VandenBroeck and Keller, Equation (23)

Percentage Accurate: 75.7% → 99.7%

Time: 7.5s

Alternatives: 29

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -4 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, t\_1\right)\\ \mathbf{elif}\;F \leq 6800000:\\ \;\;\;\;\mathsf{fma}\left(\frac{F \cdot \frac{1}{F}}{\sin B}, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 1, t\_1\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ (- x) (tan B))))
   (if (<= F -4e+67)
     (fma t_0 -1.0 t_1)
     (if (<= F 6800000.0)
       (fma
        (/ (* F (/ 1.0 F)) (sin B))
        (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F)
        t_1)
       (fma t_0 1.0 t_1)))))

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 <= -4e+67) {
		tmp = fma(t_0, -1.0, t_1);
	} else if (F <= 6800000.0) {
		tmp = fma(((F * (1.0 / F)) / sin(B)), (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F), t_1);
	} else {
		tmp = fma(t_0, 1.0, t_1);
	}
	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 <= -4e+67)
		tmp = fma(t_0, -1.0, t_1);
	elseif (F <= 6800000.0)
		tmp = fma(Float64(Float64(F * Float64(1.0 / F)) / sin(B)), Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F), t_1);
	else
		tmp = fma(t_0, 1.0, t_1);
	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, -4e+67], N[(t$95$0 * -1.0 + t$95$1), $MachinePrecision], If[LessEqual[F, 6800000.0], N[(N[(N[(F * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] + t$95$1), $MachinePrecision], N[(t$95$0 * 1.0 + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.99999999999999993e67

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

   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 55.7%

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

   if -3.99999999999999993e67 < F < 6.8e6

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{F}^{0}}}{\sin B}, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F, \frac{-x}{\tan B}\right) \]

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. inv-pow N/A

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

      6. associate-*l/ N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 84.7%

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

   5. Applied rewrites 84.7%

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

   if 6.8e6 < F

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 56.6%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -4 \cdot 10^{+67}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, t\_1\right)\\ \mathbf{elif}\;F \leq 6800000:\\ \;\;\;\;\mathsf{fma}\left(t\_0, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 1, t\_1\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ (- x) (tan B))))
   (if (<= F -4e+67)
     (fma t_0 -1.0 t_1)
     (if (<= F 6800000.0)
       (fma t_0 (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F) t_1)
       (fma t_0 1.0 t_1)))))

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 <= -4e+67) {
		tmp = fma(t_0, -1.0, t_1);
	} else if (F <= 6800000.0) {
		tmp = fma(t_0, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F), t_1);
	} else {
		tmp = fma(t_0, 1.0, t_1);
	}
	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 <= -4e+67)
		tmp = fma(t_0, -1.0, t_1);
	elseif (F <= 6800000.0)
		tmp = fma(t_0, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F), t_1);
	else
		tmp = fma(t_0, 1.0, t_1);
	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, -4e+67], N[(t$95$0 * -1.0 + t$95$1), $MachinePrecision], If[LessEqual[F, 6800000.0], 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[(t$95$0 * 1.0 + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.99999999999999993e67

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

   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 55.7%

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

   if -3.99999999999999993e67 < F < 6.8e6

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

   if 6.8e6 < F

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 56.6%

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

Alternative 3: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -1.25 \cdot 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, t\_1\right)\\ \mathbf{elif}\;F \leq 6800000:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 1, t\_1\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ (- x) (tan B))))
   (if (<= F -1.25e+93)
     (fma t_0 -1.0 t_1)
     (if (<= F 6800000.0)
       (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B)) t_1)
       (fma t_0 1.0 t_1)))))

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 <= -1.25e+93) {
		tmp = fma(t_0, -1.0, t_1);
	} else if (F <= 6800000.0) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), t_1);
	} else {
		tmp = fma(t_0, 1.0, t_1);
	}
	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 <= -1.25e+93)
		tmp = fma(t_0, -1.0, t_1);
	elseif (F <= 6800000.0)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), t_1);
	else
		tmp = fma(t_0, 1.0, t_1);
	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, -1.25e+93], N[(t$95$0 * -1.0 + t$95$1), $MachinePrecision], If[LessEqual[F, 6800000.0], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(t$95$0 * 1.0 + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.25e93

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

   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 55.7%

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

   if -1.25e93 < F < 6.8e6

   1. Initial program 75.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 84.7%

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

   if 6.8e6 < F

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 56.6%

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

Alternative 4: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, t\_1\right)\\ \mathbf{elif}\;F \leq 6800000:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 1, t\_1\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ (- x) (tan B))))
   (if (<= F -1e+15)
     (fma t_0 -1.0 t_1)
     (if (<= F 6800000.0)
       (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F (sin B))) (/ x (tan B)))
       (fma t_0 1.0 t_1)))))

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+15) {
		tmp = fma(t_0, -1.0, t_1);
	} else if (F <= 6800000.0) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / sin(B))) - (x / tan(B));
	} else {
		tmp = fma(t_0, 1.0, t_1);
	}
	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+15)
		tmp = fma(t_0, -1.0, t_1);
	elseif (F <= 6800000.0)
		tmp = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / sin(B))) - Float64(x / tan(B)));
	else
		tmp = fma(t_0, 1.0, t_1);
	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+15], N[(t$95$0 * -1.0 + t$95$1), $MachinePrecision], If[LessEqual[F, 6800000.0], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 1.0 + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e15

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

   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 55.7%

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

   if -1e15 < F < 6.8e6

   1. Initial program 75.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 75.7%

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

   3. Applied rewrites 75.8%

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

   if 6.8e6 < F

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 56.6%

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

Alternative 5: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, t\_1\right)\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, {\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5} \cdot F, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 1, t\_1\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ (- x) (tan B))))
   (if (<= F -0.5)
     (fma t_0 -1.0 t_1)
     (if (<= F 6.5e-5)
       (fma t_0 (* (pow (fma 2.0 x 2.0) -0.5) F) t_1)
       (fma t_0 1.0 t_1)))))

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 <= -0.5) {
		tmp = fma(t_0, -1.0, t_1);
	} else if (F <= 6.5e-5) {
		tmp = fma(t_0, (pow(fma(2.0, x, 2.0), -0.5) * F), t_1);
	} else {
		tmp = fma(t_0, 1.0, t_1);
	}
	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 <= -0.5)
		tmp = fma(t_0, -1.0, t_1);
	elseif (F <= 6.5e-5)
		tmp = fma(t_0, Float64((fma(2.0, x, 2.0) ^ -0.5) * F), t_1);
	else
		tmp = fma(t_0, 1.0, t_1);
	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, -0.5], N[(t$95$0 * -1.0 + t$95$1), $MachinePrecision], If[LessEqual[F, 6.5e-5], N[(t$95$0 * N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] + t$95$1), $MachinePrecision], N[(t$95$0 * 1.0 + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.5

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

   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 55.7%

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

   if -0.5 < F < 6.49999999999999943e-5

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

   4. Taylor expanded in F around 0

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

   6. Applied rewrites 55.6%

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

   if 6.49999999999999943e-5 < F

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 56.6%

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

Alternative 6: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, t\_1\right)\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5}}{\sin B}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 1, t\_1\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ (- x) (tan B))))
   (if (<= F -0.5)
     (fma t_0 -1.0 t_1)
     (if (<= F 6.5e-5)
       (fma F (/ (pow (fma x 2.0 2.0) -0.5) (sin B)) t_1)
       (fma t_0 1.0 t_1)))))

