B))) - (x / B);
	} else if (F <= 1.8e-7) {
		tmp = fma(F, (pow(fma(2.0, x, 2.0), -0.5) / sin(B)), t_0);
	} else {
		tmp = fma(F, (1.0 / (F * sin(B))), t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -4.1)
		tmp = Float64(Float64(F * Float64(Float64(-1.0 / F) / sin(B))) - Float64(x / B));
	elseif (F <= 1.8e-7)
		tmp = fma(F, Float64((fma(2.0, x, 2.0) ^ -0.5) / sin(B)), t_0);
	else
		tmp = fma(F, Float64(1.0 / Float64(F * sin(B))), 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, -4.1], N[(N[(F * N[(N[(-1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.8e-7], N[(F * N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(F * N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.0999999999999996

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   8. Applied rewrites 57.7%

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

   9. Taylor expanded in F around -inf

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

       1. lower-/.f64 35.4

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

   11. Applied rewrites 35.4%

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

   if -4.0999999999999996 < F < 1.79999999999999997e-7

   1. Initial program 76.3%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around 0

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

   6. Applied rewrites 55.9%

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

   if 1.79999999999999997e-7 < F

   1. Initial program 76.3%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 51.9

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

   6. Applied rewrites 51.9%

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

Alternative 6: 92.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -4.1:\\ \;\;\;\;F \cdot \frac{\frac{-1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {2}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{F \cdot \sin B}, \frac{-x}{\tan B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.1)
   (- (* F (/ (/ -1.0 F) (sin B))) (/ x B))
   (if (<= F 1.8e-7)
     (+ (- (* x (/ 1.0 (tan B)))) (* (/ F (sin B)) (pow 2.0 -0.5)))
     (fma F (/ 1.0 (* F (sin B))) (/ (- x) (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.1) {
		tmp = (F * ((-1.0 / F) / sin(B))) - (x / B);
	} else if (F <= 1.8e-7) {
		tmp = -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow(2.0, -0.5));
	} else {
		tmp = fma(F, (1.0 / (F * sin(B))), (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.1)
		tmp = Float64(Float64(F * Float64(Float64(-1.0 / F) / sin(B))) - Float64(x / B));
	elseif (F <= 1.8e-7)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / sin(B)) * (2.0 ^ -0.5)));
	else
		tmp = fma(F, Float64(1.0 / Float64(F * sin(B))), Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.1], N[(N[(F * N[(N[(-1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.8e-7], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.0999999999999996

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   8. Applied rewrites 57.7%

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

   9. Taylor expanded in F around -inf

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

       1. lower-/.f64 35.4

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

   11. Applied rewrites 35.4%

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

   if -4.0999999999999996 < F < 1.79999999999999997e-7

   1. Initial program 76.3%

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

   2. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval 54.6

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

   4. Applied rewrites 54.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-pow.f64 56.8

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

   7. Applied rewrites 56.8%

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

   if 1.79999999999999997e-7 < F

   1. Initial program 76.3%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 51.9

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

   6. Applied rewrites 51.9%

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

Alternative 7: 85.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := F \cdot \sin B\\ \mathbf{if}\;F \leq -9 \cdot 10^{+151}:\\ \;\;\;\;F \cdot \frac{-1}{t\_0} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -6 \cdot 10^{-73}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{B \cdot \left(1 + 0.3333333333333333 \cdot {B}^{2}\right)}\right)\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-140}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+19}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{t\_0}, \frac{-x}{\tan B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* F (sin B))))
   (if (<= F -9e+151)
     (- (* F (/ -1.0 t_0)) (/ x B))
     (if (<= F -6e-73)
       (fma
        F
        (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B))
        (/ (- x) (* B (+ 1.0 (* 0.3333333333333333 (pow B 2.0))))))
       (if (<= F 1.7e-140)
         (+
          (- (* x (/ 1.0 (tan B))))
          (* (/ F B) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
         (if (<= F 5e+19)
           (- (/ (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F) (sin B)) (/ x B))
           (fma F (/ 1.0 t_0) (/ (- x) (tan B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = F * sin(B);
	double tmp;
	if (F <= -9e+151) {
		tmp = (F * (-1.0 / t_0)) - (x / B);
	} else if (F <= -6e-73) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), (-x / (B * (1.0 + (0.3333333333333333 * pow(B, 2.0))))));
	} else if (F <= 1.7e-140) {
		tmp = -(x * (1.0 / tan(B))) + ((F / B) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else if (F <= 5e+19) {
		tmp = ((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F) / sin(B)) - (x / B);
	} else {
		tmp = fma(F, (1.0 / t_0), (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(F * sin(B))
	tmp = 0.0
	if (F <= -9e+151)
		tmp = Float64(Float64(F * Float64(-1.0 / t_0)) - Float64(x / B));
	elseif (F <= -6e-73)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), Float64(Float64(-x) / Float64(B * Float64(1.0 + Float64(0.3333333333333333 * (B ^ 2.0))))));
	elseif (F <= 1.7e-140)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / B) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))));
	elseif (F <= 5e+19)
		tmp = Float64(Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F) / sin(B)) - Float64(x / B));
	else
		tmp = fma(F, Float64(1.0 / t_0), Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -9e+151], N[(N[(F * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -6e-73], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[(B * N[(1.0 + N[(0.3333333333333333 * N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.7e-140], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / B), $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], If[LessEqual[F, 5e+19], N[(N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(F * N[(1.0 / t$95$0), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -8.9999999999999997e151

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   8. Applied rewrites 57.7%

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

   9. Taylor expanded in F around -inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 35.4

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

   11. Applied rewrites 35.4%

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

   if -8.9999999999999997e151 < F < -6e-73

   1. Initial program 76.3%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 57.8

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{B \cdot \left(1 + 0.3333333333333333 \cdot {B}^{\color{blue}{2}}\right)}\right) \]

   6. Applied rewrites 57.8%

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\color{blue}{B \cdot \left(1 + 0.3333333333333333 \cdot {B}^{2}\right)}}\right) \]

   if -6e-73 < F < 1.70000000000000004e-140

   1. Initial program 76.3%

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

   2. 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 61.8

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

   4. Applied rewrites 61.8%

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

   if 1.70000000000000004e-140 < F < 5e19

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

   8. Applied rewrites 57.8%

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

   if 5e19 < F

   1. Initial program 76.3%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 51.9

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

   6. Applied rewrites 51.9%

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

Alternative 8: 85.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}\\ t_1 := F \cdot \sin B\\ \mathbf{if}\;F \leq -4 \cdot 10^{+154}:\\ \;\;\;\;F \cdot \frac{-1}{t\_1} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -6 \cdot 10^{-73}:\\ \;\;\;\;F \cdot \frac{t\_0}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-140}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+19}:\\ \;\;\;\;\frac{t\_0 \cdot F}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{t\_1}, \frac{-x}{\tan B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (fma x 2.0 (fma F F 2.0)) -0.5)) (t_1 (* F (sin B))))
   (if (<= F -4e+154)
     (- (* F (/ -1.0 t_1)) (/ x B))
     (if (<= F -6e-73)
       (- (* F (/ t_0 (sin B))) (/ x B))
       (if (<= F 1.7e-140)
         (+
          (- (* x (/ 1.0 (tan B))))
          (* (/ F B) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
         (if (<= F 5e+19)
           (- (/ (* t_0 F) (sin B)) (/ x B))
           (fma F (/ 1.0 t_1) (/ (- 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 = F * sin(B);
	double tmp;
	if (F <= -4e+154) {
		tmp = (F * (-1.0 / t_1)) - (x / B);
	} else if (F <= -6e-73) {
		tmp = (F * (t_0 / sin(B))) - (x / B);
	} else if (F <= 1.7e-140) {
		tmp = -(x * (1.0 / tan(B))) + ((F / B) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else if (F <= 5e+19) {
		tmp = ((t_0 * F) / sin(B)) - (x / B);
	} else {
		tmp = fma(F, (1.0 / t_1), (-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(F * sin(B))
	tmp = 0.0
	if (F <= -4e+154)
		tmp = Float64(Float64(F * Float64(-1.0 / t_1)) - Float64(x / B));
	elseif (F <= -6e-73)
		tmp = Float64(Float64(F * Float64(t_0 / sin(B))) - Float64(x / B));
	elseif (F <= 1.7e-140)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / B) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))));
	elseif (F <= 5e+19)
		tmp = Float64(Float64(Float64(t_0 * F) / sin(B)) - Float64(x / B));
	else
		tmp = fma(F, Float64(1.0 / t_1), 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[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4e+154], N[(N[(F * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -6e-73], N[(N[(F * N[(t$95$0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.7e-140], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / B), $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], If[LessEqual[F, 5e+19], N[(N[(N[(t$95$0 * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(F * N[(1.0 / t$95$1), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -4.00000000000000015e154

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   8. Applied rewrites 57.7%

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

   9. Taylor expanded in F around -inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 35.4

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

   11. Applied rewrites 35.4%

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

   if -4.00000000000000015e154 < F < -6e-73

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   8. Applied rewrites 57.7%

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

   if -6e-73 < F < 1.70000000000000004e-140

   1. Initial program 76.3%

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

   2. 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 61.8

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

   4. Applied rewrites 61.8%

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

   if 1.70000000000000004e-140 < F < 5e19

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

   8. Applied rewrites 57.8%

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

   if 5e19 < F

   1. Initial program 76.3%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 51.9

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

   6. Applied rewrites 51.9%

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

Alternative 9: 85.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}\\ t_1 := F \cdot \sin B\\ \mathbf{if}\;F \leq -4 \cdot 10^{+154}:\\ \;\;\;\;F \cdot \frac{-1}{t\_1} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -6 \cdot 10^{-73}:\\ \;\;\;\;F \cdot \frac{t\_0}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-140}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot {\left(2 + 2 \cdot x\right)}^{-0.5}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+19}:\\ \;\;\;\;\frac{t\_0 \cdot F}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{t\_1}, \frac{-x}{\tan B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (fma x 2.0 (fma F F 2.0)) -0.5)) (t_1 (* F (sin B))))
   (if (<= F -4e+154)
     (- (* F (/ -1.0 t_1)) (/ x B))
     (if (<= F -6e-73)
       (- (* F (/ t_0 (sin B))) (/ x B))
       (if (<= F 1.7e-140)
         (+ (- (* x (/ 1.0 (tan B)))) (* (/ F B) (pow (+ 2.0 (* 2.0 x)) -0.5)))
         (if (<= F 5e+19)
           (- (/ (* t_0 F) (sin B)) (/ x B))
           (fma F (/ 1.0 t_1) (/ (- 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 = F * sin(B);
	double tmp;
	if (F <= -4e+154) {
		tmp = (F * (-1.0 / t_1)) - (x / B);
	} else if (F <= -6e-73) {
		tmp = (F * (t_0 / sin(B))) - (x / B);
	} else if (F <= 1.7e-140) {
		tmp = -(x * (1.0 / tan(B))) + ((F / B) * pow((2.0 + (2.0 * x)), -0.5));
	} else if (F <= 5e+19) {
		tmp = ((t_0 * F) / sin(B)) - (x / B);
	} else {
		tmp = fma(F, (1.0 / t_1), (-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(F * sin(B))
	tmp = 0.0
	if (F <= -4e+154)
		tmp = Float64(Float64(F * Float64(-1.0 / t_1)) - Float64(x / B));
	elseif (F <= -6e-73)
		tmp = Float64(Float64(F * Float64(t_0 / sin(B))) - Float64(x / B));
	elseif (F <= 1.7e-140)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / B) * (Float64(2.0 + Float64(2.0 * x)) ^ -0.5)));
	elseif (F <= 5e+19)
		tmp = Float64(Float64(Float64(t_0 * F) / sin(B)) - Float64(x / B));
	else
		tmp = fma(F, Float64(1.0 / t_1), 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[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4e+154], N[(N[(F * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -6e-73], N[(N[(F * N[(t$95$0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.7e-140], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / B), $MachinePrecision] * N[Power[N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5e+19], N[(N[(N[(t$95$0 * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(F * N[(1.0 / t$95$1), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -4.00000000000000015e154

