VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.2% → 99.6%

Time: 7.4s

Alternatives: 23

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.2% 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: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -245000:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{0.5 \cdot \frac{2 + 2 \cdot x}{{F}^{3}} - \frac{1}{F}}{\sin B}, t\_0\right)\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{F}{\sin B}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{F}, \frac{F}{\cos B}, \left(-x\right) \cdot 1\right)}{\tan B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -245000.0)
     (fma
      F
      (/ (- (* 0.5 (/ (+ 2.0 (* 2.0 x)) (pow F 3.0))) (/ 1.0 F)) (sin B))
      t_0)
     (if (<= F 1.15e+22)
       (fma (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F (sin B)) t_0)
       (/ (fma (/ 1.0 F) (/ F (cos B)) (* (- x) 1.0)) (tan B))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -245000.0) {
		tmp = fma(F, (((0.5 * ((2.0 + (2.0 * x)) / pow(F, 3.0))) - (1.0 / F)) / sin(B)), t_0);
	} else if (F <= 1.15e+22) {
		tmp = fma(pow(fma(2.0, x, fma(F, F, 2.0)), -0.5), (F / sin(B)), t_0);
	} else {
		tmp = fma((1.0 / F), (F / cos(B)), (-x * 1.0)) / tan(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -245000.0)
		tmp = fma(F, Float64(Float64(Float64(0.5 * Float64(Float64(2.0 + Float64(2.0 * x)) / (F ^ 3.0))) - Float64(1.0 / F)) / sin(B)), t_0);
	elseif (F <= 1.15e+22)
		tmp = fma((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5), Float64(F / sin(B)), t_0);
	else
		tmp = Float64(fma(Float64(1.0 / F), Float64(F / cos(B)), Float64(Float64(-x) * 1.0)) / tan(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -245000.0], N[(F * N[(N[(N[(0.5 * N[(N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision] / N[Power[F, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / F), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, 1.15e+22], N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(N[(1.0 / F), $MachinePrecision] * N[(F / N[Cos[B], $MachinePrecision]), $MachinePrecision] + N[((-x) * 1.0), $MachinePrecision]), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -245000

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in F around -inf

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

      1. metadata-eval N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-/.f64 41.0

         \[\leadsto \mathsf{fma}\left(F, \frac{0.5 \cdot \frac{2 + 2 \cdot x}{{F}^{3}} - \frac{1}{\color{blue}{F}}}{\sin B}, \frac{-x}{\tan B}\right) \]

   6. Applied rewrites 41.0%

      \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{0.5 \cdot \frac{2 + 2 \cdot x}{{F}^{3}} - \frac{1}{F}}}{\sin B}, \frac{-x}{\tan B}\right) \]

   if -245000 < F < 1.1500000000000001e22

   1. Initial program 77.2%

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

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

      5. lower-fma.f64 77.2

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

   3. Applied rewrites 77.4%

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

   if 1.1500000000000001e22 < F

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. div-add N/A

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

      5. mult-flip N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in F around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 85.6

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

   8. Applied rewrites 85.6%

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

   9. Taylor expanded in F around inf

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

       1. lower-/.f64 55.9

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

   11. Applied rewrites 55.9%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, t\_0\right)\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{F}{\sin B}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{F}, \frac{F}{\cos B}, \left(-x\right) \cdot 1\right)}{\tan B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -2e+64)
     (fma F (/ -1.0 (* F (sin B))) t_0)
     (if (<= F 1.15e+22)
       (fma (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F (sin B)) t_0)
       (/ (fma (/ 1.0 F) (/ F (cos B)) (* (- x) 1.0)) (tan B))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -2e+64) {
		tmp = fma(F, (-1.0 / (F * sin(B))), t_0);
	} else if (F <= 1.15e+22) {
		tmp = fma(pow(fma(2.0, x, fma(F, F, 2.0)), -0.5), (F / sin(B)), t_0);
	} else {
		tmp = fma((1.0 / F), (F / cos(B)), (-x * 1.0)) / tan(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -2e+64)
		tmp = fma(F, Float64(-1.0 / Float64(F * sin(B))), t_0);
	elseif (F <= 1.15e+22)
		tmp = fma((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5), Float64(F / sin(B)), t_0);
	else
		tmp = Float64(fma(Float64(1.0 / F), Float64(F / cos(B)), Float64(Float64(-x) * 1.0)) / tan(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2e+64], N[(F * N[(-1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, 1.15e+22], N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(N[(1.0 / F), $MachinePrecision] * N[(F / N[Cos[B], $MachinePrecision]), $MachinePrecision] + N[((-x) * 1.0), $MachinePrecision]), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.00000000000000004e64

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 54.6

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

   6. Applied rewrites 54.6%

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

   if -2.00000000000000004e64 < F < 1.1500000000000001e22

   1. Initial program 77.2%

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

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

      5. lower-fma.f64 77.2

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

   3. Applied rewrites 77.4%

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

   if 1.1500000000000001e22 < F

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. div-add N/A

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

      5. mult-flip N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in F around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 85.6

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

   8. Applied rewrites 85.6%

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

   9. Taylor expanded in F around inf

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

       1. lower-/.f64 55.9

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

   11. Applied rewrites 55.9%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, t\_0\right)\\ \mathbf{elif}\;F \leq 370000:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{F}, \frac{F}{\cos B}, \left(-x\right) \cdot 1\right)}{\tan B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -2e+144)
     (fma F (/ -1.0 (* F (sin B))) t_0)
     (if (<= F 370000.0)
       (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B)) t_0)
       (/ (fma (/ 1.0 F) (/ F (cos B)) (* (- x) 1.0)) (tan B))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -2e+144) {
		tmp = fma(F, (-1.0 / (F * sin(B))), t_0);
	} else if (F <= 370000.0) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), t_0);
	} else {
		tmp = fma((1.0 / F), (F / cos(B)), (-x * 1.0)) / tan(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -2e+144)
		tmp = fma(F, Float64(-1.0 / Float64(F * sin(B))), t_0);
	elseif (F <= 370000.0)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), t_0);
	else
		tmp = Float64(fma(Float64(1.0 / F), Float64(F / cos(B)), Float64(Float64(-x) * 1.0)) / tan(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2e+144], N[(F * N[(-1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, 370000.0], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(N[(1.0 / F), $MachinePrecision] * N[(F / N[Cos[B], $MachinePrecision]), $MachinePrecision] + N[((-x) * 1.0), $MachinePrecision]), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.00000000000000005e144

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 54.6

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

   6. Applied rewrites 54.6%

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

   if -2.00000000000000005e144 < F < 3.7e5

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

   if 3.7e5 < F

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. div-add N/A

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

      5. mult-flip N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in F around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 85.6

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

   8. Applied rewrites 85.6%

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

   9. Taylor expanded in F around inf

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

       1. lower-/.f64 55.9

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

   11. Applied rewrites 55.9%

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

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{+22}:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{F}, \frac{F}{\cos B}, \left(-x\right) \cdot 1\right)}{\tan B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e+64)
   (fma F (/ -1.0 (* F (sin B))) (/ (- x) (tan B)))
   (if (<= F 1.15e+22)
     (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F (sin B))) (/ x (tan B)))
     (/ (fma (/ 1.0 F) (/ F (cos B)) (* (- x) 1.0)) (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e+64) {
		tmp = fma(F, (-1.0 / (F * sin(B))), (-x / tan(B)));
	} else if (F <= 1.15e+22) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / sin(B))) - (x / tan(B));
	} else {
		tmp = fma((1.0 / F), (F / cos(B)), (-x * 1.0)) / tan(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2e+64)
		tmp = fma(F, Float64(-1.0 / Float64(F * sin(B))), Float64(Float64(-x) / tan(B)));
	elseif (F <= 1.15e+22)
		tmp = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / sin(B))) - Float64(x / tan(B)));
	else
		tmp = Float64(fma(Float64(1.0 / F), Float64(F / cos(B)), Float64(Float64(-x) * 1.0)) / tan(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2e+64], N[(F * N[(-1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.15e+22], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / F), $MachinePrecision] * N[(F / N[Cos[B], $MachinePrecision]), $MachinePrecision] + N[((-x) * 1.0), $MachinePrecision]), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.00000000000000004e64