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 <= -0.5) {
		tmp = fma(t_0, -1.0, t_1);
	} else if (F <= 6.5e-5) {
		tmp = fma(F, (pow(fma(x, 2.0, 2.0), -0.5) / sin(B)), t_1);
	} else {
		tmp = fma(t_0, 1.0, t_1);
	}
	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 <= -0.5)
		tmp = fma(t_0, -1.0, t_1);
	elseif (F <= 6.5e-5)
		tmp = fma(F, Float64((fma(x, 2.0, 2.0) ^ -0.5) / sin(B)), t_1);
	else
		tmp = fma(t_0, 1.0, t_1);
	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, -0.5], N[(t$95$0 * -1.0 + t$95$1), $MachinePrecision], If[LessEqual[F, 6.5e-5], N[(F * N[(N[Power[N[(x * 2.0 + 2.0), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(t$95$0 * 1.0 + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.5

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

   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 55.7%

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

   if -0.5 < F < 6.49999999999999943e-5

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 55.6%

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

   if 6.49999999999999943e-5 < F

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 56.6%

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

Alternative 7: 91.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ t_1 := -1 \cdot \frac{x}{B}\\ t_2 := \frac{1}{\sin B}\\ t_3 := {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\\ \mathbf{if}\;F \leq -5.2 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, -1, t\_0\right)\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{-161}:\\ \;\;\;\;\mathsf{fma}\left(\frac{F \cdot \frac{1}{F}}{\sin B}, t\_3, t\_1\right)\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{-98}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6800000:\\ \;\;\;\;\mathsf{fma}\left(\frac{F \cdot \left(\frac{1}{F \cdot F} \cdot \frac{1}{\frac{1}{F}}\right)}{\sin B}, t\_3, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, 1, t\_0\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B)))
        (t_1 (* -1.0 (/ x B)))
        (t_2 (/ 1.0 (sin B)))
        (t_3 (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F)))
   (if (<= F -5.2e+66)
     (fma t_2 -1.0 t_0)
     (if (<= F -1.3e-161)
       (fma (/ (* F (/ 1.0 F)) (sin B)) t_3 t_1)
       (if (<= F 3.8e-98)
         (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) (/ F B)) (/ x (tan B)))
         (if (<= F 6800000.0)
           (fma
            (/ (* F (* (/ 1.0 (* F F)) (/ 1.0 (/ 1.0 F)))) (sin B))
            t_3
            t_1)
           (fma t_2 1.0 t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double t_1 = -1.0 * (x / B);
	double t_2 = 1.0 / sin(B);
	double t_3 = pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F;
	double tmp;
	if (F <= -5.2e+66) {
		tmp = fma(t_2, -1.0, t_0);
	} else if (F <= -1.3e-161) {
		tmp = fma(((F * (1.0 / F)) / sin(B)), t_3, t_1);
	} else if (F <= 3.8e-98) {
		tmp = (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * (F / B)) - (x / tan(B));
	} else if (F <= 6800000.0) {
		tmp = fma(((F * ((1.0 / (F * F)) * (1.0 / (1.0 / F)))) / sin(B)), t_3, t_1);
	} else {
		tmp = fma(t_2, 1.0, t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	t_1 = Float64(-1.0 * Float64(x / B))
	t_2 = Float64(1.0 / sin(B))
	t_3 = Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F)
	tmp = 0.0
	if (F <= -5.2e+66)
		tmp = fma(t_2, -1.0, t_0);
	elseif (F <= -1.3e-161)
		tmp = fma(Float64(Float64(F * Float64(1.0 / F)) / sin(B)), t_3, t_1);
	elseif (F <= 3.8e-98)
		tmp = Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / tan(B)));
	elseif (F <= 6800000.0)
		tmp = fma(Float64(Float64(F * Float64(Float64(1.0 / Float64(F * F)) * Float64(1.0 / Float64(1.0 / F)))) / sin(B)), t_3, t_1);
	else
		tmp = fma(t_2, 1.0, t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision]}, If[LessEqual[F, -5.2e+66], N[(t$95$2 * -1.0 + t$95$0), $MachinePrecision], If[LessEqual[F, -1.3e-161], N[(N[(N[(F * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$3 + t$95$1), $MachinePrecision], If[LessEqual[F, 3.8e-98], N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6800000.0], N[(N[(N[(F * N[(N[(1.0 / N[(F * F), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$3 + t$95$1), $MachinePrecision], N[(t$95$2 * 1.0 + t$95$0), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -5.20000000000000024e66

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

   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 55.7%

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

   if -5.20000000000000024e66 < F < -1.29999999999999998e-161

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{F}^{0}}}{\sin B}, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F, \frac{-x}{\tan B}\right) \]

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. inv-pow N/A

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

      6. associate-*l/ N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 84.7%

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 57.3%

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

   8. Applied rewrites 57.3%

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

   if -1.29999999999999998e-161 < F < 3.8000000000000003e-98

   1. Initial program 75.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 61.4%

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

   4. Applied rewrites 61.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/r/ N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

   6. Applied rewrites 61.5%

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

   if 3.8000000000000003e-98 < F < 6.8e6

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{F}^{0}}}{\sin B}, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F, \frac{-x}{\tan B}\right) \]

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. inv-pow N/A

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

      6. associate-*l/ N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 84.7%

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 57.3%

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

   8. Applied rewrites 57.3%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

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

      3. *-rgt-identity N/A

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

      4. rgt-mult-inverse N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*r* N/A

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

      7. times-frac N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 45.6%

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

   10. Applied rewrites 45.6%

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

   if 6.8e6 < F

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 56.6%

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

Alternative 8: 91.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ t_1 := \frac{1}{\sin B}\\ t_2 := {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\\ \mathbf{if}\;F \leq -5.2 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, -1, t\_0\right)\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{-161}:\\ \;\;\;\;\mathsf{fma}\left(\frac{F \cdot \frac{1}{F}}{\sin B}, t\_2, -1 \cdot \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{-98}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6800000:\\ \;\;\;\;\frac{1}{\frac{\sin B}{t\_2}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, 1, t\_0\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B)))
        (t_1 (/ 1.0 (sin B)))
        (t_2 (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F)))
   (if (<= F -5.2e+66)
     (fma t_1 -1.0 t_0)
     (if (<= F -1.3e-161)
       (fma (/ (* F (/ 1.0 F)) (sin B)) t_2 (* -1.0 (/ x B)))
       (if (<= F 3.8e-98)
         (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) (/ F B)) (/ x (tan B)))
         (if (<= F 6800000.0)
           (- (/ 1.0 (/ (sin B) t_2)) (/ x B))
           (fma t_1 1.0 t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double t_1 = 1.0 / sin(B);
	double t_2 = pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F;
	double tmp;
	if (F <= -5.2e+66) {
		tmp = fma(t_1, -1.0, t_0);
	} else if (F <= -1.3e-161) {
		tmp = fma(((F * (1.0 / F)) / sin(B)), t_2, (-1.0 * (x / B)));
	} else if (F <= 3.8e-98) {
		tmp = (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * (F / B)) - (x / tan(B));
	} else if (F <= 6800000.0) {
		tmp = (1.0 / (sin(B) / t_2)) - (x / B);
	} else {
		tmp = fma(t_1, 1.0, t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	t_1 = Float64(1.0 / sin(B))
	t_2 = Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F)
	tmp = 0.0
	if (F <= -5.2e+66)
		tmp = fma(t_1, -1.0, t_0);
	elseif (F <= -1.3e-161)
		tmp = fma(Float64(Float64(F * Float64(1.0 / F)) / sin(B)), t_2, Float64(-1.0 * Float64(x / B)));
	elseif (F <= 3.8e-98)
		tmp = Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / tan(B)));
	elseif (F <= 6800000.0)
		tmp = Float64(Float64(1.0 / Float64(sin(B) / t_2)) - Float64(x / B));
	else
		tmp = fma(t_1, 1.0, t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision]}, If[LessEqual[F, -5.2e+66], N[(t$95$1 * -1.0 + t$95$0), $MachinePrecision], If[LessEqual[F, -1.3e-161], N[(N[(N[(F * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$2 + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.8e-98], N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6800000.0], N[(N[(1.0 / N[(N[Sin[B], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * 1.0 + t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -5.20000000000000024e66