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   8. Applied rewrites 57.7%

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

   9. Taylor expanded in F around -inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 35.4

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

   11. Applied rewrites 35.4%

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

   if -4.00000000000000015e154 < F < -6e-73

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   8. Applied rewrites 57.7%

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

   if -6e-73 < F < 1.70000000000000004e-140

   1. Initial program 76.3%

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

   2. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval 54.6

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

   4. Applied rewrites 54.6%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 48.4

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

   7. Applied rewrites 48.4%

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

   if 1.70000000000000004e-140 < F < 5e19

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

   8. Applied rewrites 57.8%

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

   if 5e19 < F

   1. Initial program 76.3%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 51.9

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

   6. Applied rewrites 51.9%

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

Alternative 10: 79.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}\\ t_1 := F \cdot \sin B\\ \mathbf{if}\;F \leq -4 \cdot 10^{+154}:\\ \;\;\;\;F \cdot \frac{-1}{t\_1} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -6 \cdot 10^{-73}:\\ \;\;\;\;F \cdot \frac{t\_0}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-140}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot {\left(2 + 2 \cdot x\right)}^{-0.5}\\ \mathbf{elif}\;F \leq 10^{+94}:\\ \;\;\;\;\frac{t\_0 \cdot F}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{t\_1} - \frac{x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (fma x 2.0 (fma F F 2.0)) -0.5)) (t_1 (* F (sin B))))
   (if (<= F -4e+154)
     (- (* F (/ -1.0 t_1)) (/ x B))
     (if (<= F -6e-73)
       (- (* F (/ t_0 (sin B))) (/ x B))
       (if (<= F 1.7e-140)
         (+ (- (* x (/ 1.0 (tan B)))) (* (/ F B) (pow (+ 2.0 (* 2.0 x)) -0.5)))
         (if (<= F 1e+94)
           (- (/ (* t_0 F) (sin B)) (/ x B))
           (- (* F (/ 1.0 t_1)) (/ x 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 = F * sin(B);
	double tmp;
	if (F <= -4e+154) {
		tmp = (F * (-1.0 / t_1)) - (x / B);
	} else if (F <= -6e-73) {
		tmp = (F * (t_0 / sin(B))) - (x / B);
	} else if (F <= 1.7e-140) {
		tmp = -(x * (1.0 / tan(B))) + ((F / B) * pow((2.0 + (2.0 * x)), -0.5));
	} else if (F <= 1e+94) {
		tmp = ((t_0 * F) / sin(B)) - (x / B);
	} else {
		tmp = (F * (1.0 / t_1)) - (x / 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(F * sin(B))
	tmp = 0.0
	if (F <= -4e+154)
		tmp = Float64(Float64(F * Float64(-1.0 / t_1)) - Float64(x / B));
	elseif (F <= -6e-73)
		tmp = Float64(Float64(F * Float64(t_0 / sin(B))) - Float64(x / B));
	elseif (F <= 1.7e-140)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / B) * (Float64(2.0 + Float64(2.0 * x)) ^ -0.5)));
	elseif (F <= 1e+94)
		tmp = Float64(Float64(Float64(t_0 * F) / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(F * Float64(1.0 / t_1)) - Float64(x / 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[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4e+154], N[(N[(F * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -6e-73], N[(N[(F * N[(t$95$0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.7e-140], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / B), $MachinePrecision] * N[Power[N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e+94], N[(N[(N[(t$95$0 * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(F * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -4.00000000000000015e154

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   8. Applied rewrites 57.7%

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

   9. Taylor expanded in F around -inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 35.4

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

   11. Applied rewrites 35.4%

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

   if -4.00000000000000015e154 < F < -6e-73

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   8. Applied rewrites 57.7%

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

   if -6e-73 < F < 1.70000000000000004e-140

   1. Initial program 76.3%

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

   2. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval 54.6

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

   4. Applied rewrites 54.6%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 48.4

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

   7. Applied rewrites 48.4%

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

   if 1.70000000000000004e-140 < F < 1e94

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

   8. Applied rewrites 57.8%

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

   if 1e94 < F

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   8. Applied rewrites 57.7%

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

   9. Taylor expanded in F around inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 32.3

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

   11. Applied rewrites 32.3%

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

Alternative 11: 77.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}\\ t_1 := F \cdot \sin B\\ \mathbf{if}\;F \leq -4 \cdot 10^{+154}:\\ \;\;\;\;F \cdot \frac{-1}{t\_1} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -7.4 \cdot 10^{-178}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{t\_0}{\sin B}, \frac{-x}{B}\right)\\ \mathbf{elif}\;F \leq 8.1 \cdot 10^{-143}:\\ \;\;\;\;\frac{-1 \cdot x}{\tan B}\\ \mathbf{elif}\;F \leq 10^{+94}:\\ \;\;\;\;\frac{t\_0 \cdot F}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{t\_1} - \frac{x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (fma x 2.0 (fma F F 2.0)) -0.5)) (t_1 (* F (sin B))))
   (if (<= F -4e+154)
     (- (* F (/ -1.0 t_1)) (/ x B))
     (if (<= F -7.4e-178)
       (fma F (/ t_0 (sin B)) (/ (- x) B))
       (if (<= F 8.1e-143)
         (/ (* -1.0 x) (tan B))
         (if (<= F 1e+94)
           (- (/ (* t_0 F) (sin B)) (/ x B))
           (- (* F (/ 1.0 t_1)) (/ x 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 = F * sin(B);
	double tmp;
	if (F <= -4e+154) {
		tmp = (F * (-1.0 / t_1)) - (x / B);
	} else if (F <= -7.4e-178) {
		tmp = fma(F, (t_0 / sin(B)), (-x / B));
	} else if (F <= 8.1e-143) {
		tmp = (-1.0 * x) / tan(B);
	} else if (F <= 1e+94) {
		tmp = ((t_0 * F) / sin(B)) - (x / B);
	} else {
		tmp = (F * (1.0 / t_1)) - (x / 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(F * sin(B))
	tmp = 0.0
	if (F <= -4e+154)
		tmp = Float64(Float64(F * Float64(-1.0 / t_1)) - Float64(x / B));
	elseif (F <= -7.4e-178)
		tmp = fma(F, Float64(t_0 / sin(B)), Float64(Float64(-x) / B));
	elseif (F <= 8.1e-143)
		tmp = Float64(Float64(-1.0 * x) / tan(B));
	elseif (F <= 1e+94)
		tmp = Float64(Float64(Float64(t_0 * F) / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(F * Float64(1.0 / t_1)) - Float64(x / 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[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4e+154], N[(N[(F * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7.4e-178], N[(F * N[(t$95$0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8.1e-143], N[(N[(-1.0 * x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e+94], N[(N[(N[(t$95$0 * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(F * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -4.00000000000000015e154

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   8. Applied rewrites 57.7%

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

   9. Taylor expanded in F around -inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 35.4

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

   11. Applied rewrites 35.4%

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

   if -4.00000000000000015e154 < F < -7.40000000000000008e-178

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 57.7%

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

   if -7.40000000000000008e-178 < F < 8.0999999999999998e-143

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. sub-to-fraction N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 76.4%

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

   4. Taylor expanded in F around 0

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

      1. lower-*.f64 55.6

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

   6. Applied rewrites 55.6%

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

   if 8.0999999999999998e-143 < F < 1e94

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

   8. Applied rewrites 57.8%

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

   if 1e94 < F

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   8. Applied rewrites 57.7%

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

   9. Taylor expanded in F around inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 32.3

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

   11. Applied rewrites 32.3%

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

Alternative 12: 77.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}\\ t_1 := F \cdot \sin B\\ \mathbf{if}\;F \leq -4 \cdot 10^{+154}:\\ \;\;\;\;F \cdot \frac{-1}{t\_1} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -7.4 \cdot 10^{-178}:\\ \;\;\;\;F \cdot \frac{t\_0}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 8.1 \cdot 10^{-143}:\\ \;\;\;\;\frac{-1 \cdot x}{\tan B}\\ \mathbf{elif}\;F \leq 10^{+94}:\\ \;\;\;\;\frac{t\_0 \cdot F}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{t\_1} - \frac{x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (fma x 2.0 (fma F F 2.0)) -0.5)) (t_1 (* F (sin B))))
   (if (<= F -4e+154)
     (- (* F (/ -1.0 t_1)) (/ x B))
     (if (<= F -7.4e-178)
       (- (* F (/ t_0 (sin B))) (/ x B))
       (if (<= F 8.1e-143)
         (/ (* -1.0 x) (tan B))
         (if (<= F 1e+94)
           (- (/ (* t_0 F) (sin B)) (/ x B))
           (- (* F (/ 1.0 t_1)) (/ x 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 = F * sin(B);
	double tmp;
	if (F <= -4e+154) {
		tmp = (F * (-1.0 / t_1)) - (x / B);
	} else if (F <= -7.4e-178) {
		tmp = (F * (t_0 / sin(B))) - (x / B);
	} else if (F <= 8.1e-143) {
		tmp = (-1.0 * x) / tan(B);
	} else if (F <= 1e+94) {
		tmp = ((t_0 * F) / sin(B)) - (x / B);
	} else {
		tmp = (F * (1.0 / t_1)) - (x / 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(F * sin(B))
	tmp = 0.0
	if (F <= -4e+154)
		tmp = Float64(Float64(F * Float64(-1.0 / t_1)) - Float64(x / B));
	elseif (F <= -7.4e-178)
		tmp = Float64(Float64(F * Float64(t_0 / sin(B))) - Float64(x / B));
	elseif (F <= 8.1e-143)
		tmp = Float64(Float64(-1.0 * x) / tan(B));
	elseif (F <= 1e+94)
		tmp = Float64(Float64(Float64(t_0 * F) / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(F * Float64(1.0 / t_1)) - Float64(x / 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[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4e+154], N[(N[(F * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7.4e-178], N[(N[(F * N[(t$95$0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8.1e-143], N[(N[(-1.0 * x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e+94], N[(N[(N[(t$95$0 * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(F * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -4.00000000000000015e154

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   8. Applied rewrites 57.7%

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

   9. Taylor expanded in F around -inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 35.4

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

   11. Applied rewrites 35.4%

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

   if -4.00000000000000015e154 < F < -7.40000000000000008e-178

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   8. Applied rewrites 57.7%

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

   if -7.40000000000000008e-178 < F < 8.0999999999999998e-143

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. sub-to-fraction N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 76.4%

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

   4. Taylor expanded in F around 0

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

      1. lower-*.f64 55.6

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

   6. Applied rewrites 55.6%

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

   if 8.0999999999999998e-143 < F < 1e94

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

   8. Applied rewrites 57.8%

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

   if 1e94 < F

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   8. Applied rewrites 57.7%