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 54.6

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

   6. Applied rewrites 54.6%

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

   if -2.00000000000000004e64 < F < 1.1500000000000001e22

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 77.2

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

   3. Applied rewrites 77.4%

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

   if 1.1500000000000001e22 < F

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. div-add N/A

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

      5. mult-flip N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in F around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 85.6

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

   8. Applied rewrites 85.6%

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

   9. Taylor expanded in F around inf

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

       1. lower-/.f64 55.9

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

   11. Applied rewrites 55.9%

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

Alternative 5: 96.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -0.00065:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, t\_0\right)\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5}}{\sin B}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{F}, \frac{F}{\cos B}, \left(-x\right) \cdot 1\right)}{\tan B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -0.00065)
     (fma F (/ -1.0 (* F (sin B))) t_0)
     (if (<= F 7e-52)
       (fma F (/ (pow (fma 2.0 x 2.0) -0.5) (sin B)) t_0)
       (/ (fma (/ 1.0 F) (/ F (cos B)) (* (- x) 1.0)) (tan B))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -0.00065) {
		tmp = fma(F, (-1.0 / (F * sin(B))), t_0);
	} else if (F <= 7e-52) {
		tmp = fma(F, (pow(fma(2.0, x, 2.0), -0.5) / sin(B)), t_0);
	} else {
		tmp = fma((1.0 / F), (F / cos(B)), (-x * 1.0)) / tan(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -0.00065)
		tmp = fma(F, Float64(-1.0 / Float64(F * sin(B))), t_0);
	elseif (F <= 7e-52)
		tmp = fma(F, Float64((fma(2.0, x, 2.0) ^ -0.5) / sin(B)), t_0);
	else
		tmp = Float64(fma(Float64(1.0 / F), Float64(F / cos(B)), Float64(Float64(-x) * 1.0)) / tan(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.00065], N[(F * N[(-1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, 7e-52], N[(F * N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(N[(1.0 / F), $MachinePrecision] * N[(F / N[Cos[B], $MachinePrecision]), $MachinePrecision] + N[((-x) * 1.0), $MachinePrecision]), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.4999999999999997e-4

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 54.6

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

   6. Applied rewrites 54.6%

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

   if -6.4999999999999997e-4 < F < 7.0000000000000001e-52

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in F around 0

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

   6. Applied rewrites 56.0%

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

   if 7.0000000000000001e-52 < F

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. div-add N/A

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

      5. mult-flip N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in F around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 85.6

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

   8. Applied rewrites 85.6%

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

   9. Taylor expanded in F around inf

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

       1. lower-/.f64 55.9

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

   11. Applied rewrites 55.9%

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

Alternative 6: 91.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -0.006:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, t\_0\right)\\ \mathbf{elif}\;F \leq 370000:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{2}\right)}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{F}, \frac{F}{\cos B}, \left(-x\right) \cdot 1\right)}{\tan B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -0.006)
     (fma F (/ -1.0 (* F (sin B))) t_0)
     (if (<= F 370000.0)
       (fma
        F
        (/
         (pow (fma 2.0 x (fma F F 2.0)) -0.5)
         (* B (+ 1.0 (* -0.16666666666666666 (pow B 2.0)))))
        t_0)
       (/ (fma (/ 1.0 F) (/ F (cos B)) (* (- x) 1.0)) (tan B))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -0.006) {
		tmp = fma(F, (-1.0 / (F * sin(B))), t_0);
	} else if (F <= 370000.0) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / (B * (1.0 + (-0.16666666666666666 * pow(B, 2.0))))), t_0);
	} else {
		tmp = fma((1.0 / F), (F / cos(B)), (-x * 1.0)) / tan(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -0.006)
		tmp = fma(F, Float64(-1.0 / Float64(F * sin(B))), t_0);
	elseif (F <= 370000.0)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / Float64(B * Float64(1.0 + Float64(-0.16666666666666666 * (B ^ 2.0))))), t_0);
	else
		tmp = Float64(fma(Float64(1.0 / F), Float64(F / cos(B)), Float64(Float64(-x) * 1.0)) / tan(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.006], N[(F * N[(-1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, 370000.0], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[(B * N[(1.0 + N[(-0.16666666666666666 * N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(N[(1.0 / F), $MachinePrecision] * N[(F / N[Cos[B], $MachinePrecision]), $MachinePrecision] + N[((-x) * 1.0), $MachinePrecision]), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.0060000000000000001

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 54.6

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

   6. Applied rewrites 54.6%

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

   if -0.0060000000000000001 < F < 3.7e5

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 70.6

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{\color{blue}{2}}\right)}, \frac{-x}{\tan B}\right) \]

   6. Applied rewrites 70.6%

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\color{blue}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{2}\right)}}, \frac{-x}{\tan B}\right) \]

   if 3.7e5 < F

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. div-add N/A

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

      5. mult-flip N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in F around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 85.6

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

   8. Applied rewrites 85.6%

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

   9. Taylor expanded in F around inf

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

       1. lower-/.f64 55.9

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

   11. Applied rewrites 55.9%

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

Alternative 7: 91.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -0.006:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 370000:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\mathsf{fma}\left(B \cdot B, -0.16666666666666666, 1\right) \cdot B} \cdot F - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{F}, \frac{F}{\cos B}, \left(-x\right) \cdot 1\right)}{\tan B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.006)
   (fma F (/ -1.0 (* F (sin B))) (/ (- x) (tan B)))
   (if (<= F 370000.0)
     (-
      (*
       (/
        (pow (fma 2.0 x (fma F F 2.0)) -0.5)
        (* (fma (* B B) -0.16666666666666666 1.0) B))
       F)
      (/ x (tan B)))
     (/ (fma (/ 1.0 F) (/ F (cos B)) (* (- x) 1.0)) (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.006) {
		tmp = fma(F, (-1.0 / (F * sin(B))), (-x / tan(B)));
	} else if (F <= 370000.0) {
		tmp = ((pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / (fma((B * B), -0.16666666666666666, 1.0) * B)) * F) - (x / tan(B));
	} else {
		tmp = fma((1.0 / F), (F / cos(B)), (-x * 1.0)) / tan(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.006)
		tmp = fma(F, Float64(-1.0 / Float64(F * sin(B))), Float64(Float64(-x) / tan(B)));
	elseif (F <= 370000.0)
		tmp = Float64(Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / Float64(fma(Float64(B * B), -0.16666666666666666, 1.0) * B)) * F) - Float64(x / tan(B)));
	else
		tmp = Float64(fma(Float64(1.0 / F), Float64(F / cos(B)), Float64(Float64(-x) * 1.0)) / tan(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.006], N[(F * N[(-1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 370000.0], N[(N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[(N[(N[(B * B), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / F), $MachinePrecision] * N[(F / N[Cos[B], $MachinePrecision]), $MachinePrecision] + N[((-x) * 1.0), $MachinePrecision]), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.0060000000000000001

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 54.6

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

   6. Applied rewrites 54.6%

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

   if -0.0060000000000000001 < F < 3.7e5

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 70.6

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{\color{blue}{2}}\right)}, \frac{-x}{\tan B}\right) \]

   6. Applied rewrites 70.6%

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\color{blue}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{2}\right)}}, \frac{-x}{\tan B}\right) \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

