VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.0% → 98.9%

Time: 7.9s

Alternatives: 25

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.25}\\ t_1 := \cos B \cdot x\\ \mathbf{if}\;F \leq -120000:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{fma}\left(2, x, 2\right)}{\left(F \cdot F\right) \cdot \sin B} - \frac{1 + t\_1}{\sin B}\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{-8}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \left(t\_0 \cdot t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (fma 2.0 x (fma F F 2.0)) -0.25)) (t_1 (* (cos B) x)))
   (if (<= F -120000.0)
     (-
      (/ (* 0.5 (fma 2.0 x 2.0)) (* (* F F) (sin B)))
      (/ (+ 1.0 t_1) (sin B)))
     (if (<= F 1.95e-8)
       (+ (- (* x (/ 1.0 (tan B)))) (* (/ F (sin B)) (* t_0 t_0)))
       (/ (- 1.0 t_1) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = pow(fma(2.0, x, fma(F, F, 2.0)), -0.25);
	double t_1 = cos(B) * x;
	double tmp;
	if (F <= -120000.0) {
		tmp = ((0.5 * fma(2.0, x, 2.0)) / ((F * F) * sin(B))) - ((1.0 + t_1) / sin(B));
	} else if (F <= 1.95e-8) {
		tmp = -(x * (1.0 / tan(B))) + ((F / sin(B)) * (t_0 * t_0));
	} else {
		tmp = (1.0 - t_1) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(2.0, x, fma(F, F, 2.0)) ^ -0.25
	t_1 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -120000.0)
		tmp = Float64(Float64(Float64(0.5 * fma(2.0, x, 2.0)) / Float64(Float64(F * F) * sin(B))) - Float64(Float64(1.0 + t_1) / sin(B)));
	elseif (F <= 1.95e-8)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / sin(B)) * Float64(t_0 * t_0)));
	else
		tmp = Float64(Float64(1.0 - t_1) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.25], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -120000.0], N[(N[(N[(0.5 * N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(F * F), $MachinePrecision] * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + t$95$1), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.95e-8], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$1), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.2e5

   1. Initial program 59.2%

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

   3. Taylor expanded in F around -inf

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

      1. metadata-eval N/A

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

      2. lower--.f64 N/A

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

   5. Applied rewrites 98.7%

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

   if -1.2e5 < F < 1.94999999999999992e-8

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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. lift-+.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. sqr-pow N/A

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

      9. lower-*.f64 N/A

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

   4. Applied rewrites 99.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.25} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.25}\right)} \]

   if 1.94999999999999992e-8 < F

   1. Initial program 59.6%

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

   3. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 98.4

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

   5. Applied rewrites 98.4%

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

Alternative 2: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -120000:\\ \;\;\;\;\frac{0.5 \cdot \mathsf{fma}\left(2, x, 2\right)}{\left(F \cdot F\right) \cdot \sin B} - \frac{1 + t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{-8}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -120000.0)
     (-
      (/ (* 0.5 (fma 2.0 x 2.0)) (* (* F F) (sin B)))
      (/ (+ 1.0 t_0) (sin B)))
     (if (<= F 1.95e-8)
       (+
        (- (* x (/ 1.0 (tan B))))
        (* (/ 1.0 (sqrt (fma 2.0 x (fma F F 2.0)))) (/ F (sin B))))
       (/ (- 1.0 t_0) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -120000.0) {
		tmp = ((0.5 * fma(2.0, x, 2.0)) / ((F * F) * sin(B))) - ((1.0 + t_0) / sin(B));
	} else if (F <= 1.95e-8) {
		tmp = -(x * (1.0 / tan(B))) + ((1.0 / sqrt(fma(2.0, x, fma(F, F, 2.0)))) * (F / sin(B)));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -120000.0)
		tmp = Float64(Float64(Float64(0.5 * fma(2.0, x, 2.0)) / Float64(Float64(F * F) * sin(B))) - Float64(Float64(1.0 + t_0) / sin(B)));
	elseif (F <= 1.95e-8)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(1.0 / sqrt(fma(2.0, x, fma(F, F, 2.0)))) * Float64(F / sin(B))));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -120000.0], N[(N[(N[(0.5 * N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(F * F), $MachinePrecision] * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 + t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.95e-8], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(1.0 / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.2e5