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

   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 55.7%

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

   if -5.20000000000000024e66 < F < -1.29999999999999998e-161

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{F}^{0}}}{\sin B}, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F, \frac{-x}{\tan B}\right) \]

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. inv-pow N/A

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

      6. associate-*l/ N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 84.7%

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 57.3%

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

   8. Applied rewrites 57.3%

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

   if -1.29999999999999998e-161 < F < 3.8000000000000003e-98

   1. Initial program 75.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 61.4%

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

   4. Applied rewrites 61.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/r/ N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

   6. Applied rewrites 61.5%

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

   if 3.8000000000000003e-98 < F < 6.8e6

   1. Initial program 75.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 48.5%

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

   4. Applied rewrites 48.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 48.5%

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

   6. Applied rewrites 48.5%

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

      1. lift-*.f64 N/A

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

   8. Applied rewrites 57.3%

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

   if 6.8e6 < F

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 56.6%

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

Alternative 9: 91.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\\ t_1 := \frac{1}{\sin B}\\ t_2 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -5.2 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, -1, t\_2\right)\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{-161}:\\ \;\;\;\;t\_1 \cdot t\_0 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{-98}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6800000:\\ \;\;\;\;\frac{1}{\frac{\sin B}{t\_0}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, 1, t\_2\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F))
        (t_1 (/ 1.0 (sin B)))
        (t_2 (/ (- x) (tan B))))
   (if (<= F -5.2e+66)
     (fma t_1 -1.0 t_2)
     (if (<= F -1.3e-161)
       (- (* t_1 t_0) (/ x B))
       (if (<= F 3.8e-98)
         (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) (/ F B)) (/ x (tan B)))
         (if (<= F 6800000.0)
           (- (/ 1.0 (/ (sin B) t_0)) (/ x B))
           (fma t_1 1.0 t_2)))))))

C

double code(double F, double B, double x) {
	double t_0 = pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F;
	double t_1 = 1.0 / sin(B);
	double t_2 = -x / tan(B);
	double tmp;
	if (F <= -5.2e+66) {
		tmp = fma(t_1, -1.0, t_2);
	} else if (F <= -1.3e-161) {
		tmp = (t_1 * t_0) - (x / B);
	} else if (F <= 3.8e-98) {
		tmp = (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * (F / B)) - (x / tan(B));
	} else if (F <= 6800000.0) {
		tmp = (1.0 / (sin(B) / t_0)) - (x / B);
	} else {
		tmp = fma(t_1, 1.0, t_2);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F)
	t_1 = Float64(1.0 / sin(B))
	t_2 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -5.2e+66)
		tmp = fma(t_1, -1.0, t_2);
	elseif (F <= -1.3e-161)
		tmp = Float64(Float64(t_1 * t_0) - Float64(x / B));
	elseif (F <= 3.8e-98)
		tmp = Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / tan(B)));
	elseif (F <= 6800000.0)
		tmp = Float64(Float64(1.0 / Float64(sin(B) / t_0)) - Float64(x / B));
	else
		tmp = fma(t_1, 1.0, t_2);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5.2e+66], N[(t$95$1 * -1.0 + t$95$2), $MachinePrecision], If[LessEqual[F, -1.3e-161], N[(N[(t$95$1 * t$95$0), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.8e-98], N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6800000.0], N[(N[(1.0 / N[(N[Sin[B], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * 1.0 + t$95$2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -5.20000000000000024e66

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

   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 55.7%

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

   if -5.20000000000000024e66 < F < -1.29999999999999998e-161

   1. Initial program 75.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 48.5%

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

   4. Applied rewrites 48.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 48.5%

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

   6. Applied rewrites 48.5%

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

      1. lift-*.f64 N/A

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

   8. Applied rewrites 57.3%

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

   if -1.29999999999999998e-161 < F < 3.8000000000000003e-98

   1. Initial program 75.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 61.4%

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

   4. Applied rewrites 61.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/r/ N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

   6. Applied rewrites 61.5%

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

   if 3.8000000000000003e-98 < F < 6.8e6

   1. Initial program 75.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 48.5%

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

   4. Applied rewrites 48.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 48.5%

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

   6. Applied rewrites 48.5%

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

      1. lift-*.f64 N/A

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

   8. Applied rewrites 57.3%

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

   if 6.8e6 < F

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 56.6%

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

Alternative 10: 80.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\\ t_1 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -5.2 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, -1, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{-161}:\\ \;\;\;\;t\_1 \cdot t\_0 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{-98}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{+181}:\\ \;\;\;\;\frac{t\_0}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F)) (t_1 (/ 1.0 (sin B))))
   (if (<= F -5.2e+66)
     (fma t_1 -1.0 (/ (- x) (tan B)))
     (if (<= F -1.3e-161)
       (- (* t_1 t_0) (/ x B))
       (if (<= F 3.8e-98)
         (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) (/ F B)) (/ x (tan B)))
         (if (<= F 7e+181) (- (/ t_0 (sin B)) (/ x B)) t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F;
	double t_1 = 1.0 / sin(B);
	double tmp;
	if (F <= -5.2e+66) {
		tmp = fma(t_1, -1.0, (-x / tan(B)));
	} else if (F <= -1.3e-161) {
		tmp = (t_1 * t_0) - (x / B);
	} else if (F <= 3.8e-98) {
		tmp = (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * (F / B)) - (x / tan(B));
	} else if (F <= 7e+181) {
		tmp = (t_0 / sin(B)) - (x / B);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F)
	t_1 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -5.2e+66)
		tmp = fma(t_1, -1.0, Float64(Float64(-x) / tan(B)));
	elseif (F <= -1.3e-161)
		tmp = Float64(Float64(t_1 * t_0) - Float64(x / B));
	elseif (F <= 3.8e-98)
		tmp = Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / tan(B)));
	elseif (F <= 7e+181)
		tmp = Float64(Float64(t_0 / sin(B)) - Float64(x / B));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5.2e+66], N[(t$95$1 * -1.0 + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.3e-161], N[(N[(t$95$1 * t$95$0), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.8e-98], N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7e+181], N[(N[(t$95$0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -5.20000000000000024e66

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

   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 55.7%

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

   if -5.20000000000000024e66 < F < -1.29999999999999998e-161

   1. Initial program 75.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 48.5%

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

   4. Applied rewrites 48.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 48.5%

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

   6. Applied rewrites 48.5%

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

      1. lift-*.f64 N/A

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

   8. Applied rewrites 57.3%

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

   if -1.29999999999999998e-161 < F < 3.8000000000000003e-98

   1. Initial program 75.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 61.4%

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

   4. Applied rewrites 61.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. distribute-rgt-neg-out N/A

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

      8. lift-neg.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. associate-/r/ N/A

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

      11. lift-/.f64 N/A

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

      12. lift-/.f64 N/A

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

   6. Applied rewrites 61.5%

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

   if 3.8000000000000003e-98 < F < 7.00000000000000016e181

   1. Initial program 75.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 48.5%

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

   4. Applied rewrites 48.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 48.5%

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

   6. Applied rewrites 48.5%

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

      1. lift-*.f64 N/A

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

   8. Applied rewrites 57.3%

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

   if 7.00000000000000016e181 < F

   1. Initial program 75.7%

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

   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 17.3%

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

   4. Applied rewrites 17.3%

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

Alternative 11: 74.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := -1 \cdot \frac{x \cdot \cos B}{\sin B}\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{-152}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 465000000000:\\ \;\;\;\;\frac{1}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* -1.0 (/ (* x (cos B)) (sin B)))))
   (if (<= x -1.4e-152)
     t_0
     (if (<= x 465000000000.0)
       (-
        (* (/ 1.0 (sin B)) (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F))
        (/ x B))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 * ((x * cos(B)) / sin(B));
	double tmp;
	if (x <= -1.4e-152) {
		tmp = t_0;
	} else if (x <= 465000000000.0) {
		tmp = ((1.0 / sin(B)) * (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 * Float64(Float64(x * cos(B)) / sin(B)))
	tmp = 0.0
	if (x <= -1.4e-152)
		tmp = t_0;
	elseif (x <= 465000000000.0)
		tmp = Float64(Float64(Float64(1.0 / sin(B)) * Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F)) - Float64(x / B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 * N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.4e-152], t$95$0, If[LessEqual[x, 465000000000.0], N[(N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.39999999999999992e-152 or 4.65e11 < x