   8. Applied rewrites 70.6%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\mathsf{fma}\left(B \cdot B, -0.16666666666666666, 1\right) \cdot B} \cdot F - \frac{x}{\tan B}} \]

   if 3.7e5 < F

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. div-add N/A

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

      5. mult-flip N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in F around 0

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 85.6

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

   8. Applied rewrites 85.6%

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

   9. Taylor expanded in F around inf

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

       1. lower-/.f64 55.9

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

   11. Applied rewrites 55.9%

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

Alternative 8: 91.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := F \cdot \sin B\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -0.006:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{t\_0}, t\_1\right)\\ \mathbf{elif}\;F \leq 370000:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\mathsf{fma}\left(B \cdot B, -0.16666666666666666, 1\right) \cdot B} \cdot F - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{t\_0}, t\_1\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* F (sin B))) (t_1 (/ (- x) (tan B))))
   (if (<= F -0.006)
     (fma F (/ -1.0 t_0) t_1)
     (if (<= F 370000.0)
       (-
        (*
         (/
          (pow (fma 2.0 x (fma F F 2.0)) -0.5)
          (* (fma (* B B) -0.16666666666666666 1.0) B))
         F)
        (/ x (tan B)))
       (fma F (/ 1.0 t_0) t_1)))))

C

double code(double F, double B, double x) {
	double t_0 = F * sin(B);
	double t_1 = -x / tan(B);
	double tmp;
	if (F <= -0.006) {
		tmp = fma(F, (-1.0 / t_0), t_1);
	} else if (F <= 370000.0) {
		tmp = ((pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / (fma((B * B), -0.16666666666666666, 1.0) * B)) * F) - (x / tan(B));
	} else {
		tmp = fma(F, (1.0 / t_0), t_1);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(F * sin(B))
	t_1 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -0.006)
		tmp = fma(F, Float64(-1.0 / t_0), t_1);
	elseif (F <= 370000.0)
		tmp = Float64(Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / Float64(fma(Float64(B * B), -0.16666666666666666, 1.0) * B)) * F) - Float64(x / tan(B)));
	else
		tmp = fma(F, Float64(1.0 / t_0), t_1);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.006], N[(F * N[(-1.0 / t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[F, 370000.0], N[(N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[(N[(N[(B * B), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(1.0 / t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.0060000000000000001

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 54.6

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

   6. Applied rewrites 54.6%

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

   if -0.0060000000000000001 < F < 3.7e5

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 70.6

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{\color{blue}{2}}\right)}, \frac{-x}{\tan B}\right) \]

   6. Applied rewrites 70.6%

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\color{blue}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{2}\right)}}, \frac{-x}{\tan B}\right) \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

   8. Applied rewrites 70.6%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\mathsf{fma}\left(B \cdot B, -0.16666666666666666, 1\right) \cdot B} \cdot F - \frac{x}{\tan B}} \]

   if 3.7e5 < F

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.4

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

   6. Applied rewrites 52.4%

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

Alternative 9: 91.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := F \cdot \sin B\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -0.00065:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{t\_0}, t\_1\right)\\ \mathbf{elif}\;F \leq 370000:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{F}{B}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{t\_0}, t\_1\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* F (sin B))) (t_1 (/ (- x) (tan B))))
   (if (<= F -0.00065)
     (fma F (/ -1.0 t_0) t_1)
     (if (<= F 370000.0)
       (fma (pow (fma x 2.0 (fma F F 2.0)) -0.5) (/ F B) t_1)
       (fma F (/ 1.0 t_0) t_1)))))

C

double code(double F, double B, double x) {
	double t_0 = F * sin(B);
	double t_1 = -x / tan(B);
	double tmp;
	if (F <= -0.00065) {
		tmp = fma(F, (-1.0 / t_0), t_1);
	} else if (F <= 370000.0) {
		tmp = fma(pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5), (F / B), t_1);
	} else {
		tmp = fma(F, (1.0 / t_0), t_1);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(F * sin(B))
	t_1 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -0.00065)
		tmp = fma(F, Float64(-1.0 / t_0), t_1);
	elseif (F <= 370000.0)
		tmp = fma((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5), Float64(F / B), t_1);
	else
		tmp = fma(F, Float64(1.0 / t_0), t_1);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.00065], N[(F * N[(-1.0 / t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[F, 370000.0], N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision] + t$95$1), $MachinePrecision], N[(F * N[(1.0 / t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.4999999999999997e-4

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 54.6

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

   6. Applied rewrites 54.6%

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

   if -6.4999999999999997e-4 < F < 3.7e5

   1. Initial program 77.2%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.3

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

   4. Applied rewrites 62.3%

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

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

   6. Applied rewrites 62.4%

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

   if 3.7e5 < F

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.4

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

   6. Applied rewrites 52.4%

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

Alternative 10: 85.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := F \cdot \sin B\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -0.00065:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{t\_0}, t\_1\right)\\ \mathbf{elif}\;F \leq 1550000000:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{F}{B}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{t\_0}, -1 \cdot \frac{x}{B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* F (sin B))) (t_1 (/ (- x) (tan B))))
   (if (<= F -0.00065)
     (fma F (/ -1.0 t_0) t_1)
     (if (<= F 1550000000.0)
       (fma (pow (fma x 2.0 (fma F F 2.0)) -0.5) (/ F B) t_1)
       (fma F (/ 1.0 t_0) (* -1.0 (/ x B)))))))

C

double code(double F, double B, double x) {
	double t_0 = F * sin(B);
	double t_1 = -x / tan(B);
	double tmp;
	if (F <= -0.00065) {
		tmp = fma(F, (-1.0 / t_0), t_1);
	} else if (F <= 1550000000.0) {
		tmp = fma(pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5), (F / B), t_1);
	} else {
		tmp = fma(F, (1.0 / t_0), (-1.0 * (x / B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(F * sin(B))
	t_1 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -0.00065)
		tmp = fma(F, Float64(-1.0 / t_0), t_1);
	elseif (F <= 1550000000.0)
		tmp = fma((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5), Float64(F / B), t_1);
	else
		tmp = fma(F, Float64(1.0 / t_0), Float64(-1.0 * Float64(x / B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.00065], N[(F * N[(-1.0 / t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[F, 1550000000.0], N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision] + t$95$1), $MachinePrecision], N[(F * N[(1.0 / t$95$0), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.4999999999999997e-4