   1. Initial program 59.2%

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

   3. Taylor expanded in F around -inf

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

      1. metadata-eval N/A

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

      2. lower--.f64 N/A

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

   5. Applied rewrites 98.7%

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

   if -1.2e5 < F < 1.94999999999999992e-8

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 99.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. associate-+r+ N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. pow2 N/A

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

      10. +-commutative N/A

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

      11. inv-pow N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-sqrt.f64 N/A

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

   6. Applied rewrites 99.4%

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

   if 1.94999999999999992e-8 < F

   1. Initial program 59.6%

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

   3. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 98.4

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

   5. Applied rewrites 98.4%

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

Alternative 3: 99.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{+33}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{-8}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot 1}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4e+33)
   (/ (- (fma (cos B) x 1.0)) (sin B))
   (if (<= F 1.95e-8)
     (+
      (- (* x (/ 1.0 (tan B))))
      (/ (* F 1.0) (* (sin B) (sqrt (fma F F (fma 2.0 x 2.0))))))
     (/ (- 1.0 (* (cos B) x)) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4e+33) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else if (F <= 1.95e-8) {
		tmp = -(x * (1.0 / tan(B))) + ((F * 1.0) / (sin(B) * sqrt(fma(F, F, fma(2.0, x, 2.0)))));
	} else {
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4e+33)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	elseif (F <= 1.95e-8)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F * 1.0) / Float64(sin(B) * sqrt(fma(F, F, fma(2.0, x, 2.0))))));
	else
		tmp = Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4e+33], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.95e-8], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F * 1.0), $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.9999999999999998e33

   1. Initial program 55.9%

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

   3. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. metadata-eval 99.7

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

      2. metadata-eval 99.7

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

      3. associate-*l/ 99.7

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

      4. +-commutative 99.7

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

      5. lift-neg.f64 N/A

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

      6. lift-/.f64 N/A

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

   7. Applied rewrites 99.7%

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

   if -3.9999999999999998e33 < F < 1.94999999999999992e-8

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. associate-+r+ N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. pow2 N/A

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

      10. +-commutative N/A

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

      11. inv-pow N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-sqrt.f64 N/A

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

   6. Applied rewrites 99.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sqrt.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-sin.f64 N/A

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

      8. *-commutative N/A

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

      9. pow2 N/A

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

      10. associate-+r+ N/A

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

      11. +-commutative N/A

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

      12. pow2 N/A

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

      13. frac-times N/A

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

      14. lower-/.f64 N/A

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

   8. Applied rewrites 99.5%

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

   if 1.94999999999999992e-8 < F

   1. Initial program 59.6%

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

   3. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 98.4

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

   5. Applied rewrites 98.4%

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

Alternative 4: 91.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\\ \mathbf{if}\;F \leq -4 \cdot 10^{+20}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{-132}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sqrt{t\_0}} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{-8}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + F \cdot \frac{{t\_0}^{-0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (fma 2.0 x (fma F F 2.0))))
   (if (<= F -4e+20)
     (/ (- (fma (cos B) x 1.0)) (sin B))
     (if (<= F 1.25e-132)
       (+ (- (* x (/ 1.0 (tan B)))) (* (/ 1.0 (sqrt t_0)) (/ F B)))
       (if (<= F 1.95e-8)
         (+ (- (/ x B)) (* F (/ (pow t_0 -0.5) (sin B))))
         (/ (- 1.0 (* (cos B) x)) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = fma(2.0, x, fma(F, F, 2.0));
	double tmp;
	if (F <= -4e+20) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else if (F <= 1.25e-132) {
		tmp = -(x * (1.0 / tan(B))) + ((1.0 / sqrt(t_0)) * (F / B));
	} else if (F <= 1.95e-8) {
		tmp = -(x / B) + (F * (pow(t_0, -0.5) / sin(B)));
	} else {
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(2.0, x, fma(F, F, 2.0))
	tmp = 0.0
	if (F <= -4e+20)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	elseif (F <= 1.25e-132)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(1.0 / sqrt(t_0)) * Float64(F / B)));
	elseif (F <= 1.95e-8)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(F * Float64((t_0 ^ -0.5) / sin(B))));
	else
		tmp = Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4e+20], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.25e-132], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(1.0 / N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.95e-8], N[((-N[(x / B), $MachinePrecision]) + N[(F * N[(N[Power[t$95$0, -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4e20