   1. Initial program 75.7%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.7%

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

   4. Applied rewrites 55.7%

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

   if -1.39999999999999992e-152 < x < 4.65e11

   1. Initial program 75.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 48.5%

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

   4. Applied rewrites 48.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 48.5%

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

   6. Applied rewrites 48.5%

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

      1. lift-*.f64 N/A

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

   8. Applied rewrites 57.3%

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

Alternative 12: 67.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -4 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(\frac{F \cdot \frac{1}{F}}{\sin B}, -1, -1 \cdot \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 7 \cdot 10^{+181}:\\ \;\;\;\;t\_0 \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))))
   (if (<= F -4e+111)
     (fma (/ (* F (/ 1.0 F)) (sin B)) -1.0 (* -1.0 (/ x B)))
     (if (<= F 7e+181)
       (- (* t_0 (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F)) (/ x B))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double tmp;
	if (F <= -4e+111) {
		tmp = fma(((F * (1.0 / F)) / sin(B)), -1.0, (-1.0 * (x / B)));
	} else if (F <= 7e+181) {
		tmp = (t_0 * (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -4e+111)
		tmp = fma(Float64(Float64(F * Float64(1.0 / F)) / sin(B)), -1.0, Float64(-1.0 * Float64(x / B)));
	elseif (F <= 7e+181)
		tmp = Float64(Float64(t_0 * Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F)) - Float64(x / B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4e+111], N[(N[(N[(F * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] * -1.0 + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7e+181], N[(N[(t$95$0 * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.99999999999999983e111

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{F}^{0}}}{\sin B}, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F, \frac{-x}{\tan B}\right) \]

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. inv-pow N/A

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

      6. associate-*l/ N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 84.7%

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 57.3%

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

   8. Applied rewrites 57.3%

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

   9. Taylor expanded in F around -inf

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

   11. Applied rewrites 36.1%

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

   if -3.99999999999999983e111 < F < 7.00000000000000016e181

   1. Initial program 75.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 48.5%

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

   4. Applied rewrites 48.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 48.5%

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

   6. Applied rewrites 48.5%

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

      1. lift-*.f64 N/A

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

   8. Applied rewrites 57.3%

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

   if 7.00000000000000016e181 < F

   1. Initial program 75.7%

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

   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 17.3%

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

   4. Applied rewrites 17.3%

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

Alternative 13: 67.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.25 \cdot 10^{+93}:\\ \;\;\;\;\mathsf{fma}\left(\frac{F \cdot \frac{1}{F}}{\sin B}, -1, -1 \cdot \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 7 \cdot 10^{+181}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, -\frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.25e+93)
   (fma (/ (* F (/ 1.0 F)) (sin B)) -1.0 (* -1.0 (/ x B)))
   (if (<= F 7e+181)
     (fma F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B)) (- (/ x B)))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.25e+93) {
		tmp = fma(((F * (1.0 / F)) / sin(B)), -1.0, (-1.0 * (x / B)));
	} else if (F <= 7e+181) {
		tmp = fma(F, (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B)), -(x / B));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.25e+93)
		tmp = fma(Float64(Float64(F * Float64(1.0 / F)) / sin(B)), -1.0, Float64(-1.0 * Float64(x / B)));
	elseif (F <= 7e+181)
		tmp = fma(F, Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B)), Float64(-Float64(x / B)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.25e+93], N[(N[(N[(F * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] * -1.0 + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7e+181], 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], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.25e93

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{F}^{0}}}{\sin B}, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F, \frac{-x}{\tan B}\right) \]

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. inv-pow N/A

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

      6. associate-*l/ N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 84.7%

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 57.3%

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

   8. Applied rewrites 57.3%

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

   9. Taylor expanded in F around -inf

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

   11. Applied rewrites 36.1%

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

   if -1.25e93 < F < 7.00000000000000016e181

   1. Initial program 75.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 48.5%

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

   4. Applied rewrites 48.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 57.3%

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

   if 7.00000000000000016e181 < F

   1. Initial program 75.7%

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

   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 17.3%

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

   4. Applied rewrites 17.3%

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

Alternative 14: 67.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(\frac{F \cdot \frac{1}{F}}{\sin B}, -1, -1 \cdot \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 7 \cdot 10^{+181}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4e+111)
   (fma (/ (* F (/ 1.0 F)) (sin B)) -1.0 (* -1.0 (/ x B)))
   (if (<= F 7e+181)
     (- (/ (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F) (sin B)) (/ x B))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4e+111) {
		tmp = fma(((F * (1.0 / F)) / sin(B)), -1.0, (-1.0 * (x / B)));
	} else if (F <= 7e+181) {
		tmp = ((pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F) / sin(B)) - (x / B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4e+111)
		tmp = fma(Float64(Float64(F * Float64(1.0 / F)) / sin(B)), -1.0, Float64(-1.0 * Float64(x / B)));
	elseif (F <= 7e+181)
		tmp = Float64(Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F) / sin(B)) - Float64(x / B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4e+111], N[(N[(N[(F * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] * -1.0 + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7e+181], N[(N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.99999999999999983e111

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{F}^{0}}}{\sin B}, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F, \frac{-x}{\tan B}\right) \]

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. inv-pow N/A

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

      6. associate-*l/ N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 84.7%

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 57.3%

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

   8. Applied rewrites 57.3%

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

   9. Taylor expanded in F around -inf

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

   11. Applied rewrites 36.1%

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

   if -3.99999999999999983e111 < F < 7.00000000000000016e181

   1. Initial program 75.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 48.5%

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

   4. Applied rewrites 48.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 48.5%

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

   6. Applied rewrites 48.5%

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

      1. lift-*.f64 N/A

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

   8. Applied rewrites 57.3%

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

   if 7.00000000000000016e181 < F

   1. Initial program 75.7%

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

   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 17.3%

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

   4. Applied rewrites 17.3%

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

Alternative 15: 66.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -950000:\\ \;\;\;\;\mathsf{fma}\left(\frac{F \cdot \frac{1}{F}}{\sin B}, -1, -1 \cdot \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{+144}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -950000.0)
   (fma (/ (* F (/ 1.0 F)) (sin B)) -1.0 (* -1.0 (/ x B)))
   (if (<= F 5.2e+144)
     (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) (/ F (sin B))) (/ x B))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -950000.0) {
		tmp = fma(((F * (1.0 / F)) / sin(B)), -1.0, (-1.0 * (x / B)));
	} else if (F <= 5.2e+144) {
		tmp = (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * (F / sin(B))) - (x / B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -950000.0)
		tmp = fma(Float64(Float64(F * Float64(1.0 / F)) / sin(B)), -1.0, Float64(-1.0 * Float64(x / B)));
	elseif (F <= 5.2e+144)
		tmp = Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * Float64(F / sin(B))) - Float64(x / B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -950000.0], N[(N[(N[(F * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] * -1.0 + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.2e+144], N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.5e5

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{F}^{0}}}{\sin B}, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F, \frac{-x}{\tan B}\right) \]