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 54.6

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

   6. Applied rewrites 54.6%

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

   if -6.4999999999999997e-4 < F < 1.55e9

   1. Initial program 77.2%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.3

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

   4. Applied rewrites 62.3%

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

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

   6. Applied rewrites 62.4%

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

   if 1.55e9 < F

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.7

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

   6. Applied rewrites 58.7%

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

   7. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 33.2

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

   9. Applied rewrites 33.2%

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

Alternative 11: 77.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := F \cdot \sin B\\ t_1 := -1 \cdot \frac{x}{B}\\ t_2 := \mathsf{fma}\left(F, \frac{-1}{t\_0}, t\_1\right)\\ \mathbf{if}\;F \leq -4.5 \cdot 10^{+227}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;F \leq -1.4 \cdot 10^{+146}:\\ \;\;\;\;\frac{-1 \cdot x}{\tan B}\\ \mathbf{elif}\;F \leq -0.0152:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5}, \frac{F}{B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{t\_0}, t\_1\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* F (sin B)))
        (t_1 (* -1.0 (/ x B)))
        (t_2 (fma F (/ -1.0 t_0) t_1)))
   (if (<= F -4.5e+227)
     t_2
     (if (<= F -1.4e+146)
       (/ (* -1.0 x) (tan B))
       (if (<= F -0.0152)
         t_2
         (if (<= F 1.3e-15)
           (fma (pow (fma x 2.0 2.0) -0.5) (/ F B) (/ (- x) (tan B)))
           (fma F (/ 1.0 t_0) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = F * sin(B);
	double t_1 = -1.0 * (x / B);
	double t_2 = fma(F, (-1.0 / t_0), t_1);
	double tmp;
	if (F <= -4.5e+227) {
		tmp = t_2;
	} else if (F <= -1.4e+146) {
		tmp = (-1.0 * x) / tan(B);
	} else if (F <= -0.0152) {
		tmp = t_2;
	} else if (F <= 1.3e-15) {
		tmp = fma(pow(fma(x, 2.0, 2.0), -0.5), (F / B), (-x / tan(B)));
	} else {
		tmp = fma(F, (1.0 / t_0), t_1);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(F * sin(B))
	t_1 = Float64(-1.0 * Float64(x / B))
	t_2 = fma(F, Float64(-1.0 / t_0), t_1)
	tmp = 0.0
	if (F <= -4.5e+227)
		tmp = t_2;
	elseif (F <= -1.4e+146)
		tmp = Float64(Float64(-1.0 * x) / tan(B));
	elseif (F <= -0.0152)
		tmp = t_2;
	elseif (F <= 1.3e-15)
		tmp = fma((fma(x, 2.0, 2.0) ^ -0.5), Float64(F / B), Float64(Float64(-x) / tan(B)));
	else
		tmp = fma(F, Float64(1.0 / t_0), t_1);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(F * N[(-1.0 / t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[F, -4.5e+227], t$95$2, If[LessEqual[F, -1.4e+146], N[(N[(-1.0 * x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -0.0152], t$95$2, If[LessEqual[F, 1.3e-15], N[(N[Power[N[(x * 2.0 + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(1.0 / t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.5e227 or -1.4e146 < F < -0.0152

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.7

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

   6. Applied rewrites 58.7%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 35.4

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

   9. Applied rewrites 35.4%

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

   if -4.5e227 < F < -1.4e146

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. div-add N/A

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

      5. mult-flip N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 56.0

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

   8. Applied rewrites 56.0%

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

   if -0.0152 < F < 1.30000000000000002e-15

   1. Initial program 77.2%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.3

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

   4. Applied rewrites 62.3%

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

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

   6. Applied rewrites 62.4%

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

   7. Taylor expanded in F around 0

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

   9. Applied rewrites 48.7%

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

   if 1.30000000000000002e-15 < F

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.7

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

   6. Applied rewrites 58.7%

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

   7. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 33.2

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

   9. Applied rewrites 33.2%

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

Alternative 12: 76.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{-1 \cdot x}{\tan B}\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{-29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, -1 \cdot \frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (* -1.0 x) (tan B))))
   (if (<= x -4.5e-29)
     t_0
     (if (<= x 3.5e-22)
       (fma
        F
        (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B))
        (* -1.0 (/ x B)))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 * x) / tan(B);
	double tmp;
	if (x <= -4.5e-29) {
		tmp = t_0;
	} else if (x <= 3.5e-22) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), (-1.0 * (x / B)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 * x) / tan(B))
	tmp = 0.0
	if (x <= -4.5e-29)
		tmp = t_0;
	elseif (x <= 3.5e-22)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), Float64(-1.0 * Float64(x / B)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 * x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.5e-29], t$95$0, If[LessEqual[x, 3.5e-22], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.4999999999999998e-29 or 3.50000000000000005e-22 < x

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. div-add N/A

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

      5. mult-flip N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 56.0

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

   8. Applied rewrites 56.0%

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

   if -4.4999999999999998e-29 < x < 3.50000000000000005e-22

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.7

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

   6. Applied rewrites 58.7%

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

Alternative 13: 70.4% accurate, 1.4× speedup

Math

\[\mathsf{copysign}\left(1, B\right) \cdot \begin{array}{l} \mathbf{if}\;\left|B\right| \leq 0.038:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\left|B\right| \cdot \left(1 + -0.16666666666666666 \cdot {\left(\left|B\right|\right)}^{2}\right)}, -1 \cdot \frac{x}{\left|B\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot x}{\tan \left(\left|B\right|\right)}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (*
  (copysign 1.0 B)
  (if (<= (fabs B) 0.038)
    (fma
     F
     (/
      (pow (fma 2.0 x (fma F F 2.0)) -0.5)
      (* (fabs B) (+ 1.0 (* -0.16666666666666666 (pow (fabs B) 2.0)))))
     (* -1.0 (/ x (fabs B))))
    (/ (* -1.0 x) (tan (fabs B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (fabs(B) <= 0.038) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / (fabs(B) * (1.0 + (-0.16666666666666666 * pow(fabs(B), 2.0))))), (-1.0 * (x / fabs(B))));
	} else {
		tmp = (-1.0 * x) / tan(fabs(B));
	}
	return copysign(1.0, B) * tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (abs(B) <= 0.038)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / Float64(abs(B) * Float64(1.0 + Float64(-0.16666666666666666 * (abs(B) ^ 2.0))))), Float64(-1.0 * Float64(x / abs(B))));
	else
		tmp = Float64(Float64(-1.0 * x) / tan(abs(B)));
	end
	return Float64(copysign(1.0, B) * tmp)
end

Wolfram

code[F_, B_, x_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[B]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[B], $MachinePrecision], 0.038], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[(N[Abs[B], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[Power[N[Abs[B], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(x / N[Abs[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * x), $MachinePrecision] / N[Tan[N[Abs[B], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if B < 0.0379999999999999991

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.7

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

   6. Applied rewrites 58.7%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 44.3

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{\color{blue}{2}}\right)}, -1 \cdot \frac{x}{B}\right) \]

   9. Applied rewrites 44.3%

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\color{blue}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{2}\right)}}, -1 \cdot \frac{x}{B}\right) \]

   if 0.0379999999999999991 < B

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. div-add N/A

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

      5. mult-flip N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 56.0

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

   8. Applied rewrites 56.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{\tan B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 70.4% accurate, 2.1× speedup

Math

\[\mathsf{copysign}\left(1, B\right) \cdot \begin{array}{l} \mathbf{if}\;\left|B\right| \leq 0.038:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{\left|B\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot x}{\tan \left(\left|B\right|\right)}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (*
  (copysign 1.0 B)
  (if (<= (fabs B) 0.038)
    (/ (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F) x) (fabs B))
    (/ (* -1.0 x) (tan (fabs B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (fabs(B) <= 0.038) {
		tmp = ((pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F) - x) / fabs(B);
	} else {
		tmp = (-1.0 * x) / tan(fabs(B));
	}
	return copysign(1.0, B) * tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (abs(B) <= 0.038)
		tmp = Float64(Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F) - x) / abs(B));
	else
		tmp = Float64(Float64(-1.0 * x) / tan(abs(B)));
	end
	return Float64(copysign(1.0, B) * tmp)
end

Wolfram

code[F_, B_, x_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[B]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[B], $MachinePrecision], 0.038], N[(N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / N[Abs[B], $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * x), $MachinePrecision] / N[Tan[N[Abs[B], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if B < 0.0379999999999999991

   1. Initial program 77.2%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

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

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

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

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

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

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

      13. lift-neg.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-unsound-pow.f64 N/A

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

   3. Applied rewrites 77.2%

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

   4. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.25}\right)}^{2} - x}{B}} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

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

   8. Applied rewrites 44.3%

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

   if 0.0379999999999999991 < B

   1. Initial program 77.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 85.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. div-add N/A

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

      5. mult-flip N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

   5. Applied rewrites 77.3%

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

   6. Taylor expanded in F around 0

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

      1. lower-*.f64 56.0

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

   8. Applied rewrites 56.0%

      \[\leadsto \frac{\color{blue}{-1 \cdot x}}{\tan B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 51.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\left|B\right|\right)\\ \mathsf{copysign}\left(1, B\right) \cdot \begin{array}{l} \mathbf{if}\;\left|B\right| \leq 1.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{\left|B\right|}\\ \mathbf{elif}\;\left|B\right| \leq 8.6 \cdot 10^{+74}:\\ \;\;\;\;\frac{-1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sin (fabs B))))
   (*
    (copysign 1.0 B)
    (if (<= (fabs B) 1.5e-9)
      (/ (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F) x) (fabs B))
      (if (<= (fabs B) 8.6e+74) (/ -1.0 t_0) (/ 1.0 t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = sin(fabs(B));
	double tmp;
	if (fabs(B) <= 1.5e-9) {
		tmp = ((pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F) - x) / fabs(B);
	} else if (fabs(B) <= 8.6e+74) {
		tmp = -1.0 / t_0;
	} else {
		tmp = 1.0 / t_0;
	}
	return copysign(1.0, B) * tmp;
}

