VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.7% → 99.7%

Time: 7.9s

Alternatives: 25

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, t\_1\right)\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+121}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\sin B}, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ (- x) (tan B))))
   (if (<= F -1e+81)
     (fma t_0 -1.0 t_1)
     (if (<= F 5e+121)
       (fma t_0 (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F) t_1)
       (fma -1.0 (/ (* x (cos B)) (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = -x / tan(B);
	double tmp;
	if (F <= -1e+81) {
		tmp = fma(t_0, -1.0, t_1);
	} else if (F <= 5e+121) {
		tmp = fma(t_0, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F), t_1);
	} else {
		tmp = fma(-1.0, ((x * cos(B)) / sin(B)), t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -1e+81)
		tmp = fma(t_0, -1.0, t_1);
	elseif (F <= 5e+121)
		tmp = fma(t_0, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F), t_1);
	else
		tmp = fma(-1.0, Float64(Float64(x * cos(B)) / sin(B)), t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e+81], N[(t$95$0 * -1.0 + t$95$1), $MachinePrecision], If[LessEqual[F, 5e+121], N[(t$95$0 * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] + t$95$1), $MachinePrecision], N[(-1.0 * N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.99999999999999921e80

   1. Initial program 77.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.9%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 55.6%

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

   if -9.99999999999999921e80 < F < 5.00000000000000007e121

   1. Initial program 77.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.9%

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

   if 5.00000000000000007e121 < F

   1. Initial program 77.7%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.4

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

   4. Applied rewrites 47.4%

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

   5. Taylor expanded in F around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sin.f64 55.9

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

   7. Applied rewrites 55.9%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, t\_1\right)\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{+95}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\frac{x}{\tan B \cdot F}, F, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ (- x) (tan B))))
   (if (<= F -1e+81)
     (fma t_0 -1.0 t_1)
     (if (<= F 1.8e+95)
       (fma t_0 (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F) t_1)
       (fma (- (/ x (* (tan B) F))) F t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = -x / tan(B);
	double tmp;
	if (F <= -1e+81) {
		tmp = fma(t_0, -1.0, t_1);
	} else if (F <= 1.8e+95) {
		tmp = fma(t_0, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F), t_1);
	} else {
		tmp = fma(-(x / (tan(B) * F)), F, t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -1e+81)
		tmp = fma(t_0, -1.0, t_1);
	elseif (F <= 1.8e+95)
		tmp = fma(t_0, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F), t_1);
	else
		tmp = fma(Float64(-Float64(x / Float64(tan(B) * F))), F, t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e+81], N[(t$95$0 * -1.0 + t$95$1), $MachinePrecision], If[LessEqual[F, 1.8e+95], N[(t$95$0 * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] + t$95$1), $MachinePrecision], N[((-N[(x / N[(N[Tan[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]) * F + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.99999999999999921e80

   1. Initial program 77.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.9%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 55.6%

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

   if -9.99999999999999921e80 < F < 1.79999999999999989e95

   1. Initial program 77.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.9%

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

   if 1.79999999999999989e95 < F

   1. Initial program 77.7%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.4

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

   4. Applied rewrites 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. inv-pow N/A

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

      9. pow-plus N/A

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

      10. metadata-eval N/A

          \[\leadsto \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \cdot F + \frac{{F}^{0}}{\sin B} \]

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 50.2%

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

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -8 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, t\_1\right)\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\frac{x}{\tan B \cdot F}, F, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ (- x) (tan B))))
   (if (<= F -8e+58)
     (fma t_0 -1.0 t_1)
     (if (<= F 2.2e+100)
       (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B)) t_1)
       (fma (- (/ x (* (tan B) F))) F t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = -x / tan(B);
	double tmp;
	if (F <= -8e+58) {
		tmp = fma(t_0, -1.0, t_1);
	} else if (F <= 2.2e+100) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), t_1);
	} else {
		tmp = fma(-(x / (tan(B) * F)), F, t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -8e+58)
		tmp = fma(t_0, -1.0, t_1);
	elseif (F <= 2.2e+100)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), t_1);
	else
		tmp = fma(Float64(-Float64(x / Float64(tan(B) * F))), F, t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -8e+58], N[(t$95$0 * -1.0 + t$95$1), $MachinePrecision], If[LessEqual[F, 2.2e+100], 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$1), $MachinePrecision], N[((-N[(x / N[(N[Tan[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]) * F + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.99999999999999955e58

   1. Initial program 77.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.9%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 55.6%

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

   if -7.99999999999999955e58 < F < 2.2000000000000001e100

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 85.8%

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

   if 2.2000000000000001e100 < F

   1. Initial program 77.7%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.4

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

   4. Applied rewrites 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. inv-pow N/A

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

      9. pow-plus N/A

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

      10. metadata-eval N/A

          \[\leadsto \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \cdot F + \frac{{F}^{0}}{\sin B} \]

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 50.2%

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

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -3.6 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{+90}:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\frac{x}{\tan B \cdot F}, F, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))))
   (if (<= F -3.6e+52)
     (fma t_0 -1.0 (/ (- x) (tan B)))
     (if (<= F 2.2e+90)
       (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F (sin B))) (/ x (tan B)))
       (fma (- (/ x (* (tan B) F))) F t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double tmp;
	if (F <= -3.6e+52) {
		tmp = fma(t_0, -1.0, (-x / tan(B)));
	} else if (F <= 2.2e+90) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / sin(B))) - (x / tan(B));
	} else {
		tmp = fma(-(x / (tan(B) * F)), F, t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -3.6e+52)
		tmp = fma(t_0, -1.0, Float64(Float64(-x) / tan(B)));
	elseif (F <= 2.2e+90)
		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(Float64(-Float64(x / Float64(tan(B) * F))), F, t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.6e+52], N[(t$95$0 * -1.0 + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.2e+90], 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[((-N[(x / N[(N[Tan[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]) * F + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.6e52

   1. Initial program 77.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.9%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 55.6%

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

   if -3.6e52 < F < 2.1999999999999999e90

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 77.7

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

   3. Applied rewrites 77.8%

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

   if 2.1999999999999999e90 < F

   1. Initial program 77.7%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.4

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

   4. Applied rewrites 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. inv-pow N/A

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

      9. pow-plus N/A

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

      10. metadata-eval N/A

          \[\leadsto \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \cdot F + \frac{{F}^{0}}{\sin B} \]

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 50.2%

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

Alternative 5: 97.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -2.3 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 0.0098:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\frac{x}{\tan B \cdot F}, F, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))))
   (if (<= F -2.3e+30)
     (fma t_0 -1.0 (/ (- x) (tan B)))
     (if (<= F 0.0098)
       (- (* (pow (fma 2.0 x 2.0) -0.5) (/ F (sin B))) (/ x (tan B)))
       (fma (- (/ x (* (tan B) F))) F t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double tmp;
	if (F <= -2.3e+30) {
		tmp = fma(t_0, -1.0, (-x / tan(B)));
	} else if (F <= 0.0098) {
		tmp = (pow(fma(2.0, x, 2.0), -0.5) * (F / sin(B))) - (x / tan(B));
	} else {
		tmp = fma(-(x / (tan(B) * F)), F, t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -2.3e+30)
		tmp = fma(t_0, -1.0, Float64(Float64(-x) / tan(B)));
	elseif (F <= 0.0098)
		tmp = Float64(Float64((fma(2.0, x, 2.0) ^ -0.5) * Float64(F / sin(B))) - Float64(x / tan(B)));
	else
		tmp = fma(Float64(-Float64(x / Float64(tan(B) * F))), F, t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.3e+30], N[(t$95$0 * -1.0 + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0098], N[(N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[(x / N[(N[Tan[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]) * F + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.3e30

   1. Initial program 77.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.9%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 55.6%

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

   if -2.3e30 < F < 0.0097999999999999997

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 77.7

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

   3. Applied rewrites 77.8%

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

   4. Taylor expanded in F around 0

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

   6. Applied rewrites 55.1%

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

   if 0.0097999999999999997 < F

   1. Initial program 77.7%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.4

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

   4. Applied rewrites 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. inv-pow N/A

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

      9. pow-plus N/A

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

      10. metadata-eval N/A

          \[\leadsto \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \cdot F + \frac{{F}^{0}}{\sin B} \]

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 50.2%

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

Alternative 6: 92.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -14000:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq -1.75 \cdot 10^{-85}:\\ \;\;\;\;\left(-x \cdot \frac{1}{B}\right) + \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}}\\ \mathbf{elif}\;F \leq 270000:\\ \;\;\;\;\frac{1}{\frac{B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\frac{x}{\tan B \cdot F}, F, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))))
   (if (<= F -14000.0)
     (fma t_0 -1.0 (/ (- x) (tan B)))
     (if (<= F -1.75e-85)
       (+
        (- (* x (/ 1.0 B)))
        (/ 1.0 (/ (sin B) (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F))))
       (if (<= F 270000.0)
         (-
          (/ 1.0 (/ B (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F)))
          (/ x (tan B)))
         (fma (- (/ x (* (tan B) F))) F t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double tmp;
	if (F <= -14000.0) {
		tmp = fma(t_0, -1.0, (-x / tan(B)));
	} else if (F <= -1.75e-85) {
		tmp = -(x * (1.0 / B)) + (1.0 / (sin(B) / (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F)));
	} else if (F <= 270000.0) {
		tmp = (1.0 / (B / (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F))) - (x / tan(B));
	} else {
		tmp = fma(-(x / (tan(B) * F)), F, t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -14000.0)
		tmp = fma(t_0, -1.0, Float64(Float64(-x) / tan(B)));
	elseif (F <= -1.75e-85)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / B))) + Float64(1.0 / Float64(sin(B) / Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F))));
	elseif (F <= 270000.0)
		tmp = Float64(Float64(1.0 / Float64(B / Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F))) - Float64(x / tan(B)));
	else
		tmp = fma(Float64(-Float64(x / Float64(tan(B) * F))), F, t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -14000.0], N[(t$95$0 * -1.0 + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.75e-85], N[((-N[(x * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]) + N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 270000.0], N[(N[(1.0 / N[(B / N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[(x / N[(N[Tan[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]) * F + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -14000