   1. Initial program 57.6%

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

   3. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. metadata-eval 99.7

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

      2. metadata-eval 99.7

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

      3. associate-*l/ 99.7

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

      4. +-commutative 99.7

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

      5. lift-neg.f64 N/A

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

      6. lift-/.f64 N/A

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

   7. Applied rewrites 99.7%

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

   if -4e20 < F < 1.25e-132

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. associate-+r+ N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. pow2 N/A

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

      10. +-commutative N/A

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

      11. inv-pow N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-sqrt.f64 N/A

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in B around 0

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

   9. Applied rewrites 84.6%

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

   if 1.25e-132 < F < 1.94999999999999992e-8

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. pow2 N/A

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

      10. associate-+r+ N/A

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

      11. pow2 N/A

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

      12. metadata-eval N/A

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

      13. sqrt-pow1 N/A

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

      14. lower-/.f64 N/A

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 77.6

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

   9. Applied rewrites 77.6%

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

   if 1.94999999999999992e-8 < F

   1. Initial program 59.6%

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

   3. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 98.4

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

   5. Applied rewrites 98.4%

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

Alternative 5: 91.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{if}\;F \leq -4 \cdot 10^{+20}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-132}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + t\_0 \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{-8}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + t\_0 \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt (fma 2.0 x (fma F F 2.0))))))
   (if (<= F -4e+20)
     (/ (- (fma (cos B) x 1.0)) (sin B))
     (if (<= F 1.1e-132)
       (+ (- (* x (/ 1.0 (tan B)))) (* t_0 (/ F B)))
       (if (<= F 1.95e-8)
         (+ (- (/ x B)) (* t_0 (/ F (sin B))))
         (/ (- 1.0 (* (cos B) x)) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sqrt(fma(2.0, x, fma(F, F, 2.0)));
	double tmp;
	if (F <= -4e+20) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else if (F <= 1.1e-132) {
		tmp = -(x * (1.0 / tan(B))) + (t_0 * (F / B));
	} else if (F <= 1.95e-8) {
		tmp = -(x / B) + (t_0 * (F / sin(B)));
	} else {
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sqrt(fma(2.0, x, fma(F, F, 2.0))))
	tmp = 0.0
	if (F <= -4e+20)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	elseif (F <= 1.1e-132)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(t_0 * Float64(F / B)));
	elseif (F <= 1.95e-8)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(t_0 * Float64(F / sin(B))));
	else
		tmp = Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4e+20], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.1e-132], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.95e-8], N[((-N[(x / B), $MachinePrecision]) + N[(t$95$0 * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4e20

   1. Initial program 57.6%

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

   3. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.7

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

   5. Applied rewrites 99.7%

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

      1. metadata-eval 99.7

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

      2. metadata-eval 99.7

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

      3. associate-*l/ 99.7

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

      4. +-commutative 99.7

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

      5. lift-neg.f64 N/A

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

      6. lift-/.f64 N/A

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

   7. Applied rewrites 99.7%

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

   if -4e20 < F < 1.09999999999999995e-132

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. associate-+r+ N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. pow2 N/A

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

      10. +-commutative N/A

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

      11. inv-pow N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-sqrt.f64 N/A

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in B around 0

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

   9. Applied rewrites 84.6%

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

   if 1.09999999999999995e-132 < F < 1.94999999999999992e-8

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 99.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. associate-+r+ N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. pow2 N/A

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

      10. +-commutative N/A

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

      11. inv-pow N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-sqrt.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 77.5