Julia

function code(F, B, x)
	t_0 = sin(abs(B))
	tmp = 0.0
	if (abs(B) <= 1.5e-9)
		tmp = Float64(Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F) - x) / abs(B));
	elseif (abs(B) <= 8.6e+74)
		tmp = Float64(-1.0 / t_0);
	else
		tmp = Float64(1.0 / t_0);
	end
	return Float64(copysign(1.0, B) * tmp)
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sin[N[Abs[B], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[B]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[B], $MachinePrecision], 1.5e-9], N[(N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / N[Abs[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Abs[B], $MachinePrecision], 8.6e+74], N[(-1.0 / t$95$0), $MachinePrecision], N[(1.0 / t$95$0), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if B < 1.49999999999999999e-9

   1. Initial program 77.2%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

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

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

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

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

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

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

      13. lift-neg.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-unsound-pow.f64 N/A

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

   3. Applied rewrites 77.2%

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

   4. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.25}\right)}^{2} - x}{B}} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

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

   8. Applied rewrites 44.3%

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

   if 1.49999999999999999e-9 < B < 8.60000000000000001e74

   1. Initial program 77.2%

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

   2. 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%

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

   if 8.60000000000000001e74 < B

   1. Initial program 77.2%

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

   2. 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 17.0

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

   4. Applied rewrites 17.0%

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

Alternative 16: 50.4% accurate, 2.4× speedup

Math

\[\mathsf{copysign}\left(1, B\right) \cdot \begin{array}{l} \mathbf{if}\;\left|B\right| \leq 1.5 \cdot 10^{-9}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{\left|B\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin \left(\left|B\right|\right)}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (*
  (copysign 1.0 B)
  (if (<= (fabs B) 1.5e-9)
    (/ (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F) x) (fabs B))
    (/ -1.0 (sin (fabs B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (fabs(B) <= 1.5e-9) {
		tmp = ((pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F) - x) / fabs(B);
	} else {
		tmp = -1.0 / sin(fabs(B));
	}
	return copysign(1.0, B) * tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (abs(B) <= 1.5e-9)
		tmp = Float64(Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F) - x) / abs(B));
	else
		tmp = Float64(-1.0 / sin(abs(B)));
	end
	return Float64(copysign(1.0, B) * tmp)
end

Wolfram

code[F_, B_, x_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[B]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[B], $MachinePrecision], 1.5e-9], N[(N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / N[Abs[B], $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[Sin[N[Abs[B], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if B < 1.49999999999999999e-9

   1. Initial program 77.2%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

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

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

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

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

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

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

      13. lift-neg.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-unsound-pow.f64 N/A

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

   3. Applied rewrites 77.2%

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

   4. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.25}\right)}^{2} - x}{B}} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

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

   8. Applied rewrites 44.3%

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

   if 1.49999999999999999e-9 < B

   1. Initial program 77.2%

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

   2. 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 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 50.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{B \cdot F}\\ t_1 := \frac{x}{B \cdot F}\\ \mathbf{if}\;F \leq -2.5 \cdot 10^{+154}:\\ \;\;\;\;-1 \cdot \left(F \cdot \left(t\_0 + t\_1\right)\right)\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{+152}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \mathsf{fma}\left(-1, t\_1, t\_0\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* B F))) (t_1 (/ x (* B F))))
   (if (<= F -2.5e+154)
     (* -1.0 (* F (+ t_0 t_1)))
     (if (<= F 1.2e+152)
       (/ (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F) x) B)
       (* F (fma -1.0 t_1 t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / (B * F);
	double t_1 = x / (B * F);
	double tmp;
	if (F <= -2.5e+154) {
		tmp = -1.0 * (F * (t_0 + t_1));
	} else if (F <= 1.2e+152) {
		tmp = ((pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F) - x) / B;
	} else {
		tmp = F * fma(-1.0, t_1, t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / Float64(B * F))
	t_1 = Float64(x / Float64(B * F))
	tmp = 0.0
	if (F <= -2.5e+154)
		tmp = Float64(-1.0 * Float64(F * Float64(t_0 + t_1)));
	elseif (F <= 1.2e+152)
		tmp = Float64(Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F) - x) / B);
	else
		tmp = Float64(F * fma(-1.0, t_1, t_0));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[(B * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.5e+154], N[(-1.0 * N[(F * N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.2e+152], N[(N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(F * N[(-1.0 * t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.50000000000000002e154

   1. Initial program 77.2%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

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

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

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

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

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

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

      13. lift-neg.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-unsound-pow.f64 N/A

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

   3. Applied rewrites 77.2%

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

   4. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.25}\right)}^{2} - x}{B}} \]

   8. Applied rewrites 36.1%

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

   9. Taylor expanded in F around -inf

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

       1. lower-*.f64 N/A

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

       2. lower-*.f64 N/A

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

       3. lower-+.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 27.4

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

   11. Applied rewrites 27.4%

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

   if -2.50000000000000002e154 < F < 1.2e152

   1. Initial program 77.2%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

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

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

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

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

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

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

      13. lift-neg.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. lower-unsound-pow.f64 N/A

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

   3. Applied rewrites 77.2%

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

   4. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.25}\right)}^{2} - x}{B}} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

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

   8. Applied rewrites 44.3%

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

   if 1.2e152 < F

   1. Initial program 77.2%

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

      1. lift-pow.f64 N/A

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

      2. sqr-pow N/A

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

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

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

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

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-*.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-fma.f64 N/A

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

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

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

      13. lift-neg.f64 N/A

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

      14. lift-/.f64 N/A

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

      15. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      16. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\color{blue}{\frac{-1}{2}}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      17. lower-unsound-pow.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}}\right) \]

   3. Applied rewrites 77.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{4}}\right)}^{2} - x}{B}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)}^{2} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{B} \]

      3. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{\color{blue}{B}} \]

   6. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.25}\right)}^{2} - x}{B}} \]

   8. Applied rewrites 36.1%

      \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \color{blue}{\frac{F}{B}}, \frac{-x}{B}\right) \]

   9. Taylor expanded in F around inf

      \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x}{B \cdot F} + \frac{1}{B \cdot F}\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto F \cdot \left(-1 \cdot \frac{x}{B \cdot F} + \color{blue}{\frac{1}{B \cdot F}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{\color{blue}{B \cdot F}}, \frac{1}{B \cdot F}\right) \]

       3. lower-/.f64 N/A

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot \color{blue}{F}}, \frac{1}{B \cdot F}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{B \cdot F}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{B \cdot F}\right) \]

       6. lower-*.f64 25.0

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{B \cdot F}\right) \]