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. inv-pow N/A

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

      6. associate-*l/ N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 84.7%

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 57.3%

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

   8. Applied rewrites 57.3%

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

   9. Taylor expanded in F around -inf

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

   11. Applied rewrites 36.1%

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

   if -9.5e5 < F < 5.1999999999999998e144

   1. Initial program 75.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 48.5%

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

   4. Applied rewrites 48.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 48.5%

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

   6. Applied rewrites 48.5%

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

   if 5.1999999999999998e144 < F

   1. Initial program 75.7%

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

   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 17.3%

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

   4. Applied rewrites 17.3%

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

Alternative 16: 64.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{F \cdot \frac{1}{F}}{\sin B}, -1, -1 \cdot \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 0.2:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.5)
   (fma (/ (* F (/ 1.0 F)) (sin B)) -1.0 (* -1.0 (/ x B)))
   (if (<= F 0.2)
     (- (* (pow (fma x 2.0 2.0) -0.5) (/ F (sin B))) (/ x B))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.5) {
		tmp = fma(((F * (1.0 / F)) / sin(B)), -1.0, (-1.0 * (x / B)));
	} else if (F <= 0.2) {
		tmp = (pow(fma(x, 2.0, 2.0), -0.5) * (F / sin(B))) - (x / B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.5)
		tmp = fma(Float64(Float64(F * Float64(1.0 / F)) / sin(B)), -1.0, Float64(-1.0 * Float64(x / B)));
	elseif (F <= 0.2)
		tmp = Float64(Float64((fma(x, 2.0, 2.0) ^ -0.5) * Float64(F / sin(B))) - Float64(x / B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.5], N[(N[(N[(F * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] * -1.0 + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.2], N[(N[(N[Power[N[(x * 2.0 + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.5

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{F}^{0}}}{\sin B}, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F, \frac{-x}{\tan B}\right) \]

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. inv-pow N/A

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

      6. associate-*l/ N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 84.7%

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 57.3%

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

   8. Applied rewrites 57.3%

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

   9. Taylor expanded in F around -inf

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

   11. Applied rewrites 36.1%

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

   if -0.5 < F < 0.20000000000000001

   1. Initial program 75.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 48.5%

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

   4. Applied rewrites 48.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 48.5%

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

   6. Applied rewrites 48.5%

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

   7. Taylor expanded in F around 0

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

   9. Applied rewrites 34.6%

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

   if 0.20000000000000001 < F

   1. Initial program 75.7%

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

   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 17.3%

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

   4. Applied rewrites 17.3%

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

Alternative 17: 58.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -20:\\ \;\;\;\;\mathsf{fma}\left(\frac{F \cdot \frac{1}{F}}{\sin B}, -1, -1 \cdot \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 53000000:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -20.0)
   (fma (/ (* F (/ 1.0 F)) (sin B)) -1.0 (* -1.0 (/ x B)))
   (if (<= F 53000000.0)
     (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) (/ F B)) (/ x B))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -20.0) {
		tmp = fma(((F * (1.0 / F)) / sin(B)), -1.0, (-1.0 * (x / B)));
	} else if (F <= 53000000.0) {
		tmp = (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -20.0)
		tmp = fma(Float64(Float64(F * Float64(1.0 / F)) / sin(B)), -1.0, Float64(-1.0 * Float64(x / B)));
	elseif (F <= 53000000.0)
		tmp = Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -20.0], N[(N[(N[(F * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] * -1.0 + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 53000000.0], N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -20

   1. Initial program 75.7%

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

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

      6. mult-flip N/A

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

      7. *-commutative 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 84.7%

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

      1. lift-/.f64 N/A

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

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{{F}^{0}}}{\sin B}, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F, \frac{-x}{\tan B}\right) \]

      3. metadata-eval N/A

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

      4. pow-plus N/A

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

      5. inv-pow N/A

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

      6. associate-*l/ N/A

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

      7. associate-/r* N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-/.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. associate-/r* N/A

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

      14. associate-*r/ N/A

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

      15. lower-/.f64 N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 84.7%

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 57.3%

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

   8. Applied rewrites 57.3%

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

   9. Taylor expanded in F around -inf

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

   11. Applied rewrites 36.1%

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

   if -20 < F < 5.3e7

   1. Initial program 75.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 48.5%

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

   4. Applied rewrites 48.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 48.5%

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

   6. Applied rewrites 48.5%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 35.0%

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

   9. Applied rewrites 35.0%

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

   if 5.3e7 < F

   1. Initial program 75.7%

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

   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 17.3%

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

   4. Applied rewrites 17.3%

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

Alternative 18: 52.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -5.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{-1}{F} \cdot \frac{F}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 53000000:\\ \;\;\;\;\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.8e+53)
   (- (* (/ -1.0 F) (/ F (sin B))) (/ x B))
   (if (<= F 53000000.0)
     (/ (- (* F (pow (+ 2.0 (fma 2.0 x (pow F 2.0))) -0.5)) x) B)
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.8e+53) {
		tmp = ((-1.0 / F) * (F / sin(B))) - (x / B);
	} else if (F <= 53000000.0) {
		tmp = ((F * pow((2.0 + fma(2.0, x, pow(F, 2.0))), -0.5)) - x) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.8e+53)
		tmp = Float64(Float64(Float64(-1.0 / F) * Float64(F / sin(B))) - Float64(x / B));
	elseif (F <= 53000000.0)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + fma(2.0, x, (F ^ 2.0))) ^ -0.5)) - x) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.8e+53], N[(N[(N[(-1.0 / F), $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 53000000.0], 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[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.8000000000000004e53

   1. Initial program 75.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 48.5%

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

   4. Applied rewrites 48.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 48.5%

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

   6. Applied rewrites 48.5%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 27.7%

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

   9. Applied rewrites 27.7%

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

   if -5.8000000000000004e53 < F < 5.3e7

   1. Initial program 75.7%

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

   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 43.8%

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

   if 5.3e7 < F

   1. Initial program 75.7%

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

   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 17.3%

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

   4. Applied rewrites 17.3%

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

Alternative 19: 51.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -20:\\ \;\;\;\;\frac{-1}{F} \cdot \frac{F}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 53000000:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -20.0)
   (- (* (/ -1.0 F) (/ F (sin B))) (/ x B))
   (if (<= F 53000000.0)
     (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) (/ F B)) (/ x B))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -20.0) {
		tmp = ((-1.0 / F) * (F / sin(B))) - (x / B);
	} else if (F <= 53000000.0) {
		tmp = (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -20.0)
		tmp = Float64(Float64(Float64(-1.0 / F) * Float64(F / sin(B))) - Float64(x / B));
	elseif (F <= 53000000.0)
		tmp = Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -20.0], N[(N[(N[(-1.0 / F), $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 53000000.0], N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -20

   1. Initial program 75.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 48.5%

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

   4. Applied rewrites 48.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 48.5%

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

   6. Applied rewrites 48.5%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 27.7%

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

   9. Applied rewrites 27.7%

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

   if -20 < F < 5.3e7

   1. Initial program 75.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 48.5%

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

   4. Applied rewrites 48.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 48.5%

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

   6. Applied rewrites 48.5%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 35.0%

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

   9. Applied rewrites 35.0%

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

   if 5.3e7 < F

   1. Initial program 75.7%

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

   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 17.3%

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

   4. Applied rewrites 17.3%

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

Alternative 20: 51.4% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -9.8 \cdot 10^{+58}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 53000000:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9.8e+58)
   (/ -1.0 (sin B))
   (if (<= F 53000000.0)
     (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) (/ F B)) (/ x B))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.8e+58) {
		tmp = -1.0 / sin(B);
	} else if (F <= 53000000.0) {
		tmp = (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9.8e+58)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 53000000.0)
		tmp = Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9.8e+58], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 53000000.0], N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.80000000000000037e58