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

   9. Taylor expanded in F around inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 32.3

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

   11. Applied rewrites 32.3%

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

Alternative 13: 77.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}{\sin B} - \frac{x}{B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+118}:\\ \;\;\;\;F \cdot \frac{\frac{-1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -7.4 \cdot 10^{-178}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 8.1 \cdot 10^{-143}:\\ \;\;\;\;\frac{-1 \cdot x}{\tan B}\\ \mathbf{elif}\;F \leq 10^{+94}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (- (/ (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F) (sin B)) (/ x B))))
   (if (<= F -2e+118)
     (- (* F (/ (/ -1.0 F) (sin B))) (/ x B))
     (if (<= F -7.4e-178)
       t_0
       (if (<= F 8.1e-143)
         (/ (* -1.0 x) (tan B))
         (if (<= F 1e+94) t_0 (- (* F (/ 1.0 (* F (sin B)))) (/ x 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) * F) / sin(B)) - (x / B);
	double tmp;
	if (F <= -2e+118) {
		tmp = (F * ((-1.0 / F) / sin(B))) - (x / B);
	} else if (F <= -7.4e-178) {
		tmp = t_0;
	} else if (F <= 8.1e-143) {
		tmp = (-1.0 * x) / tan(B);
	} else if (F <= 1e+94) {
		tmp = t_0;
	} else {
		tmp = (F * (1.0 / (F * sin(B)))) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F) / sin(B)) - Float64(x / B))
	tmp = 0.0
	if (F <= -2e+118)
		tmp = Float64(Float64(F * Float64(Float64(-1.0 / F) / sin(B))) - Float64(x / B));
	elseif (F <= -7.4e-178)
		tmp = t_0;
	elseif (F <= 8.1e-143)
		tmp = Float64(Float64(-1.0 * x) / tan(B));
	elseif (F <= 1e+94)
		tmp = t_0;
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * sin(B)))) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2e+118], N[(N[(F * N[(N[(-1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7.4e-178], t$95$0, If[LessEqual[F, 8.1e-143], N[(N[(-1.0 * x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e+94], t$95$0, N[(N[(F * N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.99999999999999993e118

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   8. Applied rewrites 57.7%

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

   9. Taylor expanded in F around -inf

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

       1. lower-/.f64 35.4

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

   11. Applied rewrites 35.4%

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

   if -1.99999999999999993e118 < F < -7.40000000000000008e-178 or 8.0999999999999998e-143 < F < 1e94

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

   8. Applied rewrites 57.8%

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

   if -7.40000000000000008e-178 < F < 8.0999999999999998e-143

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. sub-to-fraction N/A

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

      9. lower-/.f64 N/A

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

   3. Applied rewrites 76.4%

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

   4. Taylor expanded in F around 0

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

      1. lower-*.f64 55.6

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

   6. Applied rewrites 55.6%

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

   if 1e94 < F

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. add-flip N/A

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

      8. *-commutative N/A

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

      9. add-flip N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

   8. Applied rewrites 57.7%

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

   9. Taylor expanded in F around inf

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

       1. lower-/.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sin.f64 32.3

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

   11. Applied rewrites 32.3%

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

Alternative 14: 77.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{if}\;F \leq -6800000000:\\ \;\;\;\;F \cdot \frac{\frac{-1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -7.4 \cdot 10^{-178}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 8.1 \cdot 10^{-143}:\\ \;\;\;\;\frac{-1 \cdot x}{\tan B}\\ \mathbf{elif}\;F \leq 7000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (- (* (/ F (sin B)) (pow (fma x 2.0 (fma F F 2.0)) -0.5)) (/ x B))))
   (if (<= F -6800000000.0)
     (- (* F (/ (/ -1.0 F) (sin B))) (/ x B))
     (if (<= F -7.4e-178)
       t_0
       (if (<= F 8.1e-143)
         (/ (* -1.0 x) (tan B))
         (if (<= F 7000.0) t_0 (- (* F (/ 1.0 (* F (sin B)))) (/ x B))))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F / sin(B)) * pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5)) - (x / B);
	double tmp;
	if (F <= -6800000000.0) {
		tmp = (F * ((-1.0 / F) / sin(B))) - (x / B);
	} else if (F <= -7.4e-178) {
		tmp = t_0;
	} else if (F <= 8.1e-143) {
		tmp = (-1.0 * x) / tan(B);
	} else if (F <= 7000.0) {
		tmp = t_0;
	} else {
		tmp = (F * (1.0 / (F * sin(B)))) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F / sin(B)) * (fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5)) - Float64(x / B))
	tmp = 0.0
	if (F <= -6800000000.0)
		tmp = Float64(Float64(F * Float64(Float64(-1.0 / F) / sin(B))) - Float64(x / B));
	elseif (F <= -7.4e-178)
		tmp = t_0;
	elseif (F <= 8.1e-143)
		tmp = Float64(Float64(-1.0 * x) / tan(B));
	elseif (F <= 7000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * sin(B)))) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -6800000000.0], N[(N[(F * N[(N[(-1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7.4e-178], t$95$0, If[LessEqual[F, 8.1e-143], N[(N[(-1.0 * x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7000.0], t$95$0, N[(N[(F * N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -6.8e9

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.3%

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

   6. Applied rewrites 49.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{F \cdot \color{blue}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}{\sin B} - \frac{x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F + 2}\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      7. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{F \cdot {\left(\color{blue}{2 \cdot x} - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      9. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      10. +-commutative N/A

          \[\leadsto \frac{F \cdot {\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      11. associate-/l* N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      13. lower-/.f64 N/A

          \[\leadsto F \cdot \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

   8. Applied rewrites 57.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{B} \]

   9. Taylor expanded in F around -inf

      \[\leadsto F \cdot \frac{\color{blue}{\frac{-1}{F}}}{\sin B} - \frac{x}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 35.4

          \[\leadsto F \cdot \frac{\frac{-1}{\color{blue}{F}}}{\sin B} - \frac{x}{B} \]

   11. Applied rewrites 35.4%

       \[\leadsto F \cdot \frac{\color{blue}{\frac{-1}{F}}}{\sin B} - \frac{x}{B} \]

   if -6.8e9 < F < -7.40000000000000008e-178 or 8.0999999999999998e-143 < F < 7e3

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}} \]

   if -7.40000000000000008e-178 < F < 8.0999999999999998e-143

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in F around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{\tan B} \]
   5. Step-by-step derivation

      1. lower-*.f64 55.6

         \[\leadsto \frac{-1 \cdot \color{blue}{x}}{\tan B} \]

   6. Applied rewrites 55.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{\tan B} \]

   if 7e3 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}} - \frac{x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} - \frac{x}{B} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{F \cdot \color{blue}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}{\sin B} - \frac{x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F + 2}\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      7. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{F \cdot {\left(\color{blue}{2 \cdot x} - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      9. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      10. +-commutative N/A

          \[\leadsto \frac{F \cdot {\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      11. associate-/l* N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      13. lower-/.f64 N/A

          \[\leadsto F \cdot \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

   8. Applied rewrites 57.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{B} \]

   9. Taylor expanded in F around inf

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot \sin B}} - \frac{x}{B} \]

       2. lower-*.f64 N/A

          \[\leadsto F \cdot \frac{1}{F \cdot \color{blue}{\sin B}} - \frac{x}{B} \]

       3. lower-sin.f64 32.3

          \[\leadsto F \cdot \frac{1}{F \cdot \sin B} - \frac{x}{B} \]

   11. Applied rewrites 32.3%

       \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 76.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -0.58:\\ \;\;\;\;F \cdot \frac{\frac{-1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -7.4 \cdot 10^{-178}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 8.1 \cdot 10^{-143}:\\ \;\;\;\;\frac{-1 \cdot x}{\tan B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{F \cdot {\left(2 + 2 \cdot x\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.58)
   (- (* F (/ (/ -1.0 F) (sin B))) (/ x B))
   (if (<= F -7.4e-178)
     (- (* (/ F (sin B)) (pow (fma x 2.0 2.0) -0.5)) (/ x B))
     (if (<= F 8.1e-143)
       (/ (* -1.0 x) (tan B))
       (if (<= F 1.8e-7)
         (- (/ (* F (pow (+ 2.0 (* 2.0 x)) -0.5)) (sin B)) (/ x B))
         (- (* F (/ 1.0 (* F (sin B)))) (/ x B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.58) {
		tmp = (F * ((-1.0 / F) / sin(B))) - (x / B);
	} else if (F <= -7.4e-178) {
		tmp = ((F / sin(B)) * pow(fma(x, 2.0, 2.0), -0.5)) - (x / B);
	} else if (F <= 8.1e-143) {
		tmp = (-1.0 * x) / tan(B);
	} else if (F <= 1.8e-7) {
		tmp = ((F * pow((2.0 + (2.0 * x)), -0.5)) / sin(B)) - (x / B);
	} else {
		tmp = (F * (1.0 / (F * sin(B)))) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.58)
		tmp = Float64(Float64(F * Float64(Float64(-1.0 / F) / sin(B))) - Float64(x / B));
	elseif (F <= -7.4e-178)
		tmp = Float64(Float64(Float64(F / sin(B)) * (fma(x, 2.0, 2.0) ^ -0.5)) - Float64(x / B));
	elseif (F <= 8.1e-143)
		tmp = Float64(Float64(-1.0 * x) / tan(B));
	elseif (F <= 1.8e-7)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + Float64(2.0 * x)) ^ -0.5)) / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * sin(B)))) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.58], N[(N[(F * N[(N[(-1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7.4e-178], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(x * 2.0 + 2.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8.1e-143], N[(N[(-1.0 * x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.8e-7], N[(N[(N[(F * N[Power[N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(F * N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -0.57999999999999996

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}} - \frac{x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} - \frac{x}{B} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{F \cdot \color{blue}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}{\sin B} - \frac{x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F + 2}\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      7. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{F \cdot {\left(\color{blue}{2 \cdot x} - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      9. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      10. +-commutative N/A

          \[\leadsto \frac{F \cdot {\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      11. associate-/l* N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      13. lower-/.f64 N/A

          \[\leadsto F \cdot \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

   8. Applied rewrites 57.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{B} \]

   9. Taylor expanded in F around -inf

      \[\leadsto F \cdot \frac{\color{blue}{\frac{-1}{F}}}{\sin B} - \frac{x}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 35.4

          \[\leadsto F \cdot \frac{\frac{-1}{\color{blue}{F}}}{\sin B} - \frac{x}{B} \]

   11. Applied rewrites 35.4%

       \[\leadsto F \cdot \frac{\color{blue}{\frac{-1}{F}}}{\sin B} - \frac{x}{B} \]

   if -0.57999999999999996 < F < -7.40000000000000008e-178

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}} \]

   7. Taylor expanded in F around 0

      \[\leadsto \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{2}\right)\right)}^{\frac{-1}{2}} - \frac{x}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 36.0%

      \[\leadsto \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{2}\right)\right)}^{-0.5} - \frac{x}{B} \]

   if -7.40000000000000008e-178 < F < 8.0999999999999998e-143

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in F around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{\tan B} \]
   5. Step-by-step derivation

      1. lower-*.f64 55.6

         \[\leadsto \frac{-1 \cdot \color{blue}{x}}{\tan B} \]

   6. Applied rewrites 55.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{\tan B} \]

   if 8.0999999999999998e-143 < F < 1.79999999999999997e-7

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}} \]

   7. Taylor expanded in F around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\frac{-1}{2}}}{\color{blue}{\sin B}} - \frac{x}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin \color{blue}{B}} - \frac{x}{B} \]