   11. Applied rewrites 25.0%

       \[\leadsto F \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{B \cdot F}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 50.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{B \cdot F}\\ t_1 := \frac{x}{B \cdot F}\\ \mathbf{if}\;F \leq -1900000:\\ \;\;\;\;-1 \cdot \left(F \cdot \left(t\_0 + t\_1\right)\right)\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5}, \frac{F}{B}, \frac{-x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;F \cdot \mathsf{fma}\left(-1, t\_1, t\_0\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* B F))) (t_1 (/ x (* B F))))
   (if (<= F -1900000.0)
     (* -1.0 (* F (+ t_0 t_1)))
     (if (<= F 2.3e-26)
       (fma (pow (fma 2.0 x 2.0) -0.5) (/ F B) (/ (- x) B))
       (* F (fma -1.0 t_1 t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / (B * F);
	double t_1 = x / (B * F);
	double tmp;
	if (F <= -1900000.0) {
		tmp = -1.0 * (F * (t_0 + t_1));
	} else if (F <= 2.3e-26) {
		tmp = fma(pow(fma(2.0, x, 2.0), -0.5), (F / B), (-x / B));
	} else {
		tmp = F * fma(-1.0, t_1, t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / Float64(B * F))
	t_1 = Float64(x / Float64(B * F))
	tmp = 0.0
	if (F <= -1900000.0)
		tmp = Float64(-1.0 * Float64(F * Float64(t_0 + t_1)));
	elseif (F <= 2.3e-26)
		tmp = fma((fma(2.0, x, 2.0) ^ -0.5), Float64(F / B), Float64(Float64(-x) / B));
	else
		tmp = Float64(F * fma(-1.0, t_1, t_0));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[(B * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1900000.0], N[(-1.0 * N[(F * N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.3e-26], N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision] + N[((-x) / B), $MachinePrecision]), $MachinePrecision], N[(F * N[(-1.0 * t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.9e6

   1. Initial program 77.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. sqr-pow N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right)} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right)} \]

      4. lower-unsound-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left(\color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\color{blue}{2 \cdot x} + \left(F \cdot F + 2\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(\mathsf{fma}\left(2, x, F \cdot F + 2\right)\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{F \cdot F + 2}\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{\left(\frac{-\frac{1}{2}}{2}\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\color{blue}{\mathsf{neg}\left(\frac{1}{2}\right)}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      16. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\color{blue}{\frac{-1}{2}}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      17. lower-unsound-pow.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}}\right) \]

   3. Applied rewrites 77.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{4}}\right)}^{2} - x}{B}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)}^{2} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{B} \]

      3. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{\color{blue}{B}} \]

   6. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.25}\right)}^{2} - x}{B}} \]

   8. Applied rewrites 36.1%

      \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \color{blue}{\frac{F}{B}}, \frac{-x}{B}\right) \]

   9. Taylor expanded in F around -inf

      \[\leadsto -1 \cdot \color{blue}{\left(F \cdot \left(\frac{1}{B \cdot F} + \frac{x}{B \cdot F}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(F \cdot \color{blue}{\left(\frac{1}{B \cdot F} + \frac{x}{B \cdot F}\right)}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{B \cdot F} + \color{blue}{\frac{x}{B \cdot F}}\right)\right) \]

       3. lower-+.f64 N/A

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{B \cdot F} + \frac{x}{\color{blue}{B \cdot F}}\right)\right) \]

       4. lower-/.f64 N/A

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{B \cdot F} + \frac{x}{\color{blue}{B} \cdot F}\right)\right) \]

       5. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{B \cdot F} + \frac{x}{B \cdot F}\right)\right) \]

       6. lower-/.f64 N/A

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{B \cdot F} + \frac{x}{B \cdot \color{blue}{F}}\right)\right) \]

       7. lower-*.f64 27.4

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{B \cdot F} + \frac{x}{B \cdot F}\right)\right) \]

   11. Applied rewrites 27.4%

       \[\leadsto -1 \cdot \color{blue}{\left(F \cdot \left(\frac{1}{B \cdot F} + \frac{x}{B \cdot F}\right)\right)} \]

   if -1.9e6 < F < 2.30000000000000009e-26

   1. Initial program 77.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. sqr-pow N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right)} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right)} \]

      4. lower-unsound-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left(\color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\color{blue}{2 \cdot x} + \left(F \cdot F + 2\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(\mathsf{fma}\left(2, x, F \cdot F + 2\right)\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{F \cdot F + 2}\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{\left(\frac{-\frac{1}{2}}{2}\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\color{blue}{\mathsf{neg}\left(\frac{1}{2}\right)}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      16. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\color{blue}{\frac{-1}{2}}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      17. lower-unsound-pow.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}}\right) \]

   3. Applied rewrites 77.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{4}}\right)}^{2} - x}{B}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)}^{2} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{B} \]

      3. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{\color{blue}{B}} \]

   6. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.25}\right)}^{2} - x}{B}} \]

   8. Applied rewrites 36.1%

      \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \color{blue}{\frac{F}{B}}, \frac{-x}{B}\right) \]

   9. Taylor expanded in F around 0

      \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{\frac{-1}{2}}, \frac{F}{B}, \frac{-x}{B}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 28.5%

       \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5}, \frac{F}{B}, \frac{-x}{B}\right) \]

   if 2.30000000000000009e-26 < F

   1. Initial program 77.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. sqr-pow N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right)} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right)} \]

      4. lower-unsound-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left(\color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\color{blue}{2 \cdot x} + \left(F \cdot F + 2\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(\mathsf{fma}\left(2, x, F \cdot F + 2\right)\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{F \cdot F + 2}\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{\left(\frac{-\frac{1}{2}}{2}\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\color{blue}{\mathsf{neg}\left(\frac{1}{2}\right)}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      16. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\color{blue}{\frac{-1}{2}}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      17. lower-unsound-pow.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}}\right) \]

   3. Applied rewrites 77.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{4}}\right)}^{2} - x}{B}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)}^{2} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{B} \]

      3. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{\color{blue}{B}} \]

   6. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.25}\right)}^{2} - x}{B}} \]

   8. Applied rewrites 36.1%

      \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \color{blue}{\frac{F}{B}}, \frac{-x}{B}\right) \]

   9. Taylor expanded in F around inf

      \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x}{B \cdot F} + \frac{1}{B \cdot F}\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto F \cdot \left(-1 \cdot \frac{x}{B \cdot F} + \color{blue}{\frac{1}{B \cdot F}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{\color{blue}{B \cdot F}}, \frac{1}{B \cdot F}\right) \]

       3. lower-/.f64 N/A

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot \color{blue}{F}}, \frac{1}{B \cdot F}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{B \cdot F}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{B \cdot F}\right) \]

       6. lower-*.f64 25.0

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{B \cdot F}\right) \]

   11. Applied rewrites 25.0%

       \[\leadsto F \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{B \cdot F}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 43.9% accurate, 4.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{B \cdot F}\\ t_1 := \frac{x}{B \cdot F}\\ \mathbf{if}\;F \leq -0.017:\\ \;\;\;\;-1 \cdot \left(F \cdot \left(t\_0 + t\_1\right)\right)\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \mathsf{fma}\left(-1, t\_1, t\_0\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* B F))) (t_1 (/ x (* B F))))
   (if (<= F -0.017)
     (* -1.0 (* F (+ t_0 t_1)))
     (if (<= F 1.3e-23) (/ (- x) B) (* F (fma -1.0 t_1 t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / (B * F);
	double t_1 = x / (B * F);
	double tmp;
	if (F <= -0.017) {
		tmp = -1.0 * (F * (t_0 + t_1));
	} else if (F <= 1.3e-23) {
		tmp = -x / B;
	} else {
		tmp = F * fma(-1.0, t_1, t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / Float64(B * F))
	t_1 = Float64(x / Float64(B * F))
	tmp = 0.0
	if (F <= -0.017)
		tmp = Float64(-1.0 * Float64(F * Float64(t_0 + t_1)));
	elseif (F <= 1.3e-23)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(F * fma(-1.0, t_1, t_0));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[(B * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.017], N[(-1.0 * N[(F * N[(t$95$0 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.3e-23], N[((-x) / B), $MachinePrecision], N[(F * N[(-1.0 * t$95$1 + t$95$0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.017000000000000001

   1. Initial program 77.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. sqr-pow N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right)} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right)} \]