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

   9. Applied rewrites 77.5%

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

   if 1.94999999999999992e-8 < F

   1. Initial program 59.6%

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

   3. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 98.4

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

   5. Applied rewrites 98.4%

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

Alternative 6: 84.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ t_1 := -x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -5 \cdot 10^{+21}:\\ \;\;\;\;t\_1 + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-132}:\\ \;\;\;\;t\_1 + t\_0 \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{-8}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + t\_0 \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt (fma 2.0 x (fma F F 2.0)))))
        (t_1 (- (* x (/ 1.0 (tan B))))))
   (if (<= F -5e+21)
     (+ t_1 (/ -1.0 B))
     (if (<= F 1.1e-132)
       (+ t_1 (* t_0 (/ F B)))
       (if (<= F 1.95e-8)
         (+ (- (/ x B)) (* t_0 (/ F (sin B))))
         (/ (- 1.0 (* (cos B) x)) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sqrt(fma(2.0, x, fma(F, F, 2.0)));
	double t_1 = -(x * (1.0 / tan(B)));
	double tmp;
	if (F <= -5e+21) {
		tmp = t_1 + (-1.0 / B);
	} else if (F <= 1.1e-132) {
		tmp = t_1 + (t_0 * (F / B));
	} else if (F <= 1.95e-8) {
		tmp = -(x / B) + (t_0 * (F / sin(B)));
	} else {
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sqrt(fma(2.0, x, fma(F, F, 2.0))))
	t_1 = Float64(-Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (F <= -5e+21)
		tmp = Float64(t_1 + Float64(-1.0 / B));
	elseif (F <= 1.1e-132)
		tmp = Float64(t_1 + Float64(t_0 * Float64(F / B)));
	elseif (F <= 1.95e-8)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(t_0 * Float64(F / sin(B))));
	else
		tmp = Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[F, -5e+21], N[(t$95$1 + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.1e-132], N[(t$95$1 + N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.95e-8], N[((-N[(x / B), $MachinePrecision]) + N[(t$95$0 * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -5e21

   1. Initial program 57.5%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.8

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in F around -inf

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

      1. lower-/.f64 73.3

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

   8. Applied rewrites 73.3%

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

   if -5e21 < F < 1.09999999999999995e-132

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. associate-+r+ N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. pow2 N/A

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

      10. +-commutative N/A

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

      11. inv-pow N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-sqrt.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in B around 0

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

   9. Applied rewrites 84.5%

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

   if 1.09999999999999995e-132 < F < 1.94999999999999992e-8

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 99.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. associate-+r+ N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. pow2 N/A

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

      10. +-commutative N/A

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

      11. inv-pow N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-sqrt.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 77.5

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

   9. Applied rewrites 77.5%

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

   if 1.94999999999999992e-8 < F

   1. Initial program 59.6%

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

   3. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 98.4

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

   5. Applied rewrites 98.4%

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

Alternative 7: 73.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ t_1 := -x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -5 \cdot 10^{+21}:\\ \;\;\;\;t\_1 + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-132}:\\ \;\;\;\;t\_1 + t\_0 \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 2.65 \cdot 10^{+137}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + t\_0 \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt (fma 2.0 x (fma F F 2.0)))))
        (t_1 (- (* x (/ 1.0 (tan B))))))
   (if (<= F -5e+21)
     (+ t_1 (/ -1.0 B))
     (if (<= F 1.1e-132)
       (+ t_1 (* t_0 (/ F B)))
       (if (<= F 2.65e+137)
         (+ (- (/ x B)) (* t_0 (/ F (sin B))))
         (- (/ (* (cos B) x) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sqrt(fma(2.0, x, fma(F, F, 2.0)));
	double t_1 = -(x * (1.0 / tan(B)));
	double tmp;
	if (F <= -5e+21) {
		tmp = t_1 + (-1.0 / B);
	} else if (F <= 1.1e-132) {
		tmp = t_1 + (t_0 * (F / B));
	} else if (F <= 2.65e+137) {
		tmp = -(x / B) + (t_0 * (F / sin(B)));
	} else {
		tmp = -((cos(B) * x) / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sqrt(fma(2.0, x, fma(F, F, 2.0))))
	t_1 = Float64(-Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (F <= -5e+21)
		tmp = Float64(t_1 + Float64(-1.0 / B));
	elseif (F <= 1.1e-132)
		tmp = Float64(t_1 + Float64(t_0 * Float64(F / B)));
	elseif (F <= 2.65e+137)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(t_0 * Float64(F / sin(B))));
	else
		tmp = Float64(-Float64(Float64(cos(B) * x) / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[F, -5e+21], N[(t$95$1 + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.1e-132], N[(t$95$1 + N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.65e+137], N[((-N[(x / B), $MachinePrecision]) + N[(t$95$0 * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision])]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -5e21