      3. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      4. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      6. lower-sin.f64 36.7

         \[\leadsto \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{-0.5}}{\sin B} - \frac{x}{B} \]

   9. Applied rewrites 36.7%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + 2 \cdot x\right)}^{-0.5}}{\sin B}} - \frac{x}{B} \]

   if 1.79999999999999997e-7 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}} - \frac{x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} - \frac{x}{B} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{F \cdot \color{blue}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}{\sin B} - \frac{x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F + 2}\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      7. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{F \cdot {\left(\color{blue}{2 \cdot x} - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      9. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      10. +-commutative N/A

          \[\leadsto \frac{F \cdot {\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      11. associate-/l* N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      13. lower-/.f64 N/A

          \[\leadsto F \cdot \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

   8. Applied rewrites 57.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{B} \]

   9. Taylor expanded in F around inf

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot \sin B}} - \frac{x}{B} \]

       2. lower-*.f64 N/A

          \[\leadsto F \cdot \frac{1}{F \cdot \color{blue}{\sin B}} - \frac{x}{B} \]

       3. lower-sin.f64 32.3

          \[\leadsto F \cdot \frac{1}{F \cdot \sin B} - \frac{x}{B} \]

   11. Applied rewrites 32.3%

       \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{B} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 16: 76.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{if}\;F \leq -0.58:\\ \;\;\;\;F \cdot \frac{\frac{-1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -7.4 \cdot 10^{-178}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 8.1 \cdot 10^{-143}:\\ \;\;\;\;\frac{-1 \cdot x}{\tan B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* (/ F (sin B)) (pow (fma x 2.0 2.0) -0.5)) (/ x B))))
   (if (<= F -0.58)
     (- (* F (/ (/ -1.0 F) (sin B))) (/ x B))
     (if (<= F -7.4e-178)
       t_0
       (if (<= F 8.1e-143)
         (/ (* -1.0 x) (tan B))
         (if (<= F 1.8e-7) t_0 (- (* F (/ 1.0 (* F (sin B)))) (/ x B))))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F / sin(B)) * pow(fma(x, 2.0, 2.0), -0.5)) - (x / B);
	double tmp;
	if (F <= -0.58) {
		tmp = (F * ((-1.0 / F) / sin(B))) - (x / B);
	} else if (F <= -7.4e-178) {
		tmp = t_0;
	} else if (F <= 8.1e-143) {
		tmp = (-1.0 * x) / tan(B);
	} else if (F <= 1.8e-7) {
		tmp = t_0;
	} else {
		tmp = (F * (1.0 / (F * sin(B)))) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F / sin(B)) * (fma(x, 2.0, 2.0) ^ -0.5)) - Float64(x / B))
	tmp = 0.0
	if (F <= -0.58)
		tmp = Float64(Float64(F * Float64(Float64(-1.0 / F) / sin(B))) - Float64(x / B));
	elseif (F <= -7.4e-178)
		tmp = t_0;
	elseif (F <= 8.1e-143)
		tmp = Float64(Float64(-1.0 * x) / tan(B));
	elseif (F <= 1.8e-7)
		tmp = t_0;
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * sin(B)))) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(x * 2.0 + 2.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.58], N[(N[(F * N[(N[(-1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7.4e-178], t$95$0, If[LessEqual[F, 8.1e-143], N[(N[(-1.0 * x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.8e-7], t$95$0, N[(N[(F * N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.57999999999999996

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}} - \frac{x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} - \frac{x}{B} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{F \cdot \color{blue}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}{\sin B} - \frac{x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F + 2}\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      7. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{F \cdot {\left(\color{blue}{2 \cdot x} - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      9. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      10. +-commutative N/A

          \[\leadsto \frac{F \cdot {\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      11. associate-/l* N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      13. lower-/.f64 N/A

          \[\leadsto F \cdot \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

   8. Applied rewrites 57.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{B} \]

   9. Taylor expanded in F around -inf

      \[\leadsto F \cdot \frac{\color{blue}{\frac{-1}{F}}}{\sin B} - \frac{x}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 35.4

          \[\leadsto F \cdot \frac{\frac{-1}{\color{blue}{F}}}{\sin B} - \frac{x}{B} \]

   11. Applied rewrites 35.4%

       \[\leadsto F \cdot \frac{\color{blue}{\frac{-1}{F}}}{\sin B} - \frac{x}{B} \]

   if -0.57999999999999996 < F < -7.40000000000000008e-178 or 8.0999999999999998e-143 < F < 1.79999999999999997e-7

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}} \]

   7. Taylor expanded in F around 0

      \[\leadsto \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{2}\right)\right)}^{\frac{-1}{2}} - \frac{x}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 36.0%

      \[\leadsto \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{2}\right)\right)}^{-0.5} - \frac{x}{B} \]

   if -7.40000000000000008e-178 < F < 8.0999999999999998e-143

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in F around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{\tan B} \]
   5. Step-by-step derivation

      1. lower-*.f64 55.6

         \[\leadsto \frac{-1 \cdot \color{blue}{x}}{\tan B} \]

   6. Applied rewrites 55.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{\tan B} \]

   if 1.79999999999999997e-7 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}} - \frac{x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} - \frac{x}{B} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{F \cdot \color{blue}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}{\sin B} - \frac{x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F + 2}\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      7. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{F \cdot {\left(\color{blue}{2 \cdot x} - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      9. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      10. +-commutative N/A

          \[\leadsto \frac{F \cdot {\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      11. associate-/l* N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      13. lower-/.f64 N/A

          \[\leadsto F \cdot \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

   8. Applied rewrites 57.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{B} \]

   9. Taylor expanded in F around inf

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot \sin B}} - \frac{x}{B} \]

       2. lower-*.f64 N/A

          \[\leadsto F \cdot \frac{1}{F \cdot \color{blue}{\sin B}} - \frac{x}{B} \]

       3. lower-sin.f64 32.3

          \[\leadsto F \cdot \frac{1}{F \cdot \sin B} - \frac{x}{B} \]

   11. Applied rewrites 32.3%

       \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 17: 76.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{if}\;F \leq -0.58:\\ \;\;\;\;F \cdot \frac{\frac{-1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -7.4 \cdot 10^{-178}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 8.1 \cdot 10^{-143}:\\ \;\;\;\;\frac{-1 \cdot x}{\tan B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* F (/ (pow (fma x 2.0 2.0) -0.5) (sin B))) (/ x B))))
   (if (<= F -0.58)
     (- (* F (/ (/ -1.0 F) (sin B))) (/ x B))
     (if (<= F -7.4e-178)
       t_0
       (if (<= F 8.1e-143)
         (/ (* -1.0 x) (tan B))
         (if (<= F 1.8e-7) t_0 (- (* F (/ 1.0 (* F (sin B)))) (/ x B))))))))

C

double code(double F, double B, double x) {
	double t_0 = (F * (pow(fma(x, 2.0, 2.0), -0.5) / sin(B))) - (x / B);
	double tmp;
	if (F <= -0.58) {
		tmp = (F * ((-1.0 / F) / sin(B))) - (x / B);
	} else if (F <= -7.4e-178) {
		tmp = t_0;
	} else if (F <= 8.1e-143) {
		tmp = (-1.0 * x) / tan(B);
	} else if (F <= 1.8e-7) {
		tmp = t_0;
	} else {
		tmp = (F * (1.0 / (F * sin(B)))) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(F * Float64((fma(x, 2.0, 2.0) ^ -0.5) / sin(B))) - Float64(x / B))
	tmp = 0.0
	if (F <= -0.58)
		tmp = Float64(Float64(F * Float64(Float64(-1.0 / F) / sin(B))) - Float64(x / B));
	elseif (F <= -7.4e-178)
		tmp = t_0;
	elseif (F <= 8.1e-143)
		tmp = Float64(Float64(-1.0 * x) / tan(B));
	elseif (F <= 1.8e-7)
		tmp = t_0;
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * sin(B)))) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(F * N[(N[Power[N[(x * 2.0 + 2.0), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.58], N[(N[(F * N[(N[(-1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7.4e-178], t$95$0, If[LessEqual[F, 8.1e-143], N[(N[(-1.0 * x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.8e-7], t$95$0, N[(N[(F * N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.57999999999999996

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}} - \frac{x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} - \frac{x}{B} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{F \cdot \color{blue}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}{\sin B} - \frac{x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F + 2}\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      7. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{F \cdot {\left(\color{blue}{2 \cdot x} - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      9. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      10. +-commutative N/A

          \[\leadsto \frac{F \cdot {\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      11. associate-/l* N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      13. lower-/.f64 N/A

          \[\leadsto F \cdot \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

   8. Applied rewrites 57.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{B} \]

   9. Taylor expanded in F around -inf

      \[\leadsto F \cdot \frac{\color{blue}{\frac{-1}{F}}}{\sin B} - \frac{x}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 35.4

          \[\leadsto F \cdot \frac{\frac{-1}{\color{blue}{F}}}{\sin B} - \frac{x}{B} \]

   11. Applied rewrites 35.4%

       \[\leadsto F \cdot \frac{\color{blue}{\frac{-1}{F}}}{\sin B} - \frac{x}{B} \]

   if -0.57999999999999996 < F < -7.40000000000000008e-178 or 8.0999999999999998e-143 < F < 1.79999999999999997e-7

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}} - \frac{x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} - \frac{x}{B} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{F \cdot \color{blue}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}{\sin B} - \frac{x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F + 2}\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      7. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{F \cdot {\left(\color{blue}{2 \cdot x} - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      9. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      10. +-commutative N/A

          \[\leadsto \frac{F \cdot {\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      11. associate-/l* N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      13. lower-/.f64 N/A

          \[\leadsto F \cdot \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

   8. Applied rewrites 57.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{B} \]

   9. Taylor expanded in F around 0

      \[\leadsto F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \color{blue}{2}\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]
   10. Step-by-step derivation

   11. Applied rewrites 36.7%

       \[\leadsto F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \color{blue}{2}\right)\right)}^{-0.5}}{\sin B} - \frac{x}{B} \]

   if -7.40000000000000008e-178 < F < 8.0999999999999998e-143

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in F around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{\tan B} \]
   5. Step-by-step derivation

      1. lower-*.f64 55.6

         \[\leadsto \frac{-1 \cdot \color{blue}{x}}{\tan B} \]

   6. Applied rewrites 55.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{\tan B} \]

   if 1.79999999999999997e-7 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}} - \frac{x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} - \frac{x}{B} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{F \cdot \color{blue}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}{\sin B} - \frac{x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F + 2}\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      7. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{F \cdot {\left(\color{blue}{2 \cdot x} - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      9. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      10. +-commutative N/A

          \[\leadsto \frac{F \cdot {\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      11. associate-/l* N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      13. lower-/.f64 N/A

          \[\leadsto F \cdot \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

   8. Applied rewrites 57.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{B} \]

   9. Taylor expanded in F around inf

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot \sin B}} - \frac{x}{B} \]

       2. lower-*.f64 N/A

          \[\leadsto F \cdot \frac{1}{F \cdot \color{blue}{\sin B}} - \frac{x}{B} \]

       3. lower-sin.f64 32.3

          \[\leadsto F \cdot \frac{1}{F \cdot \sin B} - \frac{x}{B} \]