   1. Initial program 77.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.9%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 55.6%

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

   if -14000 < F < -1.74999999999999989e-85

   1. Initial program 77.7%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

   6. Applied rewrites 57.7%

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

   if -1.74999999999999989e-85 < F < 2.7e5

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 77.7

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

   3. Applied rewrites 77.8%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 62.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-flip N/A

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

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

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lift-/.f64 N/A

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

      16. lift-neg.f64 N/A

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

      17. *-commutative N/A

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

   8. Applied rewrites 70.0%

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

   if 2.7e5 < F

   1. Initial program 77.7%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.4

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

   4. Applied rewrites 47.4%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. inv-pow N/A

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

      9. pow-plus N/A

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

      10. metadata-eval N/A

          \[\leadsto \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \cdot F + \frac{{F}^{0}}{\sin B} \]

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. lower-fma.f64 N/A

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

   6. Applied rewrites 50.2%

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

Alternative 7: 92.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -14000:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, -1, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq -1.75 \cdot 10^{-85}:\\ \;\;\;\;\left(-x \cdot \frac{1}{B}\right) + \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}}\\ \mathbf{elif}\;F \leq 270000:\\ \;\;\;\;\frac{1}{\frac{B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos B, -x, 1\right)}{\sin B \cdot F} \cdot F\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -14000.0)
   (fma (/ 1.0 (sin B)) -1.0 (/ (- x) (tan B)))
   (if (<= F -1.75e-85)
     (+
      (- (* x (/ 1.0 B)))
      (/ 1.0 (/ (sin B) (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F))))
     (if (<= F 270000.0)
       (-
        (/ 1.0 (/ B (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F)))
        (/ x (tan B)))
       (* (/ (fma (cos B) (- x) 1.0) (* (sin B) F)) F)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -14000.0) {
		tmp = fma((1.0 / sin(B)), -1.0, (-x / tan(B)));
	} else if (F <= -1.75e-85) {
		tmp = -(x * (1.0 / B)) + (1.0 / (sin(B) / (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F)));
	} else if (F <= 270000.0) {
		tmp = (1.0 / (B / (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F))) - (x / tan(B));
	} else {
		tmp = (fma(cos(B), -x, 1.0) / (sin(B) * F)) * F;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -14000.0)
		tmp = fma(Float64(1.0 / sin(B)), -1.0, Float64(Float64(-x) / tan(B)));
	elseif (F <= -1.75e-85)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / B))) + Float64(1.0 / Float64(sin(B) / Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F))));
	elseif (F <= 270000.0)
		tmp = Float64(Float64(1.0 / Float64(B / Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F))) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(fma(cos(B), Float64(-x), 1.0) / Float64(sin(B) * F)) * F);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -14000.0], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * -1.0 + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.75e-85], N[((-N[(x * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]) + N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 270000.0], N[(N[(1.0 / N[(B / N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[B], $MachinePrecision] * (-x) + 1.0), $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -14000

   1. Initial program 77.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.9%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 55.6%

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

   if -14000 < F < -1.74999999999999989e-85

   1. Initial program 77.7%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

   6. Applied rewrites 57.7%

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

   if -1.74999999999999989e-85 < F < 2.7e5

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 77.7

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

   3. Applied rewrites 77.8%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 62.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-flip N/A

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

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

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lift-/.f64 N/A

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

      16. lift-neg.f64 N/A

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

      17. *-commutative N/A

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

   8. Applied rewrites 70.0%

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

   if 2.7e5 < F

   1. Initial program 77.7%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.4

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

   4. Applied rewrites 47.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.4

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

   6. Applied rewrites 50.0%

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

Alternative 8: 80.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin B \cdot F\\ \mathbf{if}\;F \leq -2 \cdot 10^{+205}:\\ \;\;\;\;F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \left|\frac{1}{t\_0}\right|\right)\\ \mathbf{elif}\;F \leq 270000:\\ \;\;\;\;\frac{1}{\frac{B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos B, -x, 1\right)}{t\_0} \cdot F\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (sin B) F)))
   (if (<= F -2e+205)
     (* F (fma -1.0 (/ x (* B F)) (fabs (/ 1.0 t_0))))
     (if (<= F 270000.0)
       (-
        (/ 1.0 (/ B (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F)))
        (/ x (tan B)))
       (* (/ (fma (cos B) (- x) 1.0) t_0) F)))))

C

double code(double F, double B, double x) {
	double t_0 = sin(B) * F;
	double tmp;
	if (F <= -2e+205) {
		tmp = F * fma(-1.0, (x / (B * F)), fabs((1.0 / t_0)));
	} else if (F <= 270000.0) {
		tmp = (1.0 / (B / (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F))) - (x / tan(B));
	} else {
		tmp = (fma(cos(B), -x, 1.0) / t_0) * F;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(sin(B) * F)
	tmp = 0.0
	if (F <= -2e+205)
		tmp = Float64(F * fma(-1.0, Float64(x / Float64(B * F)), abs(Float64(1.0 / t_0))));
	elseif (F <= 270000.0)
		tmp = Float64(Float64(1.0 / Float64(B / Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F))) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(fma(cos(B), Float64(-x), 1.0) / t_0) * F);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]}, If[LessEqual[F, -2e+205], N[(F * N[(-1.0 * N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[Abs[N[(1.0 / t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 270000.0], N[(N[(1.0 / N[(B / N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[B], $MachinePrecision] * (-x) + 1.0), $MachinePrecision] / t$95$0), $MachinePrecision] * F), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.00000000000000003e205

   1. Initial program 77.7%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.4

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

   4. Applied rewrites 47.4%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 31.3

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

   7. Applied rewrites 31.3%

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

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. lower-unsound-exp.f64 N/A

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

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

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

      6. lower-unsound-log.f64 16.6

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 16.6

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

   9. Applied rewrites 16.6%

      \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, e^{\log \left(\sin B \cdot F\right) \cdot -1}\right) \]
   10. Step-by-step derivation

       1. lift-exp.f64 N/A

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

       2. exp-fabs N/A

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

       3. lift-*.f64 N/A

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

       4. lift-log.f64 N/A

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

       5. exp-to-pow N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. inv-pow N/A

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

       10. lift-*.f64 N/A

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

       11. lift-sin.f64 N/A

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

       12. lower-fabs.f64 N/A

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

       13. lift-sin.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. lower-/.f64 32.8

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

       16. lift-*.f64 N/A

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

       17. *-commutative N/A

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

       18. lift-*.f64 32.8

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

   11. Applied rewrites 32.8%

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

   if -2.00000000000000003e205 < F < 2.7e5

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 77.7

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

   3. Applied rewrites 77.8%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 62.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-flip N/A

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

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

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lift-/.f64 N/A

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

      16. lift-neg.f64 N/A

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

      17. *-commutative N/A

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

   8. Applied rewrites 70.0%

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

   if 2.7e5 < F

   1. Initial program 77.7%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.4

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

   4. Applied rewrites 47.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.4

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

   6. Applied rewrites 50.0%

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

Alternative 9: 78.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ t_1 := \frac{1}{\frac{B}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}} - \frac{x}{\tan B}\\ \mathbf{if}\;t\_0 \leq -200000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F, \frac{-x}{B}\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+290}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \left|\frac{1}{\sin B \cdot F}\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (+
          (- (* x (/ 1.0 (tan B))))
          (*
           (/ F (sin B))
           (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))
        (t_1
         (-
          (/ 1.0 (/ B (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F)))
          (/ x (tan B)))))
   (if (<= t_0 -200000.0)
     t_1
     (if (<= t_0 10.0)
       (fma
        (/ 1.0 (sin B))
        (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F)
        (/ (- x) B))
       (if (<= t_0 4e+290)
         t_1
         (* F (fma -1.0 (/ x (* B F)) (fabs (/ 1.0 (* (sin B) F))))))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	double t_1 = (1.0 / (B / (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F))) - (x / tan(B));
	double tmp;
	if (t_0 <= -200000.0) {
		tmp = t_1;
	} else if (t_0 <= 10.0) {
		tmp = fma((1.0 / sin(B)), (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F), (-x / B));
	} else if (t_0 <= 4e+290) {
		tmp = t_1;
	} else {
		tmp = F * fma(-1.0, (x / (B * F)), fabs((1.0 / (sin(B) * F))));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))))
	t_1 = Float64(Float64(1.0 / Float64(B / Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F))) - Float64(x / tan(B)))
	tmp = 0.0
	if (t_0 <= -200000.0)
		tmp = t_1;
	elseif (t_0 <= 10.0)
		tmp = fma(Float64(1.0 / sin(B)), Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F), Float64(Float64(-x) / B));
	elseif (t_0 <= 4e+290)
		tmp = t_1;
	else
		tmp = Float64(F * fma(-1.0, Float64(x / Float64(B * F)), abs(Float64(1.0 / Float64(sin(B) * F)))));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / N[(B / N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -200000.0], t$95$1, If[LessEqual[t$95$0, 10.0], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] + N[((-x) / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+290], t$95$1, N[(F * N[(-1.0 * N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[Abs[N[(1.0 / N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < -2e5 or 10 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 4.00000000000000025e290