   1. Initial program 75.7%

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

   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 16.9%

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

   4. Applied rewrites 16.9%

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

   if -9.80000000000000037e58 < F < 5.3e7

   1. Initial program 75.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 48.5%

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

   4. Applied rewrites 48.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 48.5%

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

   6. Applied rewrites 48.5%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 35.0%

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

   9. Applied rewrites 35.0%

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

   if 5.3e7 < F

   1. Initial program 75.7%

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

   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 17.3%

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

   4. Applied rewrites 17.3%

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

Alternative 21: 33.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -4.5 \cdot 10^{-45}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{-90}:\\ \;\;\;\;\frac{-1}{B \cdot F} \cdot F\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.5e-45)
   (/ -1.0 (sin B))
   (if (<= F 2.1e-90) (* (/ -1.0 (* B F)) F) (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.5e-45) {
		tmp = -1.0 / sin(B);
	} else if (F <= 2.1e-90) {
		tmp = (-1.0 / (B * F)) * F;
	} 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 <= (-4.5d-45)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 2.1d-90) then
        tmp = ((-1.0d0) / (b * f)) * f
    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 <= -4.5e-45) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 2.1e-90) {
		tmp = (-1.0 / (B * F)) * F;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.5e-45:
		tmp = -1.0 / math.sin(B)
	elif F <= 2.1e-90:
		tmp = (-1.0 / (B * F)) * F
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.5e-45)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 2.1e-90)
		tmp = Float64(Float64(-1.0 / Float64(B * F)) * F);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -4.5e-45)
		tmp = -1.0 / sin(B);
	elseif (F <= 2.1e-90)
		tmp = (-1.0 / (B * F)) * F;
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.5e-45], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.1e-90], N[(N[(-1.0 / N[(B * F), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.4999999999999999e-45

   1. Initial program 75.7%

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

   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 16.9%

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

   4. Applied rewrites 16.9%

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

   if -4.4999999999999999e-45 < F < 2.0999999999999999e-90

   1. Initial program 75.7%

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

   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 16.9%

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

   4. Applied rewrites 16.9%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 10.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. *-inverses N/A

         \[\leadsto \frac{\frac{F}{F}}{\mathsf{neg}\left(\color{blue}{B}\right)} \]

      5. associate-/l/ N/A

         \[\leadsto \frac{F}{\color{blue}{F \cdot \left(\mathsf{neg}\left(B\right)\right)}} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{F}{\color{blue}{F \cdot \left(\mathsf{neg}\left(B\right)\right)}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{F}{F \cdot \color{blue}{\left(\mathsf{neg}\left(B\right)\right)}} \]

      8. lower-neg.f64 12.4%

         \[\leadsto \frac{F}{F \cdot \left(-B\right)} \]

   9. Applied rewrites 12.4%

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

       1. lift-/.f64 N/A

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

       2. mult-flip N/A

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

       3. lift-*.f64 N/A

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

       4. associate-/r* N/A

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

       5. lift-/.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 N/A

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

   11. Applied rewrites 12.5%

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

   if 2.0999999999999999e-90 < F

   1. Initial program 75.7%

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

   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 17.3%

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

   4. Applied rewrites 17.3%

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

Alternative 22: 21.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 7.2 \cdot 10^{+75}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, -0.00205026455026455, -0.019444444444444445\right), B \cdot B, -0.16666666666666666\right) \cdot B - \frac{B}{B \cdot B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x 7.2e+75)
   (/ -1.0 (sin B))
   (-
    (*
     (fma
      (fma (* B B) -0.00205026455026455 -0.019444444444444445)
      (* B B)
      -0.16666666666666666)
     B)
    (/ B (* B B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= 7.2e+75) {
		tmp = -1.0 / sin(B);
	} else {
		tmp = (fma(fma((B * B), -0.00205026455026455, -0.019444444444444445), (B * B), -0.16666666666666666) * B) - (B / (B * B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= 7.2e+75)
		tmp = Float64(-1.0 / sin(B));
	else
		tmp = Float64(Float64(fma(fma(Float64(B * B), -0.00205026455026455, -0.019444444444444445), Float64(B * B), -0.16666666666666666) * B) - Float64(B / Float64(B * B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, 7.2e+75], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(B * B), $MachinePrecision] * -0.00205026455026455 + -0.019444444444444445), $MachinePrecision] * N[(B * B), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * B), $MachinePrecision] - N[(B / N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.2e75

   1. Initial program 75.7%

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

   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 16.9%

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

   4. Applied rewrites 16.9%

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

   if 7.2e75 < x

   1. Initial program 75.7%

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

   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 16.9%

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

   4. Applied rewrites 16.9%

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

   5. Taylor expanded in B around 0

      \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      10. lower-pow.f64 9.9%

          \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(-0.00205026455026455 \cdot {B}^{2} - 0.019444444444444445\right) - 0.16666666666666666\right) - 1}{B} \]

   7. Applied rewrites 9.9%

      \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(-0.00205026455026455 \cdot {B}^{2} - 0.019444444444444445\right) - 0.16666666666666666\right) - 1}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      3. sub-flip N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) + \left(\mathsf{neg}\left(1\right)\right)}{B} \]

      4. metadata-eval N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) + -1}{B} \]

      5. div-add-rev N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right)}{B} + \frac{-1}{\color{blue}{B}} \]

      6. frac-add N/A

         \[\leadsto \frac{\left({B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right)\right) \cdot B + B \cdot -1}{B \cdot \color{blue}{B}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left({B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right)\right) \cdot B + B \cdot -1}{{B}^{2}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\left({B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right)\right) \cdot B + B \cdot -1}{{B}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\left({B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right)\right) \cdot B + B \cdot -1}{{B}^{\color{blue}{2}}} \]

   9. Applied rewrites 10.9%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.00205026455026455, B \cdot B, -0.019444444444444445\right), B \cdot B, -0.16666666666666666\right) \cdot B\right) \cdot B, B, B \cdot -1\right)}{B \cdot \color{blue}{B}} \]

   11. Applied rewrites 11.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, -0.00205026455026455, -0.019444444444444445\right), B \cdot B, -0.16666666666666666\right) \cdot B - \frac{B}{B \cdot B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 15.2% accurate, 3.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+78}:\\ \;\;\;\;\frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, -0.00205026455026455, -0.019444444444444445\right), B \cdot B, -0.16666666666666666\right) \cdot B - \frac{B}{B \cdot B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x 8.5e+78)
   (/ -1.0 (* B (+ 1.0 (* -0.16666666666666666 (pow B 2.0)))))
   (-
    (*
     (fma
      (fma (* B B) -0.00205026455026455 -0.019444444444444445)
      (* B B)
      -0.16666666666666666)
     B)
    (/ B (* B B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= 8.5e+78) {
		tmp = -1.0 / (B * (1.0 + (-0.16666666666666666 * pow(B, 2.0))));
	} else {
		tmp = (fma(fma((B * B), -0.00205026455026455, -0.019444444444444445), (B * B), -0.16666666666666666) * B) - (B / (B * B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= 8.5e+78)
		tmp = Float64(-1.0 / Float64(B * Float64(1.0 + Float64(-0.16666666666666666 * (B ^ 2.0)))));
	else
		tmp = Float64(Float64(fma(fma(Float64(B * B), -0.00205026455026455, -0.019444444444444445), Float64(B * B), -0.16666666666666666) * B) - Float64(B / Float64(B * B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, 8.5e+78], N[(-1.0 / N[(B * N[(1.0 + N[(-0.16666666666666666 * N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(N[(B * B), $MachinePrecision] * -0.00205026455026455 + -0.019444444444444445), $MachinePrecision] * N[(B * B), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * B), $MachinePrecision] - N[(B / N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.50000000000000079e78

   1. Initial program 75.7%

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

   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 16.9%

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

   4. Applied rewrites 16.9%

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

   5. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 10.3%

         \[\leadsto \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{2}\right)} \]