   11. Applied rewrites 32.3%

       \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 18: 70.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -11:\\ \;\;\;\;F \cdot \frac{\frac{-1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -7.4 \cdot 10^{-178}:\\ \;\;\;\;\left(1 + -1 \cdot \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{x}\right) \cdot \frac{-x}{B}\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1 \cdot x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -11.0)
   (- (* F (/ (/ -1.0 F) (sin B))) (/ x B))
   (if (<= F -7.4e-178)
     (*
      (+ 1.0 (* -1.0 (/ (* F (pow (+ 2.0 (fma 2.0 x (pow F 2.0))) -0.5)) x)))
      (/ (- x) B))
     (if (<= F 5.6e-18)
       (/ (* -1.0 x) (tan B))
       (- (* F (/ 1.0 (* F (sin B)))) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -11.0) {
		tmp = (F * ((-1.0 / F) / sin(B))) - (x / B);
	} else if (F <= -7.4e-178) {
		tmp = (1.0 + (-1.0 * ((F * pow((2.0 + fma(2.0, x, pow(F, 2.0))), -0.5)) / x))) * (-x / B);
	} else if (F <= 5.6e-18) {
		tmp = (-1.0 * x) / tan(B);
	} else {
		tmp = (F * (1.0 / (F * sin(B)))) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -11.0)
		tmp = Float64(Float64(F * Float64(Float64(-1.0 / F) / sin(B))) - Float64(x / B));
	elseif (F <= -7.4e-178)
		tmp = Float64(Float64(1.0 + Float64(-1.0 * Float64(Float64(F * (Float64(2.0 + fma(2.0, x, (F ^ 2.0))) ^ -0.5)) / x))) * Float64(Float64(-x) / B));
	elseif (F <= 5.6e-18)
		tmp = Float64(Float64(-1.0 * x) / tan(B));
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * sin(B)))) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -11.0], N[(N[(F * N[(N[(-1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7.4e-178], N[(N[(1.0 + N[(-1.0 * N[(N[(F * N[Power[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[((-x) / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.6e-18], N[(N[(-1.0 * x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(F * N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -11

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}} - \frac{x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} - \frac{x}{B} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{F \cdot \color{blue}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}{\sin B} - \frac{x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F + 2}\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      7. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{F \cdot {\left(\color{blue}{2 \cdot x} - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      9. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      10. +-commutative N/A

          \[\leadsto \frac{F \cdot {\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      11. associate-/l* N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      13. lower-/.f64 N/A

          \[\leadsto F \cdot \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

   8. Applied rewrites 57.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{B} \]

   9. Taylor expanded in F around -inf

      \[\leadsto F \cdot \frac{\color{blue}{\frac{-1}{F}}}{\sin B} - \frac{x}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 35.4

          \[\leadsto F \cdot \frac{\frac{-1}{\color{blue}{F}}}{\sin B} - \frac{x}{B} \]

   11. Applied rewrites 35.4%

       \[\leadsto F \cdot \frac{\color{blue}{\frac{-1}{F}}}{\sin B} - \frac{x}{B} \]

   if -11 < F < -7.40000000000000008e-178

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 40.4%

      \[\leadsto \color{blue}{\left(1 + \frac{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\frac{-x}{B}}\right) \cdot \frac{-x}{B}} \]

   7. Taylor expanded in B around 0

      \[\leadsto \left(1 + \color{blue}{-1 \cdot \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}}}{x}}\right) \cdot \frac{-x}{B} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(1 + -1 \cdot \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}}}{x}}\right) \cdot \frac{-x}{B} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(1 + -1 \cdot \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\color{blue}{x}}\right) \cdot \frac{-x}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(1 + -1 \cdot \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}}}{x}\right) \cdot \frac{-x}{B} \]

      4. lower-pow.f64 N/A

         \[\leadsto \left(1 + -1 \cdot \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}}}{x}\right) \cdot \frac{-x}{B} \]

      5. lower-+.f64 N/A

         \[\leadsto \left(1 + -1 \cdot \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}}}{x}\right) \cdot \frac{-x}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(1 + -1 \cdot \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}{x}\right) \cdot \frac{-x}{B} \]

      7. lower-pow.f64 44.0

         \[\leadsto \left(1 + -1 \cdot \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{x}\right) \cdot \frac{-x}{B} \]

   9. Applied rewrites 44.0%

      \[\leadsto \left(1 + \color{blue}{-1 \cdot \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{x}}\right) \cdot \frac{-x}{B} \]

   if -7.40000000000000008e-178 < F < 5.60000000000000025e-18

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in F around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{\tan B} \]
   5. Step-by-step derivation

      1. lower-*.f64 55.6

         \[\leadsto \frac{-1 \cdot \color{blue}{x}}{\tan B} \]

   6. Applied rewrites 55.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{\tan B} \]

   if 5.60000000000000025e-18 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}} - \frac{x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} - \frac{x}{B} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{F \cdot \color{blue}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}{\sin B} - \frac{x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F + 2}\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      7. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{F \cdot {\left(\color{blue}{2 \cdot x} - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      9. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      10. +-commutative N/A

          \[\leadsto \frac{F \cdot {\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      11. associate-/l* N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      13. lower-/.f64 N/A

          \[\leadsto F \cdot \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

   8. Applied rewrites 57.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{B} \]

   9. Taylor expanded in F around inf

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot \sin B}} - \frac{x}{B} \]

       2. lower-*.f64 N/A

          \[\leadsto F \cdot \frac{1}{F \cdot \color{blue}{\sin B}} - \frac{x}{B} \]

       3. lower-sin.f64 32.3

          \[\leadsto F \cdot \frac{1}{F \cdot \sin B} - \frac{x}{B} \]

   11. Applied rewrites 32.3%

       \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 19: 70.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -11:\\ \;\;\;\;F \cdot \frac{\frac{-1}{F}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -7.4 \cdot 10^{-178}:\\ \;\;\;\;\frac{F}{B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1 \cdot x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot \sin B} - \frac{x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -11.0)
   (- (* F (/ (/ -1.0 F) (sin B))) (/ x B))
   (if (<= F -7.4e-178)
     (- (* (/ F B) (pow (fma x 2.0 (fma F F 2.0)) -0.5)) (/ x B))
     (if (<= F 5.6e-18)
       (/ (* -1.0 x) (tan B))
       (- (* F (/ 1.0 (* F (sin B)))) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -11.0) {
		tmp = (F * ((-1.0 / F) / sin(B))) - (x / B);
	} else if (F <= -7.4e-178) {
		tmp = ((F / B) * pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5)) - (x / B);
	} else if (F <= 5.6e-18) {
		tmp = (-1.0 * x) / tan(B);
	} else {
		tmp = (F * (1.0 / (F * sin(B)))) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -11.0)
		tmp = Float64(Float64(F * Float64(Float64(-1.0 / F) / sin(B))) - Float64(x / B));
	elseif (F <= -7.4e-178)
		tmp = Float64(Float64(Float64(F / B) * (fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5)) - Float64(x / B));
	elseif (F <= 5.6e-18)
		tmp = Float64(Float64(-1.0 * x) / tan(B));
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * sin(B)))) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -11.0], N[(N[(F * N[(N[(-1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7.4e-178], N[(N[(N[(F / B), $MachinePrecision] * N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.6e-18], N[(N[(-1.0 * x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(F * N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -11

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}} - \frac{x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} - \frac{x}{B} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{F \cdot \color{blue}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}{\sin B} - \frac{x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F + 2}\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      7. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{F \cdot {\left(\color{blue}{2 \cdot x} - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      9. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      10. +-commutative N/A

          \[\leadsto \frac{F \cdot {\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      11. associate-/l* N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      13. lower-/.f64 N/A

          \[\leadsto F \cdot \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

   8. Applied rewrites 57.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{B} \]

   9. Taylor expanded in F around -inf

      \[\leadsto F \cdot \frac{\color{blue}{\frac{-1}{F}}}{\sin B} - \frac{x}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 35.4

          \[\leadsto F \cdot \frac{\frac{-1}{\color{blue}{F}}}{\sin B} - \frac{x}{B} \]

   11. Applied rewrites 35.4%

       \[\leadsto F \cdot \frac{\color{blue}{\frac{-1}{F}}}{\sin B} - \frac{x}{B} \]

   if -11 < F < -7.40000000000000008e-178

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}} \]

   7. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F}{B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} - \frac{x}{B} \]
   8. Step-by-step derivation

      1. lower-/.f64 35.7

         \[\leadsto \frac{F}{\color{blue}{B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B} \]

   9. Applied rewrites 35.7%

      \[\leadsto \color{blue}{\frac{F}{B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B} \]

   if -7.40000000000000008e-178 < F < 5.60000000000000025e-18

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in F around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{\tan B} \]
   5. Step-by-step derivation

      1. lower-*.f64 55.6

         \[\leadsto \frac{-1 \cdot \color{blue}{x}}{\tan B} \]

   6. Applied rewrites 55.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{\tan B} \]

   if 5.60000000000000025e-18 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}} - \frac{x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} - \frac{x}{B} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{F \cdot \color{blue}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}{\sin B} - \frac{x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F + 2}\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      7. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{F \cdot {\left(\color{blue}{2 \cdot x} - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      9. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      10. +-commutative N/A

          \[\leadsto \frac{F \cdot {\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      11. associate-/l* N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      13. lower-/.f64 N/A

          \[\leadsto F \cdot \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

   8. Applied rewrites 57.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{B} \]

   9. Taylor expanded in F around inf

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot \sin B}} - \frac{x}{B} \]

       2. lower-*.f64 N/A

          \[\leadsto F \cdot \frac{1}{F \cdot \color{blue}{\sin B}} - \frac{x}{B} \]

       3. lower-sin.f64 32.3

          \[\leadsto F \cdot \frac{1}{F \cdot \sin B} - \frac{x}{B} \]

   11. Applied rewrites 32.3%

       \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 20: 70.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := F \cdot \sin B\\ \mathbf{if}\;F \leq -11:\\ \;\;\;\;F \cdot \frac{-1}{t\_0} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -7.4 \cdot 10^{-178}:\\ \;\;\;\;\frac{F}{B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1 \cdot x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{t\_0} - \frac{x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* F (sin B))))
   (if (<= F -11.0)
     (- (* F (/ -1.0 t_0)) (/ x B))
     (if (<= F -7.4e-178)
       (- (* (/ F B) (pow (fma x 2.0 (fma F F 2.0)) -0.5)) (/ x B))
       (if (<= F 5.6e-18)
         (/ (* -1.0 x) (tan B))
         (- (* F (/ 1.0 t_0)) (/ x B)))))))

C

double code(double F, double B, double x) {
	double t_0 = F * sin(B);
	double tmp;
	if (F <= -11.0) {
		tmp = (F * (-1.0 / t_0)) - (x / B);
	} else if (F <= -7.4e-178) {
		tmp = ((F / B) * pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5)) - (x / B);
	} else if (F <= 5.6e-18) {
		tmp = (-1.0 * x) / tan(B);
	} else {
		tmp = (F * (1.0 / t_0)) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(F * sin(B))
	tmp = 0.0
	if (F <= -11.0)
		tmp = Float64(Float64(F * Float64(-1.0 / t_0)) - Float64(x / B));
	elseif (F <= -7.4e-178)
		tmp = Float64(Float64(Float64(F / B) * (fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5)) - Float64(x / B));
	elseif (F <= 5.6e-18)
		tmp = Float64(Float64(-1.0 * x) / tan(B));
	else
		tmp = Float64(Float64(F * Float64(1.0 / t_0)) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -11.0], N[(N[(F * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7.4e-178], N[(N[(N[(F / B), $MachinePrecision] * N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.6e-18], N[(N[(-1.0 * x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(F * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -11