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 77.7

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

   3. Applied rewrites 77.8%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 62.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-flip N/A

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

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

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. lift-/.f64 N/A

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

      16. lift-neg.f64 N/A

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

      17. *-commutative N/A

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

   8. Applied rewrites 70.0%

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

   if -2e5 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 10

   1. Initial program 77.7%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.7%

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

   6. Applied rewrites 57.7%

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

   if 4.00000000000000025e290 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))))))

   1. Initial program 77.7%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.4

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

   4. Applied rewrites 47.4%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 31.3

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

   7. Applied rewrites 31.3%

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

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. lower-unsound-exp.f64 N/A

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

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

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

      6. lower-unsound-log.f64 16.6

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 16.6

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

   9. Applied rewrites 16.6%

      \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, e^{\log \left(\sin B \cdot F\right) \cdot -1}\right) \]
   10. Step-by-step derivation

       1. lift-exp.f64 N/A

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

       2. exp-fabs N/A

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

       3. lift-*.f64 N/A

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

       4. lift-log.f64 N/A

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

       5. exp-to-pow N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. inv-pow N/A

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

       10. lift-*.f64 N/A

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

       11. lift-sin.f64 N/A

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

       12. lower-fabs.f64 N/A

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

       13. lift-sin.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. lower-/.f64 32.8

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

       16. lift-*.f64 N/A

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

       17. *-commutative N/A

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

       18. lift-*.f64 32.8

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

   11. Applied rewrites 32.8%

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

Alternative 10: 77.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ t_1 := {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;t\_0 \leq -200000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F, \frac{-x}{B}\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+290}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \left|\frac{1}{\sin B \cdot F}\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (+
          (- (* x (/ 1.0 (tan B))))
          (*
           (/ F (sin B))
           (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))
        (t_1
         (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F B)) (/ x (tan B)))))
   (if (<= t_0 -200000.0)
     t_1
     (if (<= t_0 10.0)
       (fma
        (/ 1.0 (sin B))
        (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F)
        (/ (- x) B))
       (if (<= t_0 4e+290)
         t_1
         (* F (fma -1.0 (/ x (* B F)) (fabs (/ 1.0 (* (sin B) F))))))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	double t_1 = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / B)) - (x / tan(B));
	double tmp;
	if (t_0 <= -200000.0) {
		tmp = t_1;
	} else if (t_0 <= 10.0) {
		tmp = fma((1.0 / sin(B)), (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F), (-x / B));
	} else if (t_0 <= 4e+290) {
		tmp = t_1;
	} else {
		tmp = F * fma(-1.0, (x / (B * F)), fabs((1.0 / (sin(B) * F))));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))))
	t_1 = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / tan(B)))
	tmp = 0.0
	if (t_0 <= -200000.0)
		tmp = t_1;
	elseif (t_0 <= 10.0)
		tmp = fma(Float64(1.0 / sin(B)), Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F), Float64(Float64(-x) / B));
	elseif (t_0 <= 4e+290)
		tmp = t_1;
	else
		tmp = Float64(F * fma(-1.0, Float64(x / Float64(B * F)), abs(Float64(1.0 / Float64(sin(B) * F)))));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -200000.0], t$95$1, If[LessEqual[t$95$0, 10.0], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] + N[((-x) / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+290], t$95$1, N[(F * N[(-1.0 * N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[Abs[N[(1.0 / N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < -2e5 or 10 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 4.00000000000000025e290

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 77.7

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

   3. Applied rewrites 77.8%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 62.0%

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

   if -2e5 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 10

   1. Initial program 77.7%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.7%

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

   6. Applied rewrites 57.7%

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

   if 4.00000000000000025e290 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))))))

   1. Initial program 77.7%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.4

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

   4. Applied rewrites 47.4%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 31.3

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

   7. Applied rewrites 31.3%

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

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. lower-unsound-exp.f64 N/A

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

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

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

      6. lower-unsound-log.f64 16.6

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 16.6

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

   9. Applied rewrites 16.6%

      \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, e^{\log \left(\sin B \cdot F\right) \cdot -1}\right) \]
   10. Step-by-step derivation

       1. lift-exp.f64 N/A

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

       2. exp-fabs N/A

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

       3. lift-*.f64 N/A

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

       4. lift-log.f64 N/A

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

       5. exp-to-pow N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. inv-pow N/A

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

       10. lift-*.f64 N/A

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

       11. lift-sin.f64 N/A

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

       12. lower-fabs.f64 N/A

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

       13. lift-sin.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. lower-/.f64 32.8

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

       16. lift-*.f64 N/A

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

       17. *-commutative N/A

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

       18. lift-*.f64 32.8

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

   11. Applied rewrites 32.8%

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

Alternative 11: 77.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ t_1 := {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;t\_0 \leq -200000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{B}\right)\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+290}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \left|\frac{1}{\sin B \cdot F}\right|\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (+
          (- (* x (/ 1.0 (tan B))))
          (*
           (/ F (sin B))
           (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))
        (t_1
         (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F B)) (/ x (tan B)))))
   (if (<= t_0 -200000.0)
     t_1
     (if (<= t_0 10.0)
       (fma F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B)) (/ (- x) B))
       (if (<= t_0 4e+290)
         t_1
         (* F (fma -1.0 (/ x (* B F)) (fabs (/ 1.0 (* (sin B) F))))))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	double t_1 = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / B)) - (x / tan(B));
	double tmp;
	if (t_0 <= -200000.0) {
		tmp = t_1;
	} else if (t_0 <= 10.0) {
		tmp = fma(F, (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B)), (-x / B));
	} else if (t_0 <= 4e+290) {
		tmp = t_1;
	} else {
		tmp = F * fma(-1.0, (x / (B * F)), fabs((1.0 / (sin(B) * F))));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))))
	t_1 = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / tan(B)))
	tmp = 0.0
	if (t_0 <= -200000.0)
		tmp = t_1;
	elseif (t_0 <= 10.0)
		tmp = fma(F, Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B)), Float64(Float64(-x) / B));
	elseif (t_0 <= 4e+290)
		tmp = t_1;
	else
		tmp = Float64(F * fma(-1.0, Float64(x / Float64(B * F)), abs(Float64(1.0 / Float64(sin(B) * F)))));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -200000.0], t$95$1, If[LessEqual[t$95$0, 10.0], N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4e+290], t$95$1, N[(F * N[(-1.0 * N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[Abs[N[(1.0 / N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < -2e5 or 10 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 4.00000000000000025e290

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 77.7

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

   3. Applied rewrites 77.8%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 62.0%

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

   if -2e5 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 10

   1. Initial program 77.7%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.7%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 4.00000000000000025e290 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))))))

   1. Initial program 77.7%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.4

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

   4. Applied rewrites 47.4%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 31.3

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

   7. Applied rewrites 31.3%

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

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. lower-unsound-exp.f64 N/A

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

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

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

      6. lower-unsound-log.f64 16.6

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 16.6

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

   9. Applied rewrites 16.6%

      \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, e^{\log \left(\sin B \cdot F\right) \cdot -1}\right) \]
   10. Step-by-step derivation

       1. lift-exp.f64 N/A

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

       2. exp-fabs N/A

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

       3. lift-*.f64 N/A

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

       4. lift-log.f64 N/A

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

       5. exp-to-pow N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. inv-pow N/A

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

       10. lift-*.f64 N/A

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

       11. lift-sin.f64 N/A

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

       12. lower-fabs.f64 N/A

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

       13. lift-sin.f64 N/A

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

       14. lift-*.f64 N/A

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

       15. lower-/.f64 32.8

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

       16. lift-*.f64 N/A

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

       17. *-commutative N/A

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

       18. lift-*.f64 32.8

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

   11. Applied rewrites 32.8%

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

Alternative 12: 74.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.75 \cdot 10^{-85}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{B}\right)\\ \mathbf{elif}\;F \leq 9.8 \cdot 10^{+42}:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\frac{x}{B \cdot F}, F, \frac{1}{\sin B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.35e+154)
   (/ -1.0 (sin B))
   (if (<= F -1.75e-85)
     (fma F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B)) (/ (- x) B))
     (if (<= F 9.8e+42)
       (- (* (pow (fma 2.0 x 2.0) -0.5) (/ F B)) (/ x (tan B)))
       (fma (- (/ x (* B F))) F (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.35e+154) {
		tmp = -1.0 / sin(B);
	} else if (F <= -1.75e-85) {
		tmp = fma(F, (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B)), (-x / B));
	} else if (F <= 9.8e+42) {
		tmp = (pow(fma(2.0, x, 2.0), -0.5) * (F / B)) - (x / tan(B));
	} else {
		tmp = fma(-(x / (B * F)), F, (1.0 / sin(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.35e+154)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= -1.75e-85)
		tmp = fma(F, Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B)), Float64(Float64(-x) / B));
	elseif (F <= 9.8e+42)
		tmp = Float64(Float64((fma(2.0, x, 2.0) ^ -0.5) * Float64(F / B)) - Float64(x / tan(B)));
	else
		tmp = fma(Float64(-Float64(x / Float64(B * F))), F, Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.35e+154], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.75e-85], N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.8e+42], N[(N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision]) * F + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.35000000000000003e154