      4. lower-unsound-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left(\color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\color{blue}{2 \cdot x} + \left(F \cdot F + 2\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(\mathsf{fma}\left(2, x, F \cdot F + 2\right)\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{F \cdot F + 2}\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{\left(\frac{-\frac{1}{2}}{2}\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\color{blue}{\mathsf{neg}\left(\frac{1}{2}\right)}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      16. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\color{blue}{\frac{-1}{2}}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      17. lower-unsound-pow.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}}\right) \]

   3. Applied rewrites 77.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{4}}\right)}^{2} - x}{B}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)}^{2} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{B} \]

      3. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{\color{blue}{B}} \]

   6. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.25}\right)}^{2} - x}{B}} \]

   8. Applied rewrites 36.1%

      \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \color{blue}{\frac{F}{B}}, \frac{-x}{B}\right) \]

   9. Taylor expanded in F around -inf

      \[\leadsto -1 \cdot \color{blue}{\left(F \cdot \left(\frac{1}{B \cdot F} + \frac{x}{B \cdot F}\right)\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(F \cdot \color{blue}{\left(\frac{1}{B \cdot F} + \frac{x}{B \cdot F}\right)}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{B \cdot F} + \color{blue}{\frac{x}{B \cdot F}}\right)\right) \]

       3. lower-+.f64 N/A

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{B \cdot F} + \frac{x}{\color{blue}{B \cdot F}}\right)\right) \]

       4. lower-/.f64 N/A

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{B \cdot F} + \frac{x}{\color{blue}{B} \cdot F}\right)\right) \]

       5. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{B \cdot F} + \frac{x}{B \cdot F}\right)\right) \]

       6. lower-/.f64 N/A

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{B \cdot F} + \frac{x}{B \cdot \color{blue}{F}}\right)\right) \]

       7. lower-*.f64 27.4

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{B \cdot F} + \frac{x}{B \cdot F}\right)\right) \]

   11. Applied rewrites 27.4%

       \[\leadsto -1 \cdot \color{blue}{\left(F \cdot \left(\frac{1}{B \cdot F} + \frac{x}{B \cdot F}\right)\right)} \]

   if -0.017000000000000001 < F < 1.3e-23

   1. Initial program 77.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. sqr-pow N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right)} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right)} \]

      4. lower-unsound-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left(\color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\color{blue}{2 \cdot x} + \left(F \cdot F + 2\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(\mathsf{fma}\left(2, x, F \cdot F + 2\right)\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{F \cdot F + 2}\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{\left(\frac{-\frac{1}{2}}{2}\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\color{blue}{\mathsf{neg}\left(\frac{1}{2}\right)}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      16. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\color{blue}{\frac{-1}{2}}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      17. lower-unsound-pow.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}}\right) \]

   3. Applied rewrites 77.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{4}}\right)}^{2} - x}{B}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)}^{2} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{B} \]

      3. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{\color{blue}{B}} \]

   6. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.25}\right)}^{2} - x}{B}} \]

   7. Taylor expanded in F around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{x}{B}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]

      2. lower-/.f64 29.7

         \[\leadsto -1 \cdot \frac{x}{B} \]

   9. Applied rewrites 29.7%

      \[\leadsto -1 \cdot \color{blue}{\frac{x}{B}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{B}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{B}\right) \]

       4. distribute-frac-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

       5. lift-neg.f64 N/A

          \[\leadsto \frac{-x}{B} \]

       6. lift-/.f64 29.7

          \[\leadsto \frac{-x}{B} \]

   11. Applied rewrites 29.7%

       \[\leadsto \frac{-x}{\color{blue}{B}} \]

   if 1.3e-23 < F

   1. Initial program 77.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. sqr-pow N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right)} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right)} \]

      4. lower-unsound-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left(\color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\color{blue}{2 \cdot x} + \left(F \cdot F + 2\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(\mathsf{fma}\left(2, x, F \cdot F + 2\right)\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{F \cdot F + 2}\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{\left(\frac{-\frac{1}{2}}{2}\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\color{blue}{\mathsf{neg}\left(\frac{1}{2}\right)}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      16. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\color{blue}{\frac{-1}{2}}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      17. lower-unsound-pow.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}}\right) \]

   3. Applied rewrites 77.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{4}}\right)}^{2} - x}{B}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)}^{2} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{B} \]

      3. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{\color{blue}{B}} \]

   6. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.25}\right)}^{2} - x}{B}} \]

   8. Applied rewrites 36.1%

      \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \color{blue}{\frac{F}{B}}, \frac{-x}{B}\right) \]

   9. Taylor expanded in F around inf

      \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x}{B \cdot F} + \frac{1}{B \cdot F}\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto F \cdot \left(-1 \cdot \frac{x}{B \cdot F} + \color{blue}{\frac{1}{B \cdot F}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{\color{blue}{B \cdot F}}, \frac{1}{B \cdot F}\right) \]

       3. lower-/.f64 N/A

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot \color{blue}{F}}, \frac{1}{B \cdot F}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{B \cdot F}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{B \cdot F}\right) \]

       6. lower-*.f64 25.0

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{B \cdot F}\right) \]

   11. Applied rewrites 25.0%

       \[\leadsto F \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{B \cdot F}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 36.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq 1.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{B \cdot F}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 1.3e-23)
   (/ (- x) B)
   (* F (fma -1.0 (/ x (* B F)) (/ 1.0 (* B F))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 1.3e-23) {
		tmp = -x / B;
	} else {
		tmp = F * fma(-1.0, (x / (B * F)), (1.0 / (B * F)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 1.3e-23)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(F * fma(-1.0, Float64(x / Float64(B * F)), Float64(1.0 / Float64(B * F))));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 1.3e-23], N[((-x) / B), $MachinePrecision], N[(F * N[(-1.0 * N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(B * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.3e-23

   1. Initial program 77.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. sqr-pow N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right)} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right)} \]

      4. lower-unsound-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left(\color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\color{blue}{2 \cdot x} + \left(F \cdot F + 2\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(\mathsf{fma}\left(2, x, F \cdot F + 2\right)\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{F \cdot F + 2}\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{\left(\frac{-\frac{1}{2}}{2}\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\color{blue}{\mathsf{neg}\left(\frac{1}{2}\right)}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      16. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\color{blue}{\frac{-1}{2}}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      17. lower-unsound-pow.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}}\right) \]

   3. Applied rewrites 77.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{4}}\right)}^{2} - x}{B}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)}^{2} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{B} \]

      3. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{\color{blue}{B}} \]

   6. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.25}\right)}^{2} - x}{B}} \]

   7. Taylor expanded in F around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{x}{B}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]

      2. lower-/.f64 29.7

         \[\leadsto -1 \cdot \frac{x}{B} \]

   9. Applied rewrites 29.7%

      \[\leadsto -1 \cdot \color{blue}{\frac{x}{B}} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]

       2. mul-1-neg N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{B}\right) \]

       3. lift-/.f64 N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{B}\right) \]

       4. distribute-frac-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

       5. lift-neg.f64 N/A

          \[\leadsto \frac{-x}{B} \]

       6. lift-/.f64 29.7

          \[\leadsto \frac{-x}{B} \]

   11. Applied rewrites 29.7%

       \[\leadsto \frac{-x}{\color{blue}{B}} \]

   if 1.3e-23 < F

   1. Initial program 77.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. sqr-pow N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right)} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right)} \]

      4. lower-unsound-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left(\color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\color{blue}{2 \cdot x} + \left(F \cdot F + 2\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(\mathsf{fma}\left(2, x, F \cdot F + 2\right)\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{F \cdot F + 2}\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{\left(\frac{-\frac{1}{2}}{2}\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\color{blue}{\mathsf{neg}\left(\frac{1}{2}\right)}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      16. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\color{blue}{\frac{-1}{2}}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      17. lower-unsound-pow.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}}\right) \]

   3. Applied rewrites 77.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{4}}\right)}^{2} - x}{B}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)}^{2} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{B} \]

      3. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{\color{blue}{B}} \]

   6. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.25}\right)}^{2} - x}{B}} \]