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}} - \frac{x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} - \frac{x}{B} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{F \cdot \color{blue}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}{\sin B} - \frac{x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F + 2}\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      7. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{F \cdot {\left(\color{blue}{2 \cdot x} - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      9. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      10. +-commutative N/A

          \[\leadsto \frac{F \cdot {\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      11. associate-/l* N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      13. lower-/.f64 N/A

          \[\leadsto F \cdot \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

   8. Applied rewrites 57.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{B} \]

   9. Taylor expanded in F around -inf

      \[\leadsto F \cdot \color{blue}{\frac{-1}{F \cdot \sin B}} - \frac{x}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto F \cdot \frac{-1}{\color{blue}{F \cdot \sin B}} - \frac{x}{B} \]

       2. lower-*.f64 N/A

          \[\leadsto F \cdot \frac{-1}{F \cdot \color{blue}{\sin B}} - \frac{x}{B} \]

       3. lower-sin.f64 35.4

          \[\leadsto F \cdot \frac{-1}{F \cdot \sin B} - \frac{x}{B} \]

   11. Applied rewrites 35.4%

       \[\leadsto F \cdot \color{blue}{\frac{-1}{F \cdot \sin B}} - \frac{x}{B} \]

   if -11 < F < -7.40000000000000008e-178

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}} \]

   7. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F}{B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} - \frac{x}{B} \]
   8. Step-by-step derivation

      1. lower-/.f64 35.7

         \[\leadsto \frac{F}{\color{blue}{B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B} \]

   9. Applied rewrites 35.7%

      \[\leadsto \color{blue}{\frac{F}{B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B} \]

   if -7.40000000000000008e-178 < F < 5.60000000000000025e-18

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in F around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{\tan B} \]
   5. Step-by-step derivation

      1. lower-*.f64 55.6

         \[\leadsto \frac{-1 \cdot \color{blue}{x}}{\tan B} \]

   6. Applied rewrites 55.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{\tan B} \]

   if 5.60000000000000025e-18 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}} - \frac{x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} - \frac{x}{B} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{F \cdot \color{blue}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}{\sin B} - \frac{x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F + 2}\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      7. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{F \cdot {\left(\color{blue}{2 \cdot x} - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      9. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      10. +-commutative N/A

          \[\leadsto \frac{F \cdot {\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      11. associate-/l* N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      13. lower-/.f64 N/A

          \[\leadsto F \cdot \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

   8. Applied rewrites 57.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{B} \]

   9. Taylor expanded in F around inf

      \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot \sin B}} - \frac{x}{B} \]

       2. lower-*.f64 N/A

          \[\leadsto F \cdot \frac{1}{F \cdot \color{blue}{\sin B}} - \frac{x}{B} \]

       3. lower-sin.f64 32.3

          \[\leadsto F \cdot \frac{1}{F \cdot \sin B} - \frac{x}{B} \]

   11. Applied rewrites 32.3%

       \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot \sin B}} - \frac{x}{B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 21: 69.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -11:\\ \;\;\;\;F \cdot \frac{-1}{F \cdot \sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -7.4 \cdot 10^{-178}:\\ \;\;\;\;\frac{F}{B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{-1 \cdot x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -11.0)
   (- (* F (/ -1.0 (* F (sin B)))) (/ x B))
   (if (<= F -7.4e-178)
     (- (* (/ F B) (pow (fma x 2.0 (fma F F 2.0)) -0.5)) (/ x B))
     (if (<= F 5.2e-17) (/ (* -1.0 x) (tan B)) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -11.0) {
		tmp = (F * (-1.0 / (F * sin(B)))) - (x / B);
	} else if (F <= -7.4e-178) {
		tmp = ((F / B) * pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5)) - (x / B);
	} else if (F <= 5.2e-17) {
		tmp = (-1.0 * x) / tan(B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -11.0)
		tmp = Float64(Float64(F * Float64(-1.0 / Float64(F * sin(B)))) - Float64(x / B));
	elseif (F <= -7.4e-178)
		tmp = Float64(Float64(Float64(F / B) * (fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5)) - Float64(x / B));
	elseif (F <= 5.2e-17)
		tmp = Float64(Float64(-1.0 * x) / tan(B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -11.0], N[(N[(F * N[(-1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7.4e-178], N[(N[(N[(F / B), $MachinePrecision] * N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.2e-17], N[(N[(-1.0 * x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -11

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}} - \frac{x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} - \frac{x}{B} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      4. lift-pow.f64 N/A

         \[\leadsto \frac{F \cdot \color{blue}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}{\sin B} - \frac{x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F + 2}\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      7. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(x \cdot 2 - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      8. *-commutative N/A

         \[\leadsto \frac{F \cdot {\left(\color{blue}{2 \cdot x} - \left(\mathsf{neg}\left(\left(F \cdot F + 2\right)\right)\right)\right)}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      9. add-flip N/A

         \[\leadsto \frac{F \cdot {\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      10. +-commutative N/A

          \[\leadsto \frac{F \cdot {\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\frac{-1}{2}}}{\sin B} - \frac{x}{B} \]

      11. associate-/l* N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      12. lower-*.f64 N/A

          \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

      13. lower-/.f64 N/A

          \[\leadsto F \cdot \color{blue}{\frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\frac{-1}{2}}}{\sin B}} - \frac{x}{B} \]

   8. Applied rewrites 57.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} - \frac{x}{B} \]

   9. Taylor expanded in F around -inf

      \[\leadsto F \cdot \color{blue}{\frac{-1}{F \cdot \sin B}} - \frac{x}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto F \cdot \frac{-1}{\color{blue}{F \cdot \sin B}} - \frac{x}{B} \]

       2. lower-*.f64 N/A

          \[\leadsto F \cdot \frac{-1}{F \cdot \color{blue}{\sin B}} - \frac{x}{B} \]

       3. lower-sin.f64 35.4

          \[\leadsto F \cdot \frac{-1}{F \cdot \sin B} - \frac{x}{B} \]

   11. Applied rewrites 35.4%

       \[\leadsto F \cdot \color{blue}{\frac{-1}{F \cdot \sin B}} - \frac{x}{B} \]

   if -11 < F < -7.40000000000000008e-178

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}} \]

   7. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F}{B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} - \frac{x}{B} \]
   8. Step-by-step derivation

      1. lower-/.f64 35.7

         \[\leadsto \frac{F}{\color{blue}{B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B} \]

   9. Applied rewrites 35.7%

      \[\leadsto \color{blue}{\frac{F}{B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B} \]

   if -7.40000000000000008e-178 < F < 5.20000000000000006e-17

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in F around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{\tan B} \]
   5. Step-by-step derivation

      1. lower-*.f64 55.6

         \[\leadsto \frac{-1 \cdot \color{blue}{x}}{\tan B} \]

   6. Applied rewrites 55.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{\tan B} \]

   if 5.20000000000000006e-17 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. 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 16.3

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 16.3%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 22: 64.5% accurate, 2.1× speedup

Math

\[\mathsf{copysign}\left(1, B\right) \cdot \begin{array}{l} \mathbf{if}\;\left|B\right| \leq 1.9 \cdot 10^{-9}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{\left|B\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot x}{\tan \left(\left|B\right|\right)}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (*
  (copysign 1.0 B)
  (if (<= (fabs B) 1.9e-9)
    (/ (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F) x) (fabs B))
    (/ (* -1.0 x) (tan (fabs B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (fabs(B) <= 1.9e-9) {
		tmp = ((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F) - x) / fabs(B);
	} else {
		tmp = (-1.0 * x) / tan(fabs(B));
	}
	return copysign(1.0, B) * tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (abs(B) <= 1.9e-9)
		tmp = Float64(Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F) - x) / abs(B));
	else
		tmp = Float64(Float64(-1.0 * x) / tan(abs(B)));
	end
	return Float64(copysign(1.0, B) * tmp)
end

Wolfram

code[F_, B_, x_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[B]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[B], $MachinePrecision], 1.9e-9], N[(N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / N[Abs[B], $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * x), $MachinePrecision] / N[Tan[N[Abs[B], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if B < 1.90000000000000006e-9

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in B around inf

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B}} - x}{\tan B} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\color{blue}{\cos B}} - x}{\tan B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos \color{blue}{B}} - x}{\tan B} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B} - x}{\tan B} \]

      11. lower-cos.f64 84.7

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B} - x}{\tan B} \]

   6. Applied rewrites 84.7%

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B}} - x}{\tan B} \]

   7. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{\color{blue}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      7. lower-pow.f64 44.1

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B} \]

   9. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

       2. *-commutative N/A

          \[\leadsto \frac{{\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       3. lift-+.f64 N/A

          \[\leadsto \frac{{\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       4. +-commutative N/A

          \[\leadsto \frac{{\left(\mathsf{fma}\left(2, x, {F}^{2}\right) + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       5. lift-fma.f64 N/A

          \[\leadsto \frac{{\left(\left(2 \cdot x + {F}^{2}\right) + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       6. associate-+r+ N/A

          \[\leadsto \frac{{\left(2 \cdot x + \left({F}^{2} + 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       7. *-commutative N/A

          \[\leadsto \frac{{\left(x \cdot 2 + \left({F}^{2} + 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       8. lift-pow.f64 N/A

          \[\leadsto \frac{{\left(x \cdot 2 + \left({F}^{2} + 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       9. pow2 N/A

          \[\leadsto \frac{{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       10. lift-fma.f64 N/A

           \[\leadsto \frac{{\left(x \cdot 2 + \mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       11. lift-fma.f64 N/A

           \[\leadsto \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       12. lift-*.f64 44.1

           \[\leadsto \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{B} \]

   11. Applied rewrites 44.1%

       \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{B}} \]

   if 1.90000000000000006e-9 < B

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in F around 0

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{\tan B} \]
   5. Step-by-step derivation

      1. lower-*.f64 55.6

         \[\leadsto \frac{-1 \cdot \color{blue}{x}}{\tan B} \]

   6. Applied rewrites 55.6%

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{\tan B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 52.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -11:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 10^{+19}:\\ \;\;\;\;\frac{F}{B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -11.0)
   (/ -1.0 (sin B))
   (if (<= F 1e+19)
     (- (* (/ F B) (pow (fma x 2.0 (fma F F 2.0)) -0.5)) (/ x B))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -11.0) {
		tmp = -1.0 / sin(B);
	} else if (F <= 1e+19) {
		tmp = ((F / B) * pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5)) - (x / B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -11.0)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 1e+19)
		tmp = Float64(Float64(Float64(F / B) * (fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5)) - Float64(x / B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -11.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e+19], N[(N[(N[(F / B), $MachinePrecision] * N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -11

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. 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 17.7

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -11 < F < 1e19

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

   4. Applied rewrites 49.3%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 49.4%

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}} \]

   7. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F}{B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} - \frac{x}{B} \]
   8. Step-by-step derivation

      1. lower-/.f64 35.7

         \[\leadsto \frac{F}{\color{blue}{B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B} \]

   9. Applied rewrites 35.7%

      \[\leadsto \color{blue}{\frac{F}{B}} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B} \]

   if 1e19 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. 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 16.3

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 16.3%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 52.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -11:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 10^{+19}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -11.0)
   (/ -1.0 (sin B))
   (if (<= F 1e+19)
     (/ (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F) x) B)
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -11.0) {
		tmp = -1.0 / sin(B);
	} else if (F <= 1e+19) {
		tmp = ((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F) - x) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -11.0)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 1e+19)
		tmp = Float64(Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F) - x) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -11.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e+19], N[(N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -11