   1. Initial program 77.7%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 17.2

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

   4. Applied rewrites 17.2%

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

   if -1.35000000000000003e154 < F < -1.74999999999999989e-85

   1. Initial program 77.7%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.7%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 -1.74999999999999989e-85 < F < 9.8000000000000004e42

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 77.7

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

   3. Applied rewrites 77.8%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 62.0%

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

   7. Taylor expanded in F around 0

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

   9. Applied rewrites 48.8%

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

   if 9.8000000000000004e42 < F

   1. Initial program 77.7%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.4

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

   4. Applied rewrites 47.4%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 31.3

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

   7. Applied rewrites 31.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. inv-pow N/A

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

      9. pow-plus N/A

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

      10. metadata-eval N/A

          \[\leadsto \left(-1 \cdot \frac{x}{B \cdot F}\right) \cdot F + \frac{{F}^{0}}{\sin B} \]

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. lower-fma.f64 N/A

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

   9. Applied rewrites 34.0%

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

Alternative 13: 73.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.8 \cdot 10^{+147}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.75 \cdot 10^{-85}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9.8 \cdot 10^{+42}:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-\frac{x}{B \cdot F}, F, \frac{1}{\sin B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.8e+147)
   (/ -1.0 (sin B))
   (if (<= F -1.75e-85)
     (- (* (/ F (sin B)) (pow (fma x 2.0 (fma F F 2.0)) -0.5)) (/ x B))
     (if (<= F 9.8e+42)
       (- (* (pow (fma 2.0 x 2.0) -0.5) (/ F B)) (/ x (tan B)))
       (fma (- (/ x (* B F))) F (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.8e+147) {
		tmp = -1.0 / sin(B);
	} else if (F <= -1.75e-85) {
		tmp = ((F / sin(B)) * pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5)) - (x / B);
	} else if (F <= 9.8e+42) {
		tmp = (pow(fma(2.0, x, 2.0), -0.5) * (F / B)) - (x / tan(B));
	} else {
		tmp = fma(-(x / (B * F)), F, (1.0 / sin(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.8e+147)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= -1.75e-85)
		tmp = Float64(Float64(Float64(F / sin(B)) * (fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5)) - Float64(x / B));
	elseif (F <= 9.8e+42)
		tmp = Float64(Float64((fma(2.0, x, 2.0) ^ -0.5) * Float64(F / B)) - Float64(x / tan(B)));
	else
		tmp = fma(Float64(-Float64(x / Float64(B * F))), F, Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.8e+147], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.75e-85], 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, 9.8e+42], N[(N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision]) * F + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.8000000000000001e147

   1. Initial program 77.7%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 17.2

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

   4. Applied rewrites 17.2%

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

   if -1.8000000000000001e147 < F < -1.74999999999999989e-85

   1. Initial program 77.7%

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

   2. Taylor expanded in B around 0

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

   4. Applied rewrites 49.7%

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

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 49.7

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

   6. Applied rewrites 49.7%

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

   if -1.74999999999999989e-85 < F < 9.8000000000000004e42

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 77.7

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

   3. Applied rewrites 77.8%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 62.0%

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

   7. Taylor expanded in F around 0

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

   9. Applied rewrites 48.8%

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

   if 9.8000000000000004e42 < F

   1. Initial program 77.7%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.4

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

   4. Applied rewrites 47.4%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 31.3

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

   7. Applied rewrites 31.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. inv-pow N/A

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

      9. pow-plus N/A

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

      10. metadata-eval N/A

          \[\leadsto \left(-1 \cdot \frac{x}{B \cdot F}\right) \cdot F + \frac{{F}^{0}}{\sin B} \]

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. lower-fma.f64 N/A

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

   9. Applied rewrites 34.0%

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

Alternative 14: 70.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{B \cdot F}\\ t_1 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -2.3 \cdot 10^{+30}:\\ \;\;\;\;F \cdot \mathsf{fma}\left(\frac{1}{\left|F\right|}, \left|t\_1\right|, t\_0\right)\\ \mathbf{elif}\;F \leq 9.8 \cdot 10^{+42}:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, F, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x (* B F)))) (t_1 (/ 1.0 (sin B))))
   (if (<= F -2.3e+30)
     (* F (fma (/ 1.0 (fabs F)) (fabs t_1) t_0))
     (if (<= F 9.8e+42)
       (- (* (pow (fma 2.0 x 2.0) -0.5) (/ F B)) (/ x (tan B)))
       (fma t_0 F t_1)))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / (B * F));
	double t_1 = 1.0 / sin(B);
	double tmp;
	if (F <= -2.3e+30) {
		tmp = F * fma((1.0 / fabs(F)), fabs(t_1), t_0);
	} else if (F <= 9.8e+42) {
		tmp = (pow(fma(2.0, x, 2.0), -0.5) * (F / B)) - (x / tan(B));
	} else {
		tmp = fma(t_0, F, t_1);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / Float64(B * F)))
	t_1 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -2.3e+30)
		tmp = Float64(F * fma(Float64(1.0 / abs(F)), abs(t_1), t_0));
	elseif (F <= 9.8e+42)
		tmp = Float64(Float64((fma(2.0, x, 2.0) ^ -0.5) * Float64(F / B)) - Float64(x / tan(B)));
	else
		tmp = fma(t_0, F, t_1);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$1 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.3e+30], N[(F * N[(N[(1.0 / N[Abs[F], $MachinePrecision]), $MachinePrecision] * N[Abs[t$95$1], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.8e+42], N[(N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * F + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.3e30

   1. Initial program 77.7%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.4

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

   4. Applied rewrites 47.4%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 31.3

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

   7. Applied rewrites 31.3%

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

      1. lift-/.f64 N/A

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

      2. inv-pow N/A

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

      3. pow-to-exp N/A

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

      4. lower-unsound-exp.f64 N/A

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

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

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

      6. lower-unsound-log.f64 16.6

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 16.6

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

   9. Applied rewrites 16.6%

      \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, e^{\log \left(\sin B \cdot F\right) \cdot -1}\right) \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

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

       2. +-commutative N/A

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

       3. lift-exp.f64 N/A

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

       4. exp-fabs N/A

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

       5. lift-*.f64 N/A

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

       6. lift-log.f64 N/A

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

       7. exp-to-pow N/A

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

       8. lift-*.f64 N/A

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

       9. *-commutative N/A

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

       10. pow-prod-down N/A

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

       11. inv-pow N/A

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

       12. lift-/.f64 N/A

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

       13. inv-pow N/A

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

       14. lift-/.f64 N/A

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

       15. fabs-mul N/A

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

   11. Applied rewrites 34.1%

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

   if -2.3e30 < F < 9.8000000000000004e42

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 77.7

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

   3. Applied rewrites 77.8%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 62.0%

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

   7. Taylor expanded in F around 0

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

   9. Applied rewrites 48.8%

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

   if 9.8000000000000004e42 < F

   1. Initial program 77.7%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.4

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

   4. Applied rewrites 47.4%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 31.3

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

   7. Applied rewrites 31.3%

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

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. distribute-rgt-in N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/r* N/A

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

      7. associate-*l/ N/A

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

      8. inv-pow N/A

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

      9. pow-plus N/A

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

      10. metadata-eval N/A

          \[\leadsto \left(-1 \cdot \frac{x}{B \cdot F}\right) \cdot F + \frac{{F}^{0}}{\sin B} \]