   7. Applied rewrites 10.3%

      \[\leadsto \frac{-1}{B \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {B}^{2}\right)}} \]

   if 8.50000000000000079e78 < x

   1. Initial program 75.7%

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

   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 16.9%

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

   4. Applied rewrites 16.9%

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

   5. Taylor expanded in B around 0

      \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      10. lower-pow.f64 9.9%

          \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(-0.00205026455026455 \cdot {B}^{2} - 0.019444444444444445\right) - 0.16666666666666666\right) - 1}{B} \]

   7. Applied rewrites 9.9%

      \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(-0.00205026455026455 \cdot {B}^{2} - 0.019444444444444445\right) - 0.16666666666666666\right) - 1}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      3. sub-flip N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) + \left(\mathsf{neg}\left(1\right)\right)}{B} \]

      4. metadata-eval N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) + -1}{B} \]

      5. div-add-rev N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right)}{B} + \frac{-1}{\color{blue}{B}} \]

      6. frac-add N/A

         \[\leadsto \frac{\left({B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right)\right) \cdot B + B \cdot -1}{B \cdot \color{blue}{B}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left({B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right)\right) \cdot B + B \cdot -1}{{B}^{2}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\left({B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right)\right) \cdot B + B \cdot -1}{{B}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\left({B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right)\right) \cdot B + B \cdot -1}{{B}^{\color{blue}{2}}} \]

   9. Applied rewrites 10.9%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.00205026455026455, B \cdot B, -0.019444444444444445\right), B \cdot B, -0.16666666666666666\right) \cdot B\right) \cdot B, B, B \cdot -1\right)}{B \cdot \color{blue}{B}} \]

   11. Applied rewrites 11.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, -0.00205026455026455, -0.019444444444444445\right), B \cdot B, -0.16666666666666666\right) \cdot B - \frac{B}{B \cdot B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 15.1% accurate, 3.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 7.5 \cdot 10^{+78}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, -0.00205026455026455, -0.019444444444444445\right), B \cdot B, -0.16666666666666666\right) \cdot B - \frac{B}{B \cdot B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x 7.5e+78)
   (/ -1.0 B)
   (-
    (*
     (fma
      (fma (* B B) -0.00205026455026455 -0.019444444444444445)
      (* B B)
      -0.16666666666666666)
     B)
    (/ B (* B B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= 7.5e+78) {
		tmp = -1.0 / B;
	} else {
		tmp = (fma(fma((B * B), -0.00205026455026455, -0.019444444444444445), (B * B), -0.16666666666666666) * B) - (B / (B * B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= 7.5e+78)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(Float64(fma(fma(Float64(B * B), -0.00205026455026455, -0.019444444444444445), Float64(B * B), -0.16666666666666666) * B) - Float64(B / Float64(B * B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, 7.5e+78], N[(-1.0 / B), $MachinePrecision], N[(N[(N[(N[(N[(B * B), $MachinePrecision] * -0.00205026455026455 + -0.019444444444444445), $MachinePrecision] * N[(B * B), $MachinePrecision] + -0.16666666666666666), $MachinePrecision] * B), $MachinePrecision] - N[(B / N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 7.49999999999999934e78

   1. Initial program 75.7%

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

   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 16.9%

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

   4. Applied rewrites 16.9%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 10.5%

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

   if 7.49999999999999934e78 < x

   1. Initial program 75.7%

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

   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 16.9%

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

   4. Applied rewrites 16.9%

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

   5. Taylor expanded in B around 0

      \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      10. lower-pow.f64 9.9%

          \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(-0.00205026455026455 \cdot {B}^{2} - 0.019444444444444445\right) - 0.16666666666666666\right) - 1}{B} \]

   7. Applied rewrites 9.9%

      \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(-0.00205026455026455 \cdot {B}^{2} - 0.019444444444444445\right) - 0.16666666666666666\right) - 1}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      3. sub-flip N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) + \left(\mathsf{neg}\left(1\right)\right)}{B} \]

      4. metadata-eval N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) + -1}{B} \]

      5. div-add-rev N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right)}{B} + \frac{-1}{\color{blue}{B}} \]

      6. frac-add N/A

         \[\leadsto \frac{\left({B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right)\right) \cdot B + B \cdot -1}{B \cdot \color{blue}{B}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left({B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right)\right) \cdot B + B \cdot -1}{{B}^{2}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\left({B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right)\right) \cdot B + B \cdot -1}{{B}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\left({B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right)\right) \cdot B + B \cdot -1}{{B}^{\color{blue}{2}}} \]

   9. Applied rewrites 10.9%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.00205026455026455, B \cdot B, -0.019444444444444445\right), B \cdot B, -0.16666666666666666\right) \cdot B\right) \cdot B, B, B \cdot -1\right)}{B \cdot \color{blue}{B}} \]

   11. Applied rewrites 11.4%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(B \cdot B, -0.00205026455026455, -0.019444444444444445\right), B \cdot B, -0.16666666666666666\right) \cdot B - \frac{B}{B \cdot B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 15.0% accurate, 4.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq 2.4 \cdot 10^{+33}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\left(-0.16666666666666666 \cdot B\right) \cdot B, B, B \cdot -1\right)}{B \cdot B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x 2.4e+33)
   (/ -1.0 B)
   (/ (fma (* (* -0.16666666666666666 B) B) B (* B -1.0)) (* B B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= 2.4e+33) {
		tmp = -1.0 / B;
	} else {
		tmp = fma(((-0.16666666666666666 * B) * B), B, (B * -1.0)) / (B * B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= 2.4e+33)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(fma(Float64(Float64(-0.16666666666666666 * B) * B), B, Float64(B * -1.0)) / Float64(B * B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, 2.4e+33], N[(-1.0 / B), $MachinePrecision], N[(N[(N[(N[(-0.16666666666666666 * B), $MachinePrecision] * B), $MachinePrecision] * B + N[(B * -1.0), $MachinePrecision]), $MachinePrecision] / N[(B * B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.4e33

   1. Initial program 75.7%

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

   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 16.9%

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

   4. Applied rewrites 16.9%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 10.5%

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

   if 2.4e33 < x

   1. Initial program 75.7%

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

   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 16.9%

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

   4. Applied rewrites 16.9%

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

   5. Taylor expanded in B around 0

      \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      4. lower-pow.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      8. lower--.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      10. lower-pow.f64 9.9%

          \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(-0.00205026455026455 \cdot {B}^{2} - 0.019444444444444445\right) - 0.16666666666666666\right) - 1}{B} \]

   7. Applied rewrites 9.9%

      \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(-0.00205026455026455 \cdot {B}^{2} - 0.019444444444444445\right) - 0.16666666666666666\right) - 1}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) - 1}{B} \]

      3. sub-flip N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) + \left(\mathsf{neg}\left(1\right)\right)}{B} \]

      4. metadata-eval N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right) + -1}{B} \]

      5. div-add-rev N/A

         \[\leadsto \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right)}{B} + \frac{-1}{\color{blue}{B}} \]

      6. frac-add N/A

         \[\leadsto \frac{\left({B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right)\right) \cdot B + B \cdot -1}{B \cdot \color{blue}{B}} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left({B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right)\right) \cdot B + B \cdot -1}{{B}^{2}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \frac{\left({B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right)\right) \cdot B + B \cdot -1}{{B}^{2}} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\left({B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-31}{15120} \cdot {B}^{2} - \frac{7}{360}\right) - \frac{1}{6}\right)\right) \cdot B + B \cdot -1}{{B}^{\color{blue}{2}}} \]

   9. Applied rewrites 10.9%

      \[\leadsto \frac{\mathsf{fma}\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.00205026455026455, B \cdot B, -0.019444444444444445\right), B \cdot B, -0.16666666666666666\right) \cdot B\right) \cdot B, B, B \cdot -1\right)}{B \cdot \color{blue}{B}} \]