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. 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 17.7

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -11 < F < 1e19

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in B around inf

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B}} - x}{\tan B} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\color{blue}{\cos B}} - x}{\tan B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos \color{blue}{B}} - x}{\tan B} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B} - x}{\tan B} \]

      11. lower-cos.f64 84.7

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B} - x}{\tan B} \]

   6. Applied rewrites 84.7%

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B}} - x}{\tan B} \]

   7. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{\color{blue}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      7. lower-pow.f64 44.1

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B} \]

   9. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

       2. *-commutative N/A

          \[\leadsto \frac{{\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       3. lift-+.f64 N/A

          \[\leadsto \frac{{\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       4. +-commutative N/A

          \[\leadsto \frac{{\left(\mathsf{fma}\left(2, x, {F}^{2}\right) + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       5. lift-fma.f64 N/A

          \[\leadsto \frac{{\left(\left(2 \cdot x + {F}^{2}\right) + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       6. associate-+r+ N/A

          \[\leadsto \frac{{\left(2 \cdot x + \left({F}^{2} + 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       7. *-commutative N/A

          \[\leadsto \frac{{\left(x \cdot 2 + \left({F}^{2} + 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       8. lift-pow.f64 N/A

          \[\leadsto \frac{{\left(x \cdot 2 + \left({F}^{2} + 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       9. pow2 N/A

          \[\leadsto \frac{{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       10. lift-fma.f64 N/A

           \[\leadsto \frac{{\left(x \cdot 2 + \mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       11. lift-fma.f64 N/A

           \[\leadsto \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       12. lift-*.f64 44.1

           \[\leadsto \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{B} \]

   11. Applied rewrites 44.1%

       \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{B}} \]

   if 1e19 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. 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 16.3

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 16.3%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 25: 52.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -11:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 10000:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -11.0)
   (/ -1.0 (sin B))
   (if (<= F 10000.0)
     (/ (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F) x) B)
     (/ (* F (fma -1.0 (/ x F) (/ 1.0 F))) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -11.0) {
		tmp = -1.0 / sin(B);
	} else if (F <= 10000.0) {
		tmp = ((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F) - x) / B;
	} else {
		tmp = (F * fma(-1.0, (x / F), (1.0 / F))) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -11.0)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 10000.0)
		tmp = Float64(Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F) - x) / B);
	else
		tmp = Float64(Float64(F * fma(-1.0, Float64(x / F), Float64(1.0 / F))) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -11.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 10000.0], N[(N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(F * N[(-1.0 * N[(x / F), $MachinePrecision] + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -11

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. 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 17.7

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -11 < F < 1e4

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in B around inf

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B}} - x}{\tan B} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\color{blue}{\cos B}} - x}{\tan B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos \color{blue}{B}} - x}{\tan B} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B} - x}{\tan B} \]

      11. lower-cos.f64 84.7

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B} - x}{\tan B} \]

   6. Applied rewrites 84.7%

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B}} - x}{\tan B} \]

   7. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{\color{blue}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      7. lower-pow.f64 44.1

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B} \]

   9. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

       2. *-commutative N/A

          \[\leadsto \frac{{\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       3. lift-+.f64 N/A

          \[\leadsto \frac{{\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       4. +-commutative N/A

          \[\leadsto \frac{{\left(\mathsf{fma}\left(2, x, {F}^{2}\right) + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       5. lift-fma.f64 N/A

          \[\leadsto \frac{{\left(\left(2 \cdot x + {F}^{2}\right) + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       6. associate-+r+ N/A

          \[\leadsto \frac{{\left(2 \cdot x + \left({F}^{2} + 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       7. *-commutative N/A

          \[\leadsto \frac{{\left(x \cdot 2 + \left({F}^{2} + 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       8. lift-pow.f64 N/A

          \[\leadsto \frac{{\left(x \cdot 2 + \left({F}^{2} + 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       9. pow2 N/A

          \[\leadsto \frac{{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       10. lift-fma.f64 N/A

           \[\leadsto \frac{{\left(x \cdot 2 + \mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       11. lift-fma.f64 N/A

           \[\leadsto \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       12. lift-*.f64 44.1

           \[\leadsto \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{B} \]

   11. Applied rewrites 44.1%

       \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{B}} \]

   if 1e4 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in B around inf

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B}} - x}{\tan B} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\color{blue}{\cos B}} - x}{\tan B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos \color{blue}{B}} - x}{\tan B} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B} - x}{\tan B} \]

      11. lower-cos.f64 84.7

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B} - x}{\tan B} \]

   6. Applied rewrites 84.7%

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B}} - x}{\tan B} \]

   7. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{\color{blue}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      7. lower-pow.f64 44.1

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B} \]

   9. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   10. Taylor expanded in F around inf

       \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

       4. lower-/.f64 28.4

          \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   12. Applied rewrites 28.4%

       \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 26: 51.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+151}:\\ \;\;\;\;\frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B}\\ \mathbf{elif}\;F \leq 10000:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5e+151)
   (/ (* -1.0 (* F (+ (/ 1.0 F) (/ x F)))) B)
   (if (<= F 10000.0)
     (/ (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F) x) B)
     (/ (* F (fma -1.0 (/ x F) (/ 1.0 F))) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5e+151) {
		tmp = (-1.0 * (F * ((1.0 / F) + (x / F)))) / B;
	} else if (F <= 10000.0) {
		tmp = ((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F) - x) / B;
	} else {
		tmp = (F * fma(-1.0, (x / F), (1.0 / F))) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5e+151)
		tmp = Float64(Float64(-1.0 * Float64(F * Float64(Float64(1.0 / F) + Float64(x / F)))) / B);
	elseif (F <= 10000.0)
		tmp = Float64(Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F) - x) / B);
	else
		tmp = Float64(Float64(F * fma(-1.0, Float64(x / F), Float64(1.0 / F))) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5e+151], N[(N[(-1.0 * N[(F * N[(N[(1.0 / F), $MachinePrecision] + N[(x / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 10000.0], N[(N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(F * N[(-1.0 * N[(x / F), $MachinePrecision] + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.0000000000000002e151

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in B around inf

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B}} - x}{\tan B} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\color{blue}{\cos B}} - x}{\tan B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos \color{blue}{B}} - x}{\tan B} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B} - x}{\tan B} \]

      11. lower-cos.f64 84.7

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B} - x}{\tan B} \]

   6. Applied rewrites 84.7%

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B}} - x}{\tan B} \]

   7. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{\color{blue}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      7. lower-pow.f64 44.1

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B} \]

   9. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   10. Taylor expanded in F around -inf

       \[\leadsto \frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B} \]

       5. lower-/.f64 28.2

          \[\leadsto \frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B} \]

   12. Applied rewrites 28.2%

       \[\leadsto \frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B} \]

   if -5.0000000000000002e151 < F < 1e4

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in B around inf

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B}} - x}{\tan B} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\color{blue}{\cos B}} - x}{\tan B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos \color{blue}{B}} - x}{\tan B} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B} - x}{\tan B} \]

      11. lower-cos.f64 84.7

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B} - x}{\tan B} \]

   6. Applied rewrites 84.7%

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B}} - x}{\tan B} \]

   7. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{\color{blue}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      7. lower-pow.f64 44.1

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B} \]

   9. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

       2. *-commutative N/A

          \[\leadsto \frac{{\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       3. lift-+.f64 N/A

          \[\leadsto \frac{{\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       4. +-commutative N/A

          \[\leadsto \frac{{\left(\mathsf{fma}\left(2, x, {F}^{2}\right) + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       5. lift-fma.f64 N/A

          \[\leadsto \frac{{\left(\left(2 \cdot x + {F}^{2}\right) + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       6. associate-+r+ N/A

          \[\leadsto \frac{{\left(2 \cdot x + \left({F}^{2} + 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       7. *-commutative N/A

          \[\leadsto \frac{{\left(x \cdot 2 + \left({F}^{2} + 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       8. lift-pow.f64 N/A

          \[\leadsto \frac{{\left(x \cdot 2 + \left({F}^{2} + 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       9. pow2 N/A

          \[\leadsto \frac{{\left(x \cdot 2 + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       10. lift-fma.f64 N/A

           \[\leadsto \frac{{\left(x \cdot 2 + \mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       11. lift-fma.f64 N/A

           \[\leadsto \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       12. lift-*.f64 44.1

           \[\leadsto \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{B} \]

   11. Applied rewrites 44.1%

       \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{B}} \]

   if 1e4 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in B around inf

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B}} - x}{\tan B} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\color{blue}{\cos B}} - x}{\tan B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos \color{blue}{B}} - x}{\tan B} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B} - x}{\tan B} \]

      11. lower-cos.f64 84.7

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B} - x}{\tan B} \]

   6. Applied rewrites 84.7%

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B}} - x}{\tan B} \]

   7. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{\color{blue}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      7. lower-pow.f64 44.1

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B} \]

   9. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   10. Taylor expanded in F around inf

       \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

       4. lower-/.f64 28.4

          \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   12. Applied rewrites 28.4%

       \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 27: 51.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -0.58:\\ \;\;\;\;\frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B}\\ \mathbf{elif}\;F \leq 0.00165:\\ \;\;\;\;\frac{F \cdot {\left(2 + 2 \cdot x\right)}^{-0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.58)
   (/ (* -1.0 (* F (+ (/ 1.0 F) (/ x F)))) B)
   (if (<= F 0.00165)
     (/ (- (* F (pow (+ 2.0 (* 2.0 x)) -0.5)) x) B)
     (/ (* F (fma -1.0 (/ x F) (/ 1.0 F))) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.58) {
		tmp = (-1.0 * (F * ((1.0 / F) + (x / F)))) / B;
	} else if (F <= 0.00165) {
		tmp = ((F * pow((2.0 + (2.0 * x)), -0.5)) - x) / B;
	} else {
		tmp = (F * fma(-1.0, (x / F), (1.0 / F))) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.58)
		tmp = Float64(Float64(-1.0 * Float64(F * Float64(Float64(1.0 / F) + Float64(x / F)))) / B);
	elseif (F <= 0.00165)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + Float64(2.0 * x)) ^ -0.5)) - x) / B);
	else
		tmp = Float64(Float64(F * fma(-1.0, Float64(x / F), Float64(1.0 / F))) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.58], N[(N[(-1.0 * N[(F * N[(N[(1.0 / F), $MachinePrecision] + N[(x / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 0.00165], N[(N[(N[(F * N[Power[N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(F * N[(-1.0 * N[(x / F), $MachinePrecision] + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.57999999999999996

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in B around inf

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B}} - x}{\tan B} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\color{blue}{\cos B}} - x}{\tan B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos \color{blue}{B}} - x}{\tan B} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B} - x}{\tan B} \]

      11. lower-cos.f64 84.7

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B} - x}{\tan B} \]

   6. Applied rewrites 84.7%

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B}} - x}{\tan B} \]

   7. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{\color{blue}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      7. lower-pow.f64 44.1

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B} \]

   9. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   10. Taylor expanded in F around -inf

       \[\leadsto \frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B} \]

       5. lower-/.f64 28.2

          \[\leadsto \frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B} \]

   12. Applied rewrites 28.2%

       \[\leadsto \frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B} \]

   if -0.57999999999999996 < F < 0.00165

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in B around inf

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B}} - x}{\tan B} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\color{blue}{\cos B}} - x}{\tan B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos \color{blue}{B}} - x}{\tan B} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B} - x}{\tan B} \]

      11. lower-cos.f64 84.7

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B} - x}{\tan B} \]