   1. Initial program 57.5%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 44.8

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

   5. Applied rewrites 44.8%

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

   6. Taylor expanded in F around -inf

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

      1. lower-/.f64 73.3

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

   8. Applied rewrites 73.3%

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

   if -5e21 < F < 1.09999999999999995e-132

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. associate-+r+ N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. pow2 N/A

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

      10. +-commutative N/A

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

      11. inv-pow N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-sqrt.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in B around 0

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

   9. Applied rewrites 84.5%

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

   if 1.09999999999999995e-132 < F < 2.64999999999999984e137

   1. Initial program 94.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 94.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. associate-+r+ N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. pow2 N/A

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

      10. +-commutative N/A

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

      11. inv-pow N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-sqrt.f64 N/A

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

   6. Applied rewrites 94.0%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 74.0

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

   9. Applied rewrites 74.0%

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

   if 2.64999999999999984e137 < F

   1. Initial program 37.1%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 50.4

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

   5. Applied rewrites 50.4%

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

Alternative 8: 73.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ t_1 := -x \cdot \frac{1}{\tan B}\\ t_2 := t\_1 + \frac{-1}{B}\\ \mathbf{if}\;F \leq -5 \cdot 10^{+21}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-132}:\\ \;\;\;\;t\_1 + t\_0 \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 2.65 \cdot 10^{+137}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + t\_0 \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{+185}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt (fma 2.0 x (fma F F 2.0)))))
        (t_1 (- (* x (/ 1.0 (tan B)))))
        (t_2 (+ t_1 (/ -1.0 B))))
   (if (<= F -5e+21)
     t_2
     (if (<= F 1.1e-132)
       (+ t_1 (* t_0 (/ F B)))
       (if (<= F 2.65e+137)
         (+ (- (/ x B)) (* t_0 (/ F (sin B))))
         (if (<= F 1.4e+185) t_2 (/ (- 1.0 x) B)))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sqrt(fma(2.0, x, fma(F, F, 2.0)));
	double t_1 = -(x * (1.0 / tan(B)));
	double t_2 = t_1 + (-1.0 / B);
	double tmp;
	if (F <= -5e+21) {
		tmp = t_2;
	} else if (F <= 1.1e-132) {
		tmp = t_1 + (t_0 * (F / B));
	} else if (F <= 2.65e+137) {
		tmp = -(x / B) + (t_0 * (F / sin(B)));
	} else if (F <= 1.4e+185) {
		tmp = t_2;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sqrt(fma(2.0, x, fma(F, F, 2.0))))
	t_1 = Float64(-Float64(x * Float64(1.0 / tan(B))))
	t_2 = Float64(t_1 + Float64(-1.0 / B))
	tmp = 0.0
	if (F <= -5e+21)
		tmp = t_2;
	elseif (F <= 1.1e-132)
		tmp = Float64(t_1 + Float64(t_0 * Float64(F / B)));
	elseif (F <= 2.65e+137)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(t_0 * Float64(F / sin(B))));
	elseif (F <= 1.4e+185)
		tmp = t_2;
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$2 = N[(t$95$1 + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5e+21], t$95$2, If[LessEqual[F, 1.1e-132], N[(t$95$1 + N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.65e+137], N[((-N[(x / B), $MachinePrecision]) + N[(t$95$0 * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4e+185], t$95$2, N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -5e21 or 2.64999999999999984e137 < F < 1.39999999999999991e185

   1. Initial program 57.1%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 45.2

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

   5. Applied rewrites 45.2%

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

   6. Taylor expanded in F around -inf

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

      1. lower-/.f64 70.3

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

   8. Applied rewrites 70.3%

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

   if -5e21 < F < 1.09999999999999995e-132

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. associate-+r+ N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. pow2 N/A

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

      10. +-commutative N/A

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

      11. inv-pow N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-sqrt.f64 N/A

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in B around 0

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

   9. Applied rewrites 84.5%

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

   if 1.09999999999999995e-132 < F < 2.64999999999999984e137

   1. Initial program 94.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 94.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. associate-+r+ N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. pow2 N/A

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

      10. +-commutative N/A

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

      11. inv-pow N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-sqrt.f64 N/A

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

   6. Applied rewrites 94.0%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 74.0

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

   9. Applied rewrites 74.0%

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

   if 1.39999999999999991e185 < F

   1. Initial program 29.8%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 25.6%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 49.2%