      11. metadata-eval N/A

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

      12. lift-/.f64 N/A

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

      13. lower-fma.f64 N/A

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

   9. Applied rewrites 34.0%

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

Alternative 15: 55.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;B \leq 7.4 \cdot 10^{-227}:\\ \;\;\;\;F \cdot \mathsf{fma}\left(\frac{1}{\left|F\right|}, \left|\frac{1}{\sin B}\right|, -\frac{x}{B \cdot F}\right)\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-14}:\\ \;\;\;\;\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{+35}:\\ \;\;\;\;\left(-\frac{t\_0}{F}\right) \cdot F\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{F} \cdot \frac{F}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= B 7.4e-227)
     (* F (fma (/ 1.0 (fabs F)) (fabs (/ 1.0 (sin B))) (- (/ x (* B F)))))
     (if (<= B 8e-14)
       (/ (- (* F (pow (+ 2.0 (fma 2.0 x (pow F 2.0))) -0.5)) x) B)
       (if (<= B 6e+35) (* (- (/ t_0 F)) F) (- (* (/ 1.0 F) (/ F B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (B <= 7.4e-227) {
		tmp = F * fma((1.0 / fabs(F)), fabs((1.0 / sin(B))), -(x / (B * F)));
	} else if (B <= 8e-14) {
		tmp = ((F * pow((2.0 + fma(2.0, x, pow(F, 2.0))), -0.5)) - x) / B;
	} else if (B <= 6e+35) {
		tmp = -(t_0 / F) * F;
	} else {
		tmp = ((1.0 / F) * (F / B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (B <= 7.4e-227)
		tmp = Float64(F * fma(Float64(1.0 / abs(F)), abs(Float64(1.0 / sin(B))), Float64(-Float64(x / Float64(B * F)))));
	elseif (B <= 8e-14)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + fma(2.0, x, (F ^ 2.0))) ^ -0.5)) - x) / B);
	elseif (B <= 6e+35)
		tmp = Float64(Float64(-Float64(t_0 / F)) * F);
	else
		tmp = Float64(Float64(Float64(1.0 / F) * Float64(F / 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[B, 7.4e-227], N[(F * N[(N[(1.0 / N[Abs[F], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + (-N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[B, 8e-14], N[(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] / B), $MachinePrecision], If[LessEqual[B, 6e+35], N[((-N[(t$95$0 / F), $MachinePrecision]) * F), $MachinePrecision], N[(N[(N[(1.0 / F), $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 7.39999999999999957e-227

   1. Initial program 77.7%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.f64 47.4

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{\color{blue}{B \cdot F}}, \frac{1}{F \cdot \sin B}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot \color{blue}{F}}, \frac{1}{F \cdot \sin B}\right) \]

      2. lower-*.f64 31.3

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{F \cdot \sin B}\right) \]

   7. Applied rewrites 31.3%

      \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{\color{blue}{B \cdot F}}, \frac{1}{F \cdot \sin B}\right) \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{F \cdot \sin B}\right) \]

      2. inv-pow N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, {\left(F \cdot \sin B\right)}^{-1}\right) \]

      3. pow-to-exp N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, e^{\log \left(F \cdot \sin B\right) \cdot -1}\right) \]

      4. lower-unsound-exp.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, e^{\log \left(F \cdot \sin B\right) \cdot -1}\right) \]

      5. lower-unsound-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, e^{\log \left(F \cdot \sin B\right) \cdot -1}\right) \]

      6. lower-unsound-log.f64 16.6

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, e^{\log \left(F \cdot \sin B\right) \cdot -1}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, e^{\log \left(F \cdot \sin B\right) \cdot -1}\right) \]

      8. *-commutative N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, e^{\log \left(\sin B \cdot F\right) \cdot -1}\right) \]

      9. lower-*.f64 16.6

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, e^{\log \left(\sin B \cdot F\right) \cdot -1}\right) \]

   9. Applied rewrites 16.6%

      \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, e^{\log \left(\sin B \cdot F\right) \cdot -1}\right) \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto F \cdot \left(-1 \cdot \frac{x}{B \cdot F} + \color{blue}{e^{\log \left(\sin B \cdot F\right) \cdot -1}}\right) \]

       2. +-commutative N/A

          \[\leadsto F \cdot \left(e^{\log \left(\sin B \cdot F\right) \cdot -1} + \color{blue}{-1 \cdot \frac{x}{B \cdot F}}\right) \]

       3. lift-exp.f64 N/A

          \[\leadsto F \cdot \left(e^{\log \left(\sin B \cdot F\right) \cdot -1} + \color{blue}{-1} \cdot \frac{x}{B \cdot F}\right) \]

       4. exp-fabs N/A

          \[\leadsto F \cdot \left(\left|e^{\log \left(\sin B \cdot F\right) \cdot -1}\right| + \color{blue}{-1} \cdot \frac{x}{B \cdot F}\right) \]

       5. lift-*.f64 N/A

          \[\leadsto F \cdot \left(\left|e^{\log \left(\sin B \cdot F\right) \cdot -1}\right| + -1 \cdot \frac{x}{B \cdot F}\right) \]

       6. lift-log.f64 N/A

          \[\leadsto F \cdot \left(\left|e^{\log \left(\sin B \cdot F\right) \cdot -1}\right| + -1 \cdot \frac{x}{B \cdot F}\right) \]

       7. exp-to-pow N/A

          \[\leadsto F \cdot \left(\left|{\left(\sin B \cdot F\right)}^{-1}\right| + -1 \cdot \frac{x}{B \cdot F}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto F \cdot \left(\left|{\left(\sin B \cdot F\right)}^{-1}\right| + -1 \cdot \frac{x}{B \cdot F}\right) \]

       9. *-commutative N/A

          \[\leadsto F \cdot \left(\left|{\left(F \cdot \sin B\right)}^{-1}\right| + -1 \cdot \frac{x}{B \cdot F}\right) \]

       10. pow-prod-down N/A

           \[\leadsto F \cdot \left(\left|{F}^{-1} \cdot {\sin B}^{-1}\right| + -1 \cdot \frac{x}{B \cdot F}\right) \]

       11. inv-pow N/A

           \[\leadsto F \cdot \left(\left|\frac{1}{F} \cdot {\sin B}^{-1}\right| + -1 \cdot \frac{x}{B \cdot F}\right) \]

       12. lift-/.f64 N/A

           \[\leadsto F \cdot \left(\left|\frac{1}{F} \cdot {\sin B}^{-1}\right| + -1 \cdot \frac{x}{B \cdot F}\right) \]

       13. inv-pow N/A

           \[\leadsto F \cdot \left(\left|\frac{1}{F} \cdot \frac{1}{\sin B}\right| + -1 \cdot \frac{x}{B \cdot F}\right) \]

       14. lift-/.f64 N/A

           \[\leadsto F \cdot \left(\left|\frac{1}{F} \cdot \frac{1}{\sin B}\right| + -1 \cdot \frac{x}{B \cdot F}\right) \]

       15. fabs-mul N/A

           \[\leadsto F \cdot \left(\left|\frac{1}{F}\right| \cdot \left|\frac{1}{\sin B}\right| + \color{blue}{-1} \cdot \frac{x}{B \cdot F}\right) \]

   11. Applied rewrites 34.1%

       \[\leadsto F \cdot \mathsf{fma}\left(\frac{1}{\left|F\right|}, \color{blue}{\left|\frac{1}{\sin B}\right|}, -\frac{x}{B \cdot F}\right) \]

   if 7.39999999999999957e-227 < B < 7.99999999999999999e-14

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 42.7%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   if 7.99999999999999999e-14 < B < 5.99999999999999981e35

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.f64 47.4

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto F \cdot \left(-1 \cdot \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin \color{blue}{B}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \]

      6. lower-sin.f64 46.1

         \[\leadsto F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \]

   7. Applied rewrites 46.1%

      \[\leadsto F \cdot \left(-1 \cdot \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \cdot \color{blue}{F} \]

      3. lower-*.f64 46.1

         \[\leadsto \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \cdot \color{blue}{F} \]

   9. Applied rewrites 48.0%

      \[\leadsto \left(-\frac{\frac{x}{\tan B}}{F}\right) \cdot \color{blue}{F} \]

   if 5.99999999999999981e35 < B

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 77.7

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

   3. Applied rewrites 77.8%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in B around 0

      \[\leadsto {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{\color{blue}{B}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

   6. Applied rewrites 62.0%

      \[\leadsto {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\color{blue}{B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{F}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. lower-/.f64 47.0

         \[\leadsto \frac{1}{\color{blue}{F}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]

   9. Applied rewrites 47.0%

      \[\leadsto \color{blue}{\frac{1}{F}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 16: 54.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;B \leq 8 \cdot 10^{-14}:\\ \;\;\;\;\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{+35}:\\ \;\;\;\;\left(-\frac{t\_0}{F}\right) \cdot F\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{F} \cdot \frac{F}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= B 8e-14)
     (/ (- (* F (pow (+ 2.0 (fma 2.0 x (pow F 2.0))) -0.5)) x) B)
     (if (<= B 6e+35) (* (- (/ t_0 F)) F) (- (* (/ 1.0 F) (/ F B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (B <= 8e-14) {
		tmp = ((F * pow((2.0 + fma(2.0, x, pow(F, 2.0))), -0.5)) - x) / B;
	} else if (B <= 6e+35) {
		tmp = -(t_0 / F) * F;
	} else {
		tmp = ((1.0 / F) * (F / B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (B <= 8e-14)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + fma(2.0, x, (F ^ 2.0))) ^ -0.5)) - x) / B);
	elseif (B <= 6e+35)
		tmp = Float64(Float64(-Float64(t_0 / F)) * F);
	else
		tmp = Float64(Float64(Float64(1.0 / F) * Float64(F / 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[B, 8e-14], N[(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] / B), $MachinePrecision], If[LessEqual[B, 6e+35], N[((-N[(t$95$0 / F), $MachinePrecision]) * F), $MachinePrecision], N[(N[(N[(1.0 / F), $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 7.99999999999999999e-14

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 42.7%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   if 7.99999999999999999e-14 < B < 5.99999999999999981e35

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.f64 47.4

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto F \cdot \left(-1 \cdot \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin \color{blue}{B}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \]

      6. lower-sin.f64 46.1

         \[\leadsto F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \]