   8. Applied rewrites 36.1%

      \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \color{blue}{\frac{F}{B}}, \frac{-x}{B}\right) \]

   9. Taylor expanded in F around inf

      \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x}{B \cdot F} + \frac{1}{B \cdot F}\right)} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto F \cdot \left(-1 \cdot \frac{x}{B \cdot F} + \color{blue}{\frac{1}{B \cdot F}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{\color{blue}{B \cdot F}}, \frac{1}{B \cdot F}\right) \]

       3. lower-/.f64 N/A

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot \color{blue}{F}}, \frac{1}{B \cdot F}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{B \cdot F}\right) \]

       5. lower-/.f64 N/A

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{B \cdot F}\right) \]

       6. lower-*.f64 25.0

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{B \cdot F}\right) \]

   11. Applied rewrites 25.0%

       \[\leadsto F \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{B \cdot F}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 29.7% accurate, 21.7× speedup

Math

\[\frac{-x}{B} \]

FPCore

(FPCore (F B x) :precision binary64 (/ (- x) B))

C

double code(double F, double B, double x) {
	return -x / B;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -x / b
end function

Java

public static double code(double F, double B, double x) {
	return -x / B;
}

Python

def code(F, B, x):
	return -x / B

Julia

function code(F, B, x)
	return Float64(Float64(-x) / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -x / B;
end

Wolfram

code[F_, B_, x_] := N[((-x) / B), $MachinePrecision]

Derivation

1. Initial program 77.2%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

   2. sqr-pow N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right)} \]

   3. lower-unsound-*.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right)} \]

   4. lower-unsound-pow.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left(\color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   5. lift-+.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   6. +-commutative N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\color{blue}{2 \cdot x} + \left(F \cdot F + 2\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(\mathsf{fma}\left(2, x, F \cdot F + 2\right)\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   9. lift-+.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{F \cdot F + 2}\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   12. lower-unsound-/.f64 N/A

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{\left(\frac{-\frac{1}{2}}{2}\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   13. lift-neg.f64 N/A

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\color{blue}{\mathsf{neg}\left(\frac{1}{2}\right)}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   14. lift-/.f64 N/A

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   15. metadata-eval N/A

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   16. metadata-eval N/A

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\color{blue}{\frac{-1}{2}}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   17. lower-unsound-pow.f64 N/A

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}}\right) \]

3. Applied rewrites 77.2%

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \]

4. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{4}}\right)}^{2} - x}{B}} \]
5. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)}^{2} - x}{B} \]

   2. metadata-eval N/A

      \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{B} \]

   3. metadata-eval N/A

      \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{B} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{\color{blue}{B}} \]

6. Applied rewrites 44.3%

   \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.25}\right)}^{2} - x}{B}} \]

7. Taylor expanded in F around 0

   \[\leadsto -1 \cdot \color{blue}{\frac{x}{B}} \]
8. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]

   2. lower-/.f64 29.7

      \[\leadsto -1 \cdot \frac{x}{B} \]

9. Applied rewrites 29.7%

   \[\leadsto -1 \cdot \color{blue}{\frac{x}{B}} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]

    2. mul-1-neg N/A

       \[\leadsto \mathsf{neg}\left(\frac{x}{B}\right) \]

    3. lift-/.f64 N/A

       \[\leadsto \mathsf{neg}\left(\frac{x}{B}\right) \]

    4. distribute-frac-neg N/A

       \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

    5. lift-neg.f64 N/A

       \[\leadsto \frac{-x}{B} \]

    6. lift-/.f64 29.7

       \[\leadsto \frac{-x}{B} \]

11. Applied rewrites 29.7%

    \[\leadsto \frac{-x}{\color{blue}{B}} \]
12. Add Preprocessing

Alternative 22: 17.5% accurate, 14.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq 1100:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (if (<= F 1100.0) (/ -1.0 B) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 1100.0) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= 1100.0d0) then
        tmp = (-1.0d0) / b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 1100.0) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 1100.0:
		tmp = -1.0 / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 1100.0)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 1100.0)
		tmp = -1.0 / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 1100.0], N[(-1.0 / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1100

   1. Initial program 77.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. 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%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-1}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 10.4%

      \[\leadsto \frac{-1}{B} \]

   if 1100 < F

   1. Initial program 77.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. sqr-pow N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right)} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right)} \]

      4. lower-unsound-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left(\color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      5. lift-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      6. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\color{blue}{2 \cdot x} + \left(F \cdot F + 2\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(\mathsf{fma}\left(2, x, F \cdot F + 2\right)\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      9. lift-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{F \cdot F + 2}\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      10. lift-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      12. lower-unsound-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{\left(\frac{-\frac{1}{2}}{2}\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      13. lift-neg.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\color{blue}{\mathsf{neg}\left(\frac{1}{2}\right)}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      14. lift-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      15. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      16. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\color{blue}{\frac{-1}{2}}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

      17. lower-unsound-pow.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}}\right) \]

   3. Applied rewrites 77.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{4}}\right)}^{2} - x}{B}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)}^{2} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{B} \]

      3. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{\color{blue}{B}} \]

   6. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.25}\right)}^{2} - x}{B}} \]

   8. Applied rewrites 36.1%

      \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \color{blue}{\frac{F}{B}}, \frac{-x}{B}\right) \]

   9. Taylor expanded in F around inf

      \[\leadsto \frac{1}{\color{blue}{B}} \]
   10. Step-by-step derivation

       1. lower-/.f64 10.0

          \[\leadsto \frac{1}{B} \]

   11. Applied rewrites 10.0%

       \[\leadsto \frac{1}{\color{blue}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 10.0% 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 77.2%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

   2. sqr-pow N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right)} \]

   3. lower-unsound-*.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right)} \]

   4. lower-unsound-pow.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left(\color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   5. lift-+.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   6. +-commutative N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\color{blue}{2 \cdot x} + \left(F \cdot F + 2\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\color{blue}{\left(\mathsf{fma}\left(2, x, F \cdot F + 2\right)\right)}}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   9. lift-+.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{F \cdot F + 2}\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   10. lift-*.f64 N/A

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   12. lower-unsound-/.f64 N/A

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{\left(\frac{-\frac{1}{2}}{2}\right)}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   13. lift-neg.f64 N/A

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\color{blue}{\mathsf{neg}\left(\frac{1}{2}\right)}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   14. lift-/.f64 N/A

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   15. metadata-eval N/A

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   16. metadata-eval N/A

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\color{blue}{\frac{-1}{2}}}{2}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}\right) \]

   17. lower-unsound-pow.f64 N/A

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-\frac{1}{2}}{2}\right)}}\right) \]

3. Applied rewrites 77.2%

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{-0.5}{2}\right)} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\left(\frac{-0.5}{2}\right)}\right)} \]

4. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{4}}\right)}^{2} - x}{B}} \]
5. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\frac{-1}{2}}{2}\right)}\right)}^{2} - x}{B} \]

   2. metadata-eval N/A

      \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{B} \]

   3. metadata-eval N/A

      \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{B} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{F \cdot {\left({\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\frac{\mathsf{neg}\left(\frac{1}{2}\right)}{2}\right)}\right)}^{2} - x}{\color{blue}{B}} \]

6. Applied rewrites 44.3%

   \[\leadsto \color{blue}{\frac{F \cdot {\left({\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.25}\right)}^{2} - x}{B}} \]

8. Applied rewrites 36.1%

   \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \color{blue}{\frac{F}{B}}, \frac{-x}{B}\right) \]

9. Taylor expanded in F around inf

   \[\leadsto \frac{1}{\color{blue}{B}} \]
10. Step-by-step derivation

    1. lower-/.f64 10.0

       \[\leadsto \frac{1}{B} \]

11. Applied rewrites 10.0%

    \[\leadsto \frac{1}{\color{blue}{B}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025175 
(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))))))