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

Alternative 9: 73.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -x \cdot \frac{1}{\tan B}\\ t_1 := t\_0 + \frac{-1}{B}\\ t_2 := \mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\\ \mathbf{if}\;F \leq -4 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-132}:\\ \;\;\;\;t\_0 + \frac{F}{B} \cdot \sqrt{\frac{1}{t\_2}}\\ \mathbf{elif}\;F \leq 2.65 \cdot 10^{+137}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{1}{\sqrt{t\_2}} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{+185}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* x (/ 1.0 (tan B)))))
        (t_1 (+ t_0 (/ -1.0 B)))
        (t_2 (fma 2.0 x (fma F F 2.0))))
   (if (<= F -4e+20)
     t_1
     (if (<= F 1.1e-132)
       (+ t_0 (* (/ F B) (sqrt (/ 1.0 t_2))))
       (if (<= F 2.65e+137)
         (+ (- (/ x B)) (* (/ 1.0 (sqrt t_2)) (/ F (sin B))))
         (if (<= F 1.4e+185) t_1 (/ (- 1.0 x) B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B)));
	double t_1 = t_0 + (-1.0 / B);
	double t_2 = fma(2.0, x, fma(F, F, 2.0));
	double tmp;
	if (F <= -4e+20) {
		tmp = t_1;
	} else if (F <= 1.1e-132) {
		tmp = t_0 + ((F / B) * sqrt((1.0 / t_2)));
	} else if (F <= 2.65e+137) {
		tmp = -(x / B) + ((1.0 / sqrt(t_2)) * (F / sin(B)));
	} else if (F <= 1.4e+185) {
		tmp = t_1;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x * Float64(1.0 / tan(B))))
	t_1 = Float64(t_0 + Float64(-1.0 / B))
	t_2 = fma(2.0, x, fma(F, F, 2.0))
	tmp = 0.0
	if (F <= -4e+20)
		tmp = t_1;
	elseif (F <= 1.1e-132)
		tmp = Float64(t_0 + Float64(Float64(F / B) * sqrt(Float64(1.0 / t_2))));
	elseif (F <= 2.65e+137)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(1.0 / sqrt(t_2)) * Float64(F / sin(B))));
	elseif (F <= 1.4e+185)
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$1 = N[(t$95$0 + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4e+20], t$95$1, If[LessEqual[F, 1.1e-132], N[(t$95$0 + N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.65e+137], N[((-N[(x / B), $MachinePrecision]) + N[(N[(1.0 / N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4e+185], t$95$1, N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4e20 or 2.64999999999999984e137 < F < 1.39999999999999991e185

   1. Initial program 57.2%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 45.3

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in F around -inf

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

      1. lower-/.f64 70.3

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

   8. Applied rewrites 70.3%

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

   if -4e20 < F < 1.09999999999999995e-132

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 84.6

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

   5. Applied rewrites 84.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. unpow-1 N/A

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

      6. lower-/.f64 N/A

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

      7. pow2 N/A

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

      8. associate-+r+ N/A

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

      9. pow2 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-fma.f64 84.6

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

   7. Applied rewrites 84.6%

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

   if 1.09999999999999995e-132 < F < 2.64999999999999984e137

   1. Initial program 94.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 94.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. associate-+r+ N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. pow2 N/A

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

      10. +-commutative N/A

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

      11. inv-pow N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-sqrt.f64 N/A

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

   6. Applied rewrites 94.0%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 74.0

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

   9. Applied rewrites 74.0%

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

   if 1.39999999999999991e185 < F

   1. Initial program 29.8%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 25.6%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 49.2%

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

Alternative 10: 74.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B}\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{-35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1000000000:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ (- (* x (/ 1.0 (tan B)))) (/ -1.0 B))))
   (if (<= x -1.1e-35)
     t_0
     (if (<= x 1000000000.0)
       (+
        (- (/ x B))
        (* (/ 1.0 (sqrt (fma 2.0 x (fma F F 2.0)))) (/ F (sin B))))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B))) + (-1.0 / B);
	double tmp;
	if (x <= -1.1e-35) {
		tmp = t_0;
	} else if (x <= 1000000000.0) {
		tmp = -(x / B) + ((1.0 / sqrt(fma(2.0, x, fma(F, F, 2.0)))) * (F / sin(B)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(-1.0 / B))
	tmp = 0.0
	if (x <= -1.1e-35)
		tmp = t_0;
	elseif (x <= 1000000000.0)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(1.0 / sqrt(fma(2.0, x, fma(F, F, 2.0)))) * Float64(F / sin(B))));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.1e-35], t$95$0, If[LessEqual[x, 1000000000.0], N[((-N[(x / B), $MachinePrecision]) + N[(N[(1.0 / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.09999999999999997e-35 or 1e9 < x