   7. Applied rewrites 46.1%

      \[\leadsto F \cdot \left(-1 \cdot \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \cdot \color{blue}{F} \]

      3. lower-*.f64 46.1

         \[\leadsto \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \cdot \color{blue}{F} \]

   9. Applied rewrites 48.0%

      \[\leadsto \left(-\frac{\frac{x}{\tan B}}{F}\right) \cdot \color{blue}{F} \]

   if 5.99999999999999981e35 < B

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 77.7

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

   3. Applied rewrites 77.8%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in B around 0

      \[\leadsto {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{\color{blue}{B}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

   6. Applied rewrites 62.0%

      \[\leadsto {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\color{blue}{B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{F}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]
   8. Step-by-step derivation

      1. lower-/.f64 47.0

         \[\leadsto \frac{1}{\color{blue}{F}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]

   9. Applied rewrites 47.0%

      \[\leadsto \color{blue}{\frac{1}{F}} \cdot \frac{F}{B} - \frac{x}{\tan B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 50.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 8 \cdot 10^{-14}:\\ \;\;\;\;\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\frac{x}{\tan B}}{F}\right) \cdot F\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 8e-14)
   (/ (- (* F (pow (+ 2.0 (fma 2.0 x (pow F 2.0))) -0.5)) x) B)
   (* (- (/ (/ x (tan B)) F)) F)))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 8e-14) {
		tmp = ((F * pow((2.0 + fma(2.0, x, pow(F, 2.0))), -0.5)) - x) / B;
	} else {
		tmp = -((x / tan(B)) / F) * F;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 8e-14)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + fma(2.0, x, (F ^ 2.0))) ^ -0.5)) - x) / B);
	else
		tmp = Float64(Float64(-Float64(Float64(x / tan(B)) / F)) * F);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 8e-14], N[(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] / B), $MachinePrecision], N[((-N[(N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]) * F), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 7.99999999999999999e-14

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 42.7%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   if 7.99999999999999999e-14 < B

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.f64 47.4

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto F \cdot \left(-1 \cdot \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin \color{blue}{B}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \]

      6. lower-sin.f64 46.1

         \[\leadsto F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \]

   7. Applied rewrites 46.1%

      \[\leadsto F \cdot \left(-1 \cdot \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \cdot \color{blue}{F} \]

      3. lower-*.f64 46.1

         \[\leadsto \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \cdot \color{blue}{F} \]

   9. Applied rewrites 48.0%

      \[\leadsto \left(-\frac{\frac{x}{\tan B}}{F}\right) \cdot \color{blue}{F} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 49.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 9.2 \cdot 10^{-220}:\\ \;\;\;\;\frac{-1 \cdot x}{B}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-14}:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\frac{x}{\tan B}}{F}\right) \cdot F\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 9.2e-220)
   (/ (* -1.0 x) B)
   (if (<= B 8e-14)
     (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F B)) (/ x B))
     (* (- (/ (/ x (tan B)) F)) F))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 9.2e-220) {
		tmp = (-1.0 * x) / B;
	} else if (B <= 8e-14) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = -((x / tan(B)) / F) * F;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 9.2e-220)
		tmp = Float64(Float64(-1.0 * x) / B);
	elseif (B <= 8e-14)
		tmp = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	else
		tmp = Float64(Float64(-Float64(Float64(x / tan(B)) / F)) * F);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 9.2e-220], N[(N[(-1.0 * x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[B, 8e-14], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[((-N[(N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]) * F), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 9.19999999999999922e-220

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.f64 47.4

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{\color{blue}{B}} \]
   6. 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-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      5. lower-/.f64 28.0

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   7. Applied rewrites 28.0%

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{\color{blue}{B}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{-1 \cdot x}{B} \]
   9. Step-by-step derivation

      1. lower-*.f64 28.8

         \[\leadsto \frac{-1 \cdot x}{B} \]

   10. Applied rewrites 28.8%

       \[\leadsto \frac{-1 \cdot x}{B} \]

   if 9.19999999999999922e-220 < B < 7.99999999999999999e-14

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 77.7

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

   3. Applied rewrites 77.8%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in B around 0

      \[\leadsto {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{\color{blue}{B}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

   6. Applied rewrites 62.0%

      \[\leadsto {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\color{blue}{B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0

      \[\leadsto {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{B} - \color{blue}{\frac{x}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 34.7

         \[\leadsto {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\color{blue}{B}} \]

   9. Applied rewrites 34.7%

      \[\leadsto {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \color{blue}{\frac{x}{B}} \]

   if 7.99999999999999999e-14 < B

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.f64 47.4

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto F \cdot \left(-1 \cdot \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin \color{blue}{B}}\right) \]

      4. lower-cos.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \]

      6. lower-sin.f64 46.1

         \[\leadsto F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \]

   7. Applied rewrites 46.1%

      \[\leadsto F \cdot \left(-1 \cdot \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \cdot \color{blue}{F} \]

      3. lower-*.f64 46.1

         \[\leadsto \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \cdot \color{blue}{F} \]

   9. Applied rewrites 48.0%

      \[\leadsto \left(-\frac{\frac{x}{\tan B}}{F}\right) \cdot \color{blue}{F} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 46.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 4 \cdot 10^{+28}:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{+188}:\\ \;\;\;\;\frac{1 + \mathsf{fma}\left(-1, x, {B}^{2} \cdot \left(0.16666666666666666 + -1 \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)\right)\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8.5e+43)
   (/ -1.0 (sin B))
   (if (<= F 4e+28)
     (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F B)) (/ x B))
     (if (<= F 8e+188)
       (/
        (+
         1.0
         (fma
          -1.0
          x
          (*
           (pow B 2.0)
           (+
            0.16666666666666666
            (* -1.0 (- (* -0.5 x) (* -0.16666666666666666 x)))))))
        B)
       (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.5e+43) {
		tmp = -1.0 / sin(B);
	} else if (F <= 4e+28) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 8e+188) {
		tmp = (1.0 + fma(-1.0, x, (pow(B, 2.0) * (0.16666666666666666 + (-1.0 * ((-0.5 * x) - (-0.16666666666666666 * x))))))) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8.5e+43)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 4e+28)
		tmp = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	elseif (F <= 8e+188)
		tmp = Float64(Float64(1.0 + fma(-1.0, x, Float64((B ^ 2.0) * Float64(0.16666666666666666 + Float64(-1.0 * Float64(Float64(-0.5 * x) - Float64(-0.16666666666666666 * x))))))) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8.5e+43], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4e+28], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8e+188], N[(N[(1.0 + N[(-1.0 * x + N[(N[Power[B, 2.0], $MachinePrecision] * N[(0.16666666666666666 + N[(-1.0 * N[(N[(-0.5 * x), $MachinePrecision] - N[(-0.16666666666666666 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -8.5e43

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.2

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -8.5e43 < F < 3.99999999999999983e28

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 77.7

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

   3. Applied rewrites 77.8%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in B around 0

      \[\leadsto {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{\color{blue}{B}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

   6. Applied rewrites 62.0%

      \[\leadsto {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\color{blue}{B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0

      \[\leadsto {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{B} - \color{blue}{\frac{x}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 34.7

         \[\leadsto {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\color{blue}{B}} \]

   9. Applied rewrites 34.7%

      \[\leadsto {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \color{blue}{\frac{x}{B}} \]

   if 3.99999999999999983e28 < F < 8.0000000000000002e188

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.f64 47.4

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{\color{blue}{B \cdot F}}, \frac{1}{F \cdot \sin B}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot \color{blue}{F}}, \frac{1}{F \cdot \sin B}\right) \]

      2. lower-*.f64 31.3

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{F \cdot \sin B}\right) \]

   7. Applied rewrites 31.3%

      \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{\color{blue}{B \cdot F}}, \frac{1}{F \cdot \sin B}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{F \cdot \sin B}\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \frac{x}{B \cdot F} + \color{blue}{\frac{1}{F \cdot \sin B}}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(-1 \cdot \frac{x}{B \cdot F}\right) \cdot F + \color{blue}{\frac{1}{F \cdot \sin B} \cdot F} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-1 \cdot \frac{x}{B \cdot F}\right) \cdot F + \frac{1}{F \cdot \sin B} \cdot F \]

      5. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot \frac{x}{B \cdot F}\right) \cdot F + \frac{1}{F \cdot \sin B} \cdot F \]

      6. associate-/r* N/A

         \[\leadsto \left(-1 \cdot \frac{x}{B \cdot F}\right) \cdot F + \frac{\frac{1}{F}}{\sin B} \cdot F \]

      7. associate-*l/ N/A

         \[\leadsto \left(-1 \cdot \frac{x}{B \cdot F}\right) \cdot F + \frac{\frac{1}{F} \cdot F}{\color{blue}{\sin B}} \]

      8. inv-pow N/A

         \[\leadsto \left(-1 \cdot \frac{x}{B \cdot F}\right) \cdot F + \frac{{F}^{-1} \cdot F}{\sin B} \]

      9. pow-plus N/A

         \[\leadsto \left(-1 \cdot \frac{x}{B \cdot F}\right) \cdot F + \frac{{F}^{\left(-1 + 1\right)}}{\sin \color{blue}{B}} \]