   6. Applied rewrites 84.7%

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B}} - x}{\tan B} \]

   7. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{\color{blue}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      7. lower-pow.f64 44.1

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B} \]

   9. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   10. Taylor expanded in F around 0

       \[\leadsto \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{\frac{-1}{2}} - x}{B} \]
   11. Step-by-step derivation

       1. lower-*.f64 29.9

          \[\leadsto \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{-0.5} - x}{B} \]

   12. Applied rewrites 29.9%

       \[\leadsto \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{-0.5} - x}{B} \]

   if 0.00165 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in B around inf

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B}} - x}{\tan B} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\color{blue}{\cos B}} - x}{\tan B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos \color{blue}{B}} - x}{\tan B} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B} - x}{\tan B} \]

      11. lower-cos.f64 84.7

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B} - x}{\tan B} \]

   6. Applied rewrites 84.7%

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B}} - x}{\tan B} \]

   7. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{\color{blue}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      7. lower-pow.f64 44.1

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B} \]

   9. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   10. Taylor expanded in F around inf

       \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

       4. lower-/.f64 28.4

          \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   12. Applied rewrites 28.4%

       \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 28: 44.2% accurate, 4.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B}\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{-18}:\\ \;\;\;\;-1 \cdot \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.3e-15)
   (/ (* -1.0 (* F (+ (/ 1.0 F) (/ x F)))) B)
   (if (<= F 1.45e-18)
     (* -1.0 (/ x B))
     (/ (* F (fma -1.0 (/ x F) (/ 1.0 F))) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.3e-15) {
		tmp = (-1.0 * (F * ((1.0 / F) + (x / F)))) / B;
	} else if (F <= 1.45e-18) {
		tmp = -1.0 * (x / B);
	} else {
		tmp = (F * fma(-1.0, (x / F), (1.0 / F))) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.3e-15)
		tmp = Float64(Float64(-1.0 * Float64(F * Float64(Float64(1.0 / F) + Float64(x / F)))) / B);
	elseif (F <= 1.45e-18)
		tmp = Float64(-1.0 * Float64(x / B));
	else
		tmp = Float64(Float64(F * fma(-1.0, Float64(x / F), Float64(1.0 / F))) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.3e-15], N[(N[(-1.0 * N[(F * N[(N[(1.0 / F), $MachinePrecision] + N[(x / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.45e-18], N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(F * N[(-1.0 * N[(x / F), $MachinePrecision] + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.30000000000000002e-15

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in B around inf

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B}} - x}{\tan B} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\color{blue}{\cos B}} - x}{\tan B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos \color{blue}{B}} - x}{\tan B} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B} - x}{\tan B} \]

      11. lower-cos.f64 84.7

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B} - x}{\tan B} \]

   6. Applied rewrites 84.7%

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B}} - x}{\tan B} \]

   7. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{\color{blue}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      7. lower-pow.f64 44.1

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B} \]

   9. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   10. Taylor expanded in F around -inf

       \[\leadsto \frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B} \]

       2. lower-*.f64 N/A

          \[\leadsto \frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B} \]

       3. lower-+.f64 N/A

          \[\leadsto \frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B} \]

       4. lower-/.f64 N/A

          \[\leadsto \frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B} \]

       5. lower-/.f64 28.2

          \[\leadsto \frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B} \]

   12. Applied rewrites 28.2%

       \[\leadsto \frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B} \]

   if -1.30000000000000002e-15 < F < 1.45e-18

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in B around inf

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B}} - x}{\tan B} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\color{blue}{\cos B}} - x}{\tan B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos \color{blue}{B}} - x}{\tan B} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B} - x}{\tan B} \]

      11. lower-cos.f64 84.7

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B} - x}{\tan B} \]

   6. Applied rewrites 84.7%

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B}} - x}{\tan B} \]

   7. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{\color{blue}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      7. lower-pow.f64 44.1

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B} \]

   9. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   10. Taylor expanded in F around 0

       \[\leadsto -1 \cdot \color{blue}{\frac{x}{B}} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]

       2. lower-/.f64 29.4

          \[\leadsto -1 \cdot \frac{x}{B} \]

   12. Applied rewrites 29.4%

       \[\leadsto -1 \cdot \color{blue}{\frac{x}{B}} \]

   if 1.45e-18 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in B around inf

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B}} - x}{\tan B} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\color{blue}{\cos B}} - x}{\tan B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos \color{blue}{B}} - x}{\tan B} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B} - x}{\tan B} \]

      11. lower-cos.f64 84.7

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B} - x}{\tan B} \]

   6. Applied rewrites 84.7%

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B}} - x}{\tan B} \]

   7. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{\color{blue}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      7. lower-pow.f64 44.1

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B} \]

   9. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   10. Taylor expanded in F around inf

       \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

       4. lower-/.f64 28.4

          \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   12. Applied rewrites 28.4%

       \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 29: 37.9% accurate, 4.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.45 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{-18}:\\ \;\;\;\;-1 \cdot \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.45e-9)
   (/ -1.0 B)
   (if (<= F 1.45e-18)
     (* -1.0 (/ x B))
     (/ (* F (fma -1.0 (/ x F) (/ 1.0 F))) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.45e-9) {
		tmp = -1.0 / B;
	} else if (F <= 1.45e-18) {
		tmp = -1.0 * (x / B);
	} else {
		tmp = (F * fma(-1.0, (x / F), (1.0 / F))) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.45e-9)
		tmp = Float64(-1.0 / B);
	elseif (F <= 1.45e-18)
		tmp = Float64(-1.0 * Float64(x / B));
	else
		tmp = Float64(Float64(F * fma(-1.0, Float64(x / F), Float64(1.0 / F))) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.45e-9], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 1.45e-18], N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(F * N[(-1.0 * N[(x / F), $MachinePrecision] + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.44999999999999996e-9

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. 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 17.7

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-1}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 10.6%

      \[\leadsto \frac{-1}{B} \]

   if -1.44999999999999996e-9 < F < 1.45e-18

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in B around inf

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B}} - x}{\tan B} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\color{blue}{\cos B}} - x}{\tan B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos \color{blue}{B}} - x}{\tan B} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B} - x}{\tan B} \]

      11. lower-cos.f64 84.7

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B} - x}{\tan B} \]

   6. Applied rewrites 84.7%

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B}} - x}{\tan B} \]

   7. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{\color{blue}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      7. lower-pow.f64 44.1

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B} \]

   9. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   10. Taylor expanded in F around 0

       \[\leadsto -1 \cdot \color{blue}{\frac{x}{B}} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]

       2. lower-/.f64 29.4

          \[\leadsto -1 \cdot \frac{x}{B} \]

   12. Applied rewrites 29.4%

       \[\leadsto -1 \cdot \color{blue}{\frac{x}{B}} \]

   if 1.45e-18 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in B around inf

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B}} - x}{\tan B} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\color{blue}{\cos B}} - x}{\tan B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos \color{blue}{B}} - x}{\tan B} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B} - x}{\tan B} \]

      11. lower-cos.f64 84.7

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B} - x}{\tan B} \]

   6. Applied rewrites 84.7%

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B}} - x}{\tan B} \]

   7. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{\color{blue}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      7. lower-pow.f64 44.1

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B} \]

   9. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   10. Taylor expanded in F around inf

       \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

       2. lower-fma.f64 N/A

          \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

       4. lower-/.f64 28.4

          \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   12. Applied rewrites 28.4%

       \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 30: 30.5% accurate, 10.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.45 \cdot 10^{-9}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.45e-9) (/ -1.0 B) (* -1.0 (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.45e-9) {
		tmp = -1.0 / B;
	} else {
		tmp = -1.0 * (x / B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.45d-9)) then
        tmp = (-1.0d0) / b
    else
        tmp = (-1.0d0) * (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.45e-9) {
		tmp = -1.0 / B;
	} else {
		tmp = -1.0 * (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.45e-9:
		tmp = -1.0 / B
	else:
		tmp = -1.0 * (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.45e-9)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(-1.0 * Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.45e-9)
		tmp = -1.0 / B;
	else
		tmp = -1.0 * (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.45e-9], N[(-1.0 / B), $MachinePrecision], N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.44999999999999996e-9

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. 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 17.7

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-1}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 10.6%

      \[\leadsto \frac{-1}{B} \]

   if -1.44999999999999996e-9 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in B around inf

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B}} - x}{\tan B} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\color{blue}{\cos B}} - x}{\tan B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos \color{blue}{B}} - x}{\tan B} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B} - x}{\tan B} \]

      11. lower-cos.f64 84.7

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B} - x}{\tan B} \]

   6. Applied rewrites 84.7%

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B}} - x}{\tan B} \]

   7. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{\color{blue}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      7. lower-pow.f64 44.1

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B} \]

   9. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   10. Taylor expanded in F around 0

       \[\leadsto -1 \cdot \color{blue}{\frac{x}{B}} \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]

       2. lower-/.f64 29.4

          \[\leadsto -1 \cdot \frac{x}{B} \]

   12. Applied rewrites 29.4%

       \[\leadsto -1 \cdot \color{blue}{\frac{x}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 31: 18.0% accurate, 14.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq 2 \cdot 10^{-134}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (if (<= F 2e-134) (/ -1.0 B) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 2e-134) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= 2d-134) then
        tmp = (-1.0d0) / b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 2e-134) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 2e-134:
		tmp = -1.0 / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 2e-134)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 2e-134)
		tmp = -1.0 / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 2e-134], N[(-1.0 / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 2.00000000000000008e-134

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. 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 17.7

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-1}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 10.6%

      \[\leadsto \frac{-1}{B} \]

   if 2.00000000000000008e-134 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{x \cdot \frac{1}{\tan B}} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \color{blue}{\frac{1}{\tan B}} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \color{blue}{\frac{x}{\tan B}} \]

      8. sub-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

      9. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \cdot \tan B - x}{\tan B}} \]

   3. Applied rewrites 76.4%

      \[\leadsto \color{blue}{\frac{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right) \cdot \tan B - x}{\tan B}} \]

   4. Taylor expanded in B around inf

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B}} - x}{\tan B} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\color{blue}{\cos B}} - x}{\tan B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos \color{blue}{B}} - x}{\tan B} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      8. lower-pow.f64 N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\cos B} - x}{\tan B} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}{\cos B} - x}{\tan B} \]

      11. lower-cos.f64 84.7

          \[\leadsto \frac{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B} - x}{\tan B} \]

   6. Applied rewrites 84.7%

      \[\leadsto \frac{\color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5}}{\cos B}} - x}{\tan B} \]

   7. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{\color{blue}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      7. lower-pow.f64 44.1

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B} \]

   9. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   10. Taylor expanded in F around inf

       \[\leadsto \frac{1}{\color{blue}{B}} \]
   11. Step-by-step derivation

       1. lower-/.f64 10.1

          \[\leadsto \frac{1}{B} \]

   12. Applied rewrites 10.1%

       \[\leadsto \frac{1}{\color{blue}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 32: 10.6% accurate, 26.5× speedup

Math

\[\frac{-1}{B} \]

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 76.3%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in F around -inf

   \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

   2. lower-sin.f64 17.7

      \[\leadsto \frac{-1}{\sin B} \]

4. Applied rewrites 17.7%

   \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

5. Taylor expanded in B around 0

   \[\leadsto \frac{-1}{B} \]
6. Step-by-step derivation

7. Applied rewrites 10.6%

   \[\leadsto \frac{-1}{B} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025174 
(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))))))