   1. Initial program 83.2%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 82.3

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

   5. Applied rewrites 82.3%

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

   6. Taylor expanded in F around -inf

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

      1. lower-/.f64 94.2

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

   8. Applied rewrites 94.2%

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

   if -1.09999999999999997e-35 < x < 1e9

   1. Initial program 72.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

   4. Applied rewrites 72.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. associate-+r+ N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. pow2 N/A

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

      10. +-commutative N/A

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

      11. inv-pow N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. pow2 N/A

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

      17. lower-sqrt.f64 N/A

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

   6. Applied rewrites 72.2%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 59.9

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

   9. Applied rewrites 59.9%

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

Alternative 11: 55.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}}\\ \mathbf{if}\;B \leq 0.0064:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, t\_0, \left(B \cdot B\right) \cdot \mathsf{fma}\left(0.16666666666666666 \cdot F, t\_0, 0.3333333333333333 \cdot x\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt (+ 2.0 (fma 2.0 x (* F F)))))))
   (if (<= B 0.0064)
     (/
      (-
       (fma
        F
        t_0
        (*
         (* B B)
         (fma (* 0.16666666666666666 F) t_0 (* 0.3333333333333333 x))))
       x)
      B)
     (+ (- (* x (/ 1.0 (tan B)))) (/ -1.0 B)))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sqrt((2.0 + fma(2.0, x, (F * F))));
	double tmp;
	if (B <= 0.0064) {
		tmp = (fma(F, t_0, ((B * B) * fma((0.16666666666666666 * F), t_0, (0.3333333333333333 * x)))) - x) / B;
	} else {
		tmp = -(x * (1.0 / tan(B))) + (-1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sqrt(Float64(2.0 + fma(2.0, x, Float64(F * F)))))
	tmp = 0.0
	if (B <= 0.0064)
		tmp = Float64(Float64(fma(F, t_0, Float64(Float64(B * B) * fma(Float64(0.16666666666666666 * F), t_0, Float64(0.3333333333333333 * x)))) - x) / B);
	else
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(-1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[N[(2.0 + N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 0.0064], N[(N[(N[(F * t$95$0 + N[(N[(B * B), $MachinePrecision] * N[(N[(0.16666666666666666 * F), $MachinePrecision] * t$95$0 + N[(0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 0.00640000000000000031

   1. Initial program 74.3%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 57.2%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

      2. lower-neg.f64 38.5

         \[\leadsto \frac{-x}{B} \]

   8. Applied rewrites 38.5%

      \[\leadsto \frac{-x}{B} \]

   9. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{\color{blue}{B}} \]

   11. Applied rewrites 57.3%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}}, \left(B \cdot B\right) \cdot \mathsf{fma}\left(0.16666666666666666 \cdot F, \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}}, 0.3333333333333333 \cdot x\right)\right) - x}{B}} \]