      10. metadata-eval N/A

          \[\leadsto \left(-1 \cdot \frac{x}{B \cdot F}\right) \cdot F + \frac{{F}^{0}}{\sin B} \]

      11. metadata-eval N/A

          \[\leadsto \left(-1 \cdot \frac{x}{B \cdot F}\right) \cdot F + \frac{1}{\sin \color{blue}{B}} \]

      12. lift-/.f64 N/A

          \[\leadsto \left(-1 \cdot \frac{x}{B \cdot F}\right) \cdot F + \frac{1}{\color{blue}{\sin B}} \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{x}{B \cdot F}, \color{blue}{F}, \frac{1}{\sin B}\right) \]

   9. Applied rewrites 34.0%

      \[\leadsto \mathsf{fma}\left(-\frac{x}{B \cdot F}, \color{blue}{F}, \frac{1}{\sin B}\right) \]

   10. Taylor expanded in B around 0

       \[\leadsto \frac{1 + \left(-1 \cdot x + {B}^{2} \cdot \left(\frac{1}{6} + -1 \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right)\right)\right)}{\color{blue}{B}} \]
   11. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{1 + \left(-1 \cdot x + {B}^{2} \cdot \left(\frac{1}{6} + -1 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x - \frac{-1}{6} \cdot x\right)\right)\right)}{B} \]

       2. metadata-eval N/A

          \[\leadsto \frac{1 + \left(-1 \cdot x + {B}^{2} \cdot \left(\frac{1}{6} + -1 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x - \frac{-1}{6} \cdot x\right)\right)\right)}{B} \]

       3. lower-/.f64 N/A

          \[\leadsto \frac{1 + \left(-1 \cdot x + {B}^{2} \cdot \left(\frac{1}{6} + -1 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x - \frac{-1}{6} \cdot x\right)\right)\right)}{B} \]

   12. Applied rewrites 28.7%

       \[\leadsto \frac{1 + \mathsf{fma}\left(-1, x, {B}^{2} \cdot \left(0.16666666666666666 + -1 \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)\right)\right)}{\color{blue}{B}} \]

   if 8.0000000000000002e188 < F

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 16.8

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 16.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 20: 43.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{+88}:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{+188}:\\ \;\;\;\;\mathsf{fma}\left(-\frac{x}{B \cdot F}, F, \frac{1 + 0.16666666666666666 \cdot {B}^{2}}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8.5e+43)
   (/ -1.0 (sin B))
   (if (<= F 8.5e+88)
     (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F B)) (/ x B))
     (if (<= F 8e+188)
       (fma
        (- (/ x (* B F)))
        F
        (/ (+ 1.0 (* 0.16666666666666666 (pow B 2.0))) B))
       (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.5e+43) {
		tmp = -1.0 / sin(B);
	} else if (F <= 8.5e+88) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 8e+188) {
		tmp = fma(-(x / (B * F)), F, ((1.0 + (0.16666666666666666 * pow(B, 2.0))) / B));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8.5e+43)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 8.5e+88)
		tmp = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	elseif (F <= 8e+188)
		tmp = fma(Float64(-Float64(x / Float64(B * F))), F, Float64(Float64(1.0 + Float64(0.16666666666666666 * (B ^ 2.0))) / B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8.5e+43], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8.5e+88], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8e+188], N[((-N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision]) * F + N[(N[(1.0 + N[(0.16666666666666666 * N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -8.5e43

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.2

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -8.5e43 < F < 8.5000000000000005e88

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 77.7

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

   3. Applied rewrites 77.8%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}} \]

   4. Taylor expanded in B around 0

      \[\leadsto {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{\color{blue}{B}} - \frac{x}{\tan B} \]
   5. Step-by-step derivation

   6. Applied rewrites 62.0%

      \[\leadsto {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\color{blue}{B}} - \frac{x}{\tan B} \]

   7. Taylor expanded in B around 0

      \[\leadsto {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{B} - \color{blue}{\frac{x}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 34.7

         \[\leadsto {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\color{blue}{B}} \]

   9. Applied rewrites 34.7%

      \[\leadsto {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \color{blue}{\frac{x}{B}} \]

   if 8.5000000000000005e88 < F < 8.0000000000000002e188

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.f64 47.4

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{\color{blue}{B \cdot F}}, \frac{1}{F \cdot \sin B}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot \color{blue}{F}}, \frac{1}{F \cdot \sin B}\right) \]

      2. lower-*.f64 31.3

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{F \cdot \sin B}\right) \]

   7. Applied rewrites 31.3%

      \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{\color{blue}{B \cdot F}}, \frac{1}{F \cdot \sin B}\right) \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{F \cdot \sin B}\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \frac{x}{B \cdot F} + \color{blue}{\frac{1}{F \cdot \sin B}}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(-1 \cdot \frac{x}{B \cdot F}\right) \cdot F + \color{blue}{\frac{1}{F \cdot \sin B} \cdot F} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(-1 \cdot \frac{x}{B \cdot F}\right) \cdot F + \frac{1}{F \cdot \sin B} \cdot F \]

      5. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot \frac{x}{B \cdot F}\right) \cdot F + \frac{1}{F \cdot \sin B} \cdot F \]

      6. associate-/r* N/A

         \[\leadsto \left(-1 \cdot \frac{x}{B \cdot F}\right) \cdot F + \frac{\frac{1}{F}}{\sin B} \cdot F \]

      7. associate-*l/ N/A

         \[\leadsto \left(-1 \cdot \frac{x}{B \cdot F}\right) \cdot F + \frac{\frac{1}{F} \cdot F}{\color{blue}{\sin B}} \]

      8. inv-pow N/A

         \[\leadsto \left(-1 \cdot \frac{x}{B \cdot F}\right) \cdot F + \frac{{F}^{-1} \cdot F}{\sin B} \]

      9. pow-plus N/A

         \[\leadsto \left(-1 \cdot \frac{x}{B \cdot F}\right) \cdot F + \frac{{F}^{\left(-1 + 1\right)}}{\sin \color{blue}{B}} \]

      10. metadata-eval N/A

          \[\leadsto \left(-1 \cdot \frac{x}{B \cdot F}\right) \cdot F + \frac{{F}^{0}}{\sin B} \]

      11. metadata-eval N/A

          \[\leadsto \left(-1 \cdot \frac{x}{B \cdot F}\right) \cdot F + \frac{1}{\sin \color{blue}{B}} \]

      12. lift-/.f64 N/A

          \[\leadsto \left(-1 \cdot \frac{x}{B \cdot F}\right) \cdot F + \frac{1}{\color{blue}{\sin B}} \]

      13. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{x}{B \cdot F}, \color{blue}{F}, \frac{1}{\sin B}\right) \]

   9. Applied rewrites 34.0%

      \[\leadsto \mathsf{fma}\left(-\frac{x}{B \cdot F}, \color{blue}{F}, \frac{1}{\sin B}\right) \]

   10. Taylor expanded in B around 0

       \[\leadsto \mathsf{fma}\left(-\frac{x}{B \cdot F}, F, \frac{1 + \frac{1}{6} \cdot {B}^{2}}{B}\right) \]
   11. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(-\frac{x}{B \cdot F}, F, \frac{1 + \frac{1}{6} \cdot {B}^{2}}{B}\right) \]

       2. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(-\frac{x}{B \cdot F}, F, \frac{1 + \frac{1}{6} \cdot {B}^{2}}{B}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-\frac{x}{B \cdot F}, F, \frac{1 + \frac{1}{6} \cdot {B}^{2}}{B}\right) \]

       4. lower-pow.f64 26.6

          \[\leadsto \mathsf{fma}\left(-\frac{x}{B \cdot F}, F, \frac{1 + 0.16666666666666666 \cdot {B}^{2}}{B}\right) \]

   12. Applied rewrites 26.6%

       \[\leadsto \mathsf{fma}\left(-\frac{x}{B \cdot F}, F, \frac{1 + 0.16666666666666666 \cdot {B}^{2}}{B}\right) \]

   if 8.0000000000000002e188 < F

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 16.8

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 16.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 21: 42.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{+30}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-40}:\\ \;\;\;\;\frac{-1 \cdot x}{B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{+188}:\\ \;\;\;\;\frac{-1 \cdot \left(x - 1\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.3e+30)
   (/ -1.0 (sin B))
   (if (<= F 8e-40)
     (/ (* -1.0 x) B)
     (if (<= F 8e+188) (/ (* -1.0 (- x 1.0)) B) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.3e+30) {
		tmp = -1.0 / sin(B);
	} else if (F <= 8e-40) {
		tmp = (-1.0 * x) / B;
	} else if (F <= 8e+188) {
		tmp = (-1.0 * (x - 1.0)) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.3d+30)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 8d-40) then
        tmp = ((-1.0d0) * x) / b
    else if (f <= 8d+188) then
        tmp = ((-1.0d0) * (x - 1.0d0)) / b
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.3e+30) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 8e-40) {
		tmp = (-1.0 * x) / B;
	} else if (F <= 8e+188) {
		tmp = (-1.0 * (x - 1.0)) / B;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.3e+30:
		tmp = -1.0 / math.sin(B)
	elif F <= 8e-40:
		tmp = (-1.0 * x) / B
	elif F <= 8e+188:
		tmp = (-1.0 * (x - 1.0)) / B
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.3e+30)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 8e-40)
		tmp = Float64(Float64(-1.0 * x) / B);
	elseif (F <= 8e+188)
		tmp = Float64(Float64(-1.0 * Float64(x - 1.0)) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.3e+30)
		tmp = -1.0 / sin(B);
	elseif (F <= 8e-40)
		tmp = (-1.0 * x) / B;
	elseif (F <= 8e+188)
		tmp = (-1.0 * (x - 1.0)) / B;
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.3e+30], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8e-40], N[(N[(-1.0 * x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 8e+188], N[(N[(-1.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.3e30