   10. Taylor expanded in B around 0

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

   12. Applied rewrites 11.1%

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

Alternative 26: 12.5% accurate, 10.8× speedup

Math

\[F \cdot \frac{\frac{-1}{F}}{B} \]

FPCore

(FPCore (F B x) :precision binary64 (* F (/ (/ -1.0 F) B)))

C

double code(double F, double B, double x) {
	return F * ((-1.0 / F) / 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 = f * (((-1.0d0) / f) / b)
end function

Java

public static double code(double F, double B, double x) {
	return F * ((-1.0 / F) / B);
}

Python

def code(F, B, x):
	return F * ((-1.0 / F) / B)

Julia

function code(F, B, x)
	return Float64(F * Float64(Float64(-1.0 / F) / B))
end

MATLAB

function tmp = code(F, B, x)
	tmp = F * ((-1.0 / F) / B);
end

Wolfram

code[F_, B_, x_] := N[(F * N[(N[(-1.0 / F), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.7%

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

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 16.9%

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

4. Applied rewrites 16.9%

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

5. Taylor expanded in B around 0

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

7. Applied rewrites 10.5%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. rgt-mult-inverse N/A

      \[\leadsto \frac{F \cdot \frac{1}{F}}{\mathsf{neg}\left(\color{blue}{B}\right)} \]

   5. lift-/.f64 N/A

      \[\leadsto \frac{F \cdot \frac{1}{F}}{\mathsf{neg}\left(B\right)} \]

   6. associate-/l* N/A

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\mathsf{neg}\left(B\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto F \cdot \color{blue}{\frac{\frac{1}{F}}{\mathsf{neg}\left(B\right)}} \]

   8. lift-/.f64 N/A

      \[\leadsto F \cdot \frac{\frac{1}{F}}{\mathsf{neg}\left(\color{blue}{B}\right)} \]

   9. frac-2neg N/A

      \[\leadsto F \cdot \frac{\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(F\right)}}{\mathsf{neg}\left(\color{blue}{B}\right)} \]

   10. distribute-frac-neg N/A

       \[\leadsto F \cdot \frac{\mathsf{neg}\left(\frac{1}{\mathsf{neg}\left(F\right)}\right)}{\mathsf{neg}\left(\color{blue}{B}\right)} \]

   11. frac-2neg-rev N/A

       \[\leadsto F \cdot \frac{\frac{1}{\mathsf{neg}\left(F\right)}}{\color{blue}{B}} \]

   12. lower-/.f64 N/A

       \[\leadsto F \cdot \frac{\frac{1}{\mathsf{neg}\left(F\right)}}{\color{blue}{B}} \]

   13. metadata-eval N/A

       \[\leadsto F \cdot \frac{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(F\right)}}{B} \]

   14. frac-2neg-rev N/A

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

   15. lower-/.f64 12.5%

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

9. Applied rewrites 12.5%

   \[\leadsto F \cdot \color{blue}{\frac{\frac{-1}{F}}{B}} \]
10. Add Preprocessing

Alternative 27: 12.5% accurate, 11.3× speedup

Math

\[\frac{-1}{B \cdot F} \cdot F \]

FPCore

(FPCore (F B x) :precision binary64 (* (/ -1.0 (* B F)) F))

C

double code(double F, double B, double x) {
	return (-1.0 / (B * F)) * F;
}

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 * f)) * f
end function

Java

public static double code(double F, double B, double x) {
	return (-1.0 / (B * F)) * F;
}

Python

def code(F, B, x):
	return (-1.0 / (B * F)) * F

Julia

function code(F, B, x)
	return Float64(Float64(-1.0 / Float64(B * F)) * F)
end

MATLAB

function tmp = code(F, B, x)
	tmp = (-1.0 / (B * F)) * F;
end

Wolfram

code[F_, B_, x_] := N[(N[(-1.0 / N[(B * F), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]

Derivation

1. Initial program 75.7%

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

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 16.9%

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

4. Applied rewrites 16.9%

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

5. Taylor expanded in B around 0

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

7. Applied rewrites 10.5%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. metadata-eval N/A

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

   4. *-inverses N/A

      \[\leadsto \frac{\frac{F}{F}}{\mathsf{neg}\left(\color{blue}{B}\right)} \]

   5. associate-/l/ N/A

      \[\leadsto \frac{F}{\color{blue}{F \cdot \left(\mathsf{neg}\left(B\right)\right)}} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{F}{\color{blue}{F \cdot \left(\mathsf{neg}\left(B\right)\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{F}{F \cdot \color{blue}{\left(\mathsf{neg}\left(B\right)\right)}} \]

   8. lower-neg.f64 12.4%

      \[\leadsto \frac{F}{F \cdot \left(-B\right)} \]

9. Applied rewrites 12.4%

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

    1. lift-/.f64 N/A

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

    2. mult-flip N/A

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

    3. lift-*.f64 N/A

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

    4. associate-/r* N/A

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

    5. lift-/.f64 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f64 N/A

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

11. Applied rewrites 12.5%

    \[\leadsto \frac{-1}{B \cdot F} \cdot \color{blue}{F} \]
12. Add Preprocessing

Alternative 28: 12.4% accurate, 14.0× speedup

Math

\[\frac{F}{F \cdot \left(-B\right)} \]

FPCore

(FPCore (F B x) :precision binary64 (/ F (* F (- B))))

C

double code(double F, double B, double x) {
	return F / (F * -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 = f / (f * -b)
end function

Java

public static double code(double F, double B, double x) {
	return F / (F * -B);
}

Python

def code(F, B, x):
	return F / (F * -B)

Julia

function code(F, B, x)
	return Float64(F / Float64(F * Float64(-B)))
end

MATLAB

function tmp = code(F, B, x)
	tmp = F / (F * -B);
end

Wolfram

code[F_, B_, x_] := N[(F / N[(F * (-B)), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 75.7%

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

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 16.9%

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

4. Applied rewrites 16.9%

   \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

5. Taylor expanded in B around 0

   \[\leadsto \frac{-1}{B} \]
6. Step-by-step derivation

7. Applied rewrites 10.5%

   \[\leadsto \frac{-1}{B} \]
8. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \frac{-1}{\color{blue}{B}} \]

   2. frac-2neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(-1\right)}{\color{blue}{\mathsf{neg}\left(B\right)}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{1}{\mathsf{neg}\left(\color{blue}{B}\right)} \]

   4. *-inverses N/A

      \[\leadsto \frac{\frac{F}{F}}{\mathsf{neg}\left(\color{blue}{B}\right)} \]

   5. associate-/l/ N/A

      \[\leadsto \frac{F}{\color{blue}{F \cdot \left(\mathsf{neg}\left(B\right)\right)}} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{F}{\color{blue}{F \cdot \left(\mathsf{neg}\left(B\right)\right)}} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{F}{F \cdot \color{blue}{\left(\mathsf{neg}\left(B\right)\right)}} \]

   8. lower-neg.f64 12.4%

      \[\leadsto \frac{F}{F \cdot \left(-B\right)} \]

9. Applied rewrites 12.4%

   \[\leadsto \frac{F}{\color{blue}{F \cdot \left(-B\right)}} \]
10. Add Preprocessing

Alternative 29: 10.5% accurate, 26.5× speedup

Math

\[\frac{-1}{B} \]

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 75.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

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 16.9%

      \[\leadsto \frac{-1}{\sin B} \]

4. Applied rewrites 16.9%

   \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

5. Taylor expanded in B around 0

   \[\leadsto \frac{-1}{B} \]
6. Step-by-step derivation

7. Applied rewrites 10.5%

   \[\leadsto \frac{-1}{B} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025185 
(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))))))