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.2

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -2.3e30 < F < 7.9999999999999994e-40

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.f64 47.4

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{\color{blue}{B}} \]
   6. 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-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      5. lower-/.f64 28.0

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   7. Applied rewrites 28.0%

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{\color{blue}{B}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{-1 \cdot x}{B} \]
   9. Step-by-step derivation

      1. lower-*.f64 28.8

         \[\leadsto \frac{-1 \cdot x}{B} \]

   10. Applied rewrites 28.8%

       \[\leadsto \frac{-1 \cdot x}{B} \]

   if 7.9999999999999994e-40 < F < 8.0000000000000002e188

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.f64 47.4

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{\color{blue}{B}} \]
   6. 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-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      5. lower-/.f64 28.0

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   7. Applied rewrites 28.0%

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{\color{blue}{B}} \]

   8. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]

      2. lower--.f64 29.0

         \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]

   10. Applied rewrites 29.0%

       \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]

   if 8.0000000000000002e188 < F

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 16.8

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 16.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 22: 42.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{+30}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-40}:\\ \;\;\;\;\frac{-1 \cdot x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \left(x - 1\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.3e+30)
   (/ -1.0 (sin B))
   (if (<= F 8e-40) (/ (* -1.0 x) B) (/ (* -1.0 (- x 1.0)) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.3e+30) {
		tmp = -1.0 / sin(B);
	} else if (F <= 8e-40) {
		tmp = (-1.0 * x) / B;
	} else {
		tmp = (-1.0 * (x - 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 <= (-2.3d+30)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 8d-40) then
        tmp = ((-1.0d0) * x) / b
    else
        tmp = ((-1.0d0) * (x - 1.0d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.3e+30) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 8e-40) {
		tmp = (-1.0 * x) / B;
	} else {
		tmp = (-1.0 * (x - 1.0)) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.3e+30:
		tmp = -1.0 / math.sin(B)
	elif F <= 8e-40:
		tmp = (-1.0 * x) / B
	else:
		tmp = (-1.0 * (x - 1.0)) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.3e+30)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 8e-40)
		tmp = Float64(Float64(-1.0 * x) / B);
	else
		tmp = Float64(Float64(-1.0 * Float64(x - 1.0)) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.3e+30)
		tmp = -1.0 / sin(B);
	elseif (F <= 8e-40)
		tmp = (-1.0 * x) / B;
	else
		tmp = (-1.0 * (x - 1.0)) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.3e+30], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8e-40], N[(N[(-1.0 * x), $MachinePrecision] / B), $MachinePrecision], N[(N[(-1.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.3e30

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.2

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -2.3e30 < F < 7.9999999999999994e-40

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.f64 47.4

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{\color{blue}{B}} \]
   6. 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-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      5. lower-/.f64 28.0

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   7. Applied rewrites 28.0%

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{\color{blue}{B}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{-1 \cdot x}{B} \]
   9. Step-by-step derivation

      1. lower-*.f64 28.8

         \[\leadsto \frac{-1 \cdot x}{B} \]

   10. Applied rewrites 28.8%

       \[\leadsto \frac{-1 \cdot x}{B} \]

   if 7.9999999999999994e-40 < F

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.f64 47.4

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{\color{blue}{B}} \]
   6. 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-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      5. lower-/.f64 28.0

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   7. Applied rewrites 28.0%

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{\color{blue}{B}} \]

   8. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]

      2. lower--.f64 29.0

         \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]

   10. Applied rewrites 29.0%

       \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 36.2% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.6 \cdot 10^{+34}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-40}:\\ \;\;\;\;\frac{-1 \cdot x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \left(x - 1\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.6e+34)
   (/ -1.0 B)
   (if (<= F 8e-40) (/ (* -1.0 x) B) (/ (* -1.0 (- x 1.0)) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.6e+34) {
		tmp = -1.0 / B;
	} else if (F <= 8e-40) {
		tmp = (-1.0 * x) / B;
	} else {
		tmp = (-1.0 * (x - 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 <= (-5.6d+34)) then
        tmp = (-1.0d0) / b
    else if (f <= 8d-40) then
        tmp = ((-1.0d0) * x) / b
    else
        tmp = ((-1.0d0) * (x - 1.0d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.6e+34) {
		tmp = -1.0 / B;
	} else if (F <= 8e-40) {
		tmp = (-1.0 * x) / B;
	} else {
		tmp = (-1.0 * (x - 1.0)) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5.6e+34:
		tmp = -1.0 / B
	elif F <= 8e-40:
		tmp = (-1.0 * x) / B
	else:
		tmp = (-1.0 * (x - 1.0)) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.6e+34)
		tmp = Float64(-1.0 / B);
	elseif (F <= 8e-40)
		tmp = Float64(Float64(-1.0 * x) / B);
	else
		tmp = Float64(Float64(-1.0 * Float64(x - 1.0)) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5.6e+34)
		tmp = -1.0 / B;
	elseif (F <= 8e-40)
		tmp = (-1.0 * x) / B;
	else
		tmp = (-1.0 * (x - 1.0)) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.6e+34], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 8e-40], N[(N[(-1.0 * x), $MachinePrecision] / B), $MachinePrecision], N[(N[(-1.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.60000000000000016e34

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.2

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-1}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 10.1%

      \[\leadsto \frac{-1}{B} \]

   if -5.60000000000000016e34 < F < 7.9999999999999994e-40

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.f64 47.4

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{\color{blue}{B}} \]
   6. 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-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      5. lower-/.f64 28.0

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   7. Applied rewrites 28.0%

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{\color{blue}{B}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{-1 \cdot x}{B} \]
   9. Step-by-step derivation

      1. lower-*.f64 28.8

         \[\leadsto \frac{-1 \cdot x}{B} \]

   10. Applied rewrites 28.8%

       \[\leadsto \frac{-1 \cdot x}{B} \]

   if 7.9999999999999994e-40 < F

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.f64 47.4

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{\color{blue}{B}} \]
   6. 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-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      5. lower-/.f64 28.0

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   7. Applied rewrites 28.0%

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{\color{blue}{B}} \]

   8. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]

      2. lower--.f64 29.0

         \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]

   10. Applied rewrites 29.0%

       \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 30.2% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1 \cdot x}{B}\\ \mathbf{if}\;x \leq -6 \cdot 10^{-154}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.6 \cdot 10^{-101}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (* -1.0 x) B)))
   (if (<= x -6e-154) t_0 (if (<= x 5.6e-101) (/ -1.0 B) t_0))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 * x) / B;
	double tmp;
	if (x <= -6e-154) {
		tmp = t_0;
	} else if (x <= 5.6e-101) {
		tmp = -1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) * x) / b
    if (x <= (-6d-154)) then
        tmp = t_0
    else if (x <= 5.6d-101) then
        tmp = (-1.0d0) / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 * x) / B;
	double tmp;
	if (x <= -6e-154) {
		tmp = t_0;
	} else if (x <= 5.6e-101) {
		tmp = -1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 * x) / B
	tmp = 0
	if x <= -6e-154:
		tmp = t_0
	elif x <= 5.6e-101:
		tmp = -1.0 / B
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 * x) / B)
	tmp = 0.0
	if (x <= -6e-154)
		tmp = t_0;
	elseif (x <= 5.6e-101)
		tmp = Float64(-1.0 / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 * x) / B;
	tmp = 0.0;
	if (x <= -6e-154)
		tmp = t_0;
	elseif (x <= 5.6e-101)
		tmp = -1.0 / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 * x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[x, -6e-154], t$95$0, If[LessEqual[x, 5.6e-101], N[(-1.0 / B), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.0000000000000005e-154 or 5.59999999999999978e-101 < x

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.f64 47.4

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.4%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{\color{blue}{B}} \]
   6. 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-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      5. lower-/.f64 28.0

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   7. Applied rewrites 28.0%

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{\color{blue}{B}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{-1 \cdot x}{B} \]
   9. Step-by-step derivation

      1. lower-*.f64 28.8

         \[\leadsto \frac{-1 \cdot x}{B} \]

   10. Applied rewrites 28.8%

       \[\leadsto \frac{-1 \cdot x}{B} \]

   if -6.0000000000000005e-154 < x < 5.59999999999999978e-101

   1. Initial program 77.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.2

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-1}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 10.1%

      \[\leadsto \frac{-1}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 10.1% accurate, 26.5× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 77.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in F around -inf

   \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

   2. lower-sin.f64 17.2

      \[\leadsto \frac{-1}{\sin B} \]

4. Applied rewrites 17.2%

   \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

5. Taylor expanded in B around 0

   \[\leadsto \frac{-1}{B} \]
6. Step-by-step derivation

7. Applied rewrites 10.1%

   \[\leadsto \frac{-1}{B} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025157 
(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))))))