   if 0.00640000000000000031 < B

   1. Initial program 84.9%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 55.8

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

   5. Applied rewrites 55.8%

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

   6. Taylor expanded in F around -inf

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

      1. lower-/.f64 50.6

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

   8. Applied rewrites 50.6%

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

Alternative 12: 57.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}}\\ \mathbf{if}\;F \leq -4 \cdot 10^{+20}:\\ \;\;\;\;-\frac{1 + x}{\sin B}\\ \mathbf{elif}\;F \leq 4 \cdot 10^{+86}:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, t\_0, \left(B \cdot B\right) \cdot \mathsf{fma}\left(0.16666666666666666 \cdot F, t\_0, 0.3333333333333333 \cdot x\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt (+ 2.0 (fma 2.0 x (* F F)))))))
   (if (<= F -4e+20)
     (- (/ (+ 1.0 x) (sin B)))
     (if (<= F 4e+86)
       (/
        (-
         (fma
          F
          t_0
          (*
           (* B B)
           (fma (* 0.16666666666666666 F) t_0 (* 0.3333333333333333 x))))
         x)
        B)
       (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sqrt((2.0 + fma(2.0, x, (F * F))));
	double tmp;
	if (F <= -4e+20) {
		tmp = -((1.0 + x) / sin(B));
	} else if (F <= 4e+86) {
		tmp = (fma(F, t_0, ((B * B) * fma((0.16666666666666666 * F), t_0, (0.3333333333333333 * x)))) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sqrt(Float64(2.0 + fma(2.0, x, Float64(F * F)))))
	tmp = 0.0
	if (F <= -4e+20)
		tmp = Float64(-Float64(Float64(1.0 + x) / sin(B)));
	elseif (F <= 4e+86)
		tmp = Float64(Float64(fma(F, t_0, Float64(Float64(B * B) * fma(Float64(0.16666666666666666 * F), t_0, Float64(0.3333333333333333 * x)))) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[N[(2.0 + N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4e+20], (-N[(N[(1.0 + x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 4e+86], N[(N[(N[(F * t$95$0 + N[(N[(B * B), $MachinePrecision] * N[(N[(0.16666666666666666 * F), $MachinePrecision] * t$95$0 + N[(0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4e20

   1. Initial program 57.6%

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

   3. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      7. lower-*.f64 N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      8. lower-cos.f64 N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      9. lift-sin.f64 99.7

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{-\frac{1 + \cos B \cdot x}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto -\frac{1 + x}{\sin B} \]
   7. Step-by-step derivation

   8. Applied rewrites 77.9%

      \[\leadsto -\frac{1 + x}{\sin B} \]

   if -4e20 < F < 4.0000000000000001e86

   1. Initial program 98.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

      2. lower-neg.f64 33.0

         \[\leadsto \frac{-x}{B} \]

   8. Applied rewrites 33.0%

      \[\leadsto \frac{-x}{B} \]

   9. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{\color{blue}{B}} \]

   11. Applied rewrites 50.8%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(F, \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}}, \left(B \cdot B\right) \cdot \mathsf{fma}\left(0.16666666666666666 \cdot F, \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}}, 0.3333333333333333 \cdot x\right)\right) - x}{B}} \]

   if 4.0000000000000001e86 < F

   1. Initial program 47.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 33.0%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 49.4%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 50.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.2 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(B \cdot B\right) \cdot \left(\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(-0.16666666666666666, x \cdot -0.3333333333333333, 0.008333333333333333 \cdot x\right) - \mathsf{fma}\left(0.041666666666666664, x, 0.019444444444444445\right), -0.16666666666666666 \cdot x\right) - \mathsf{fma}\left(-0.5, x, 0.16666666666666666\right)\right) - \left(1 + x\right)}{B}\\ \mathbf{elif}\;F \leq -9 \cdot 10^{+22}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{+140}:\\ \;\;\;\;\left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(-0.0021164021164021165 \cdot \left(B \cdot B\right) - 0.022222222222222223\right) - 0.3333333333333333, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.2e+99)
   (/
    (-
     (*
      (* B B)
      (-
       (fma
        (* B B)
        (-
         (fma
          -0.16666666666666666
          (* x -0.3333333333333333)
          (* 0.008333333333333333 x))
         (fma 0.041666666666666664 x 0.019444444444444445))
        (* -0.16666666666666666 x))
       (fma -0.5 x 0.16666666666666666)))
     (+ 1.0 x))
    B)
   (if (<= F -9e+22)
     (/ -1.0 (sin B))
     (if (<= F 1.8e+140)
       (+
        (-
         (*
          x
          (/
           (fma
            (* B B)
            (-
             (*
              (* B B)
              (- (* -0.0021164021164021165 (* B B)) 0.022222222222222223))
             0.3333333333333333)
            1.0)
           B)))
        (/ (* F 1.0) (* B (sqrt (fma F F (fma 2.0 x 2.0))))))
       (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.2e+99) {
		tmp = (((B * B) * (fma((B * B), (fma(-0.16666666666666666, (x * -0.3333333333333333), (0.008333333333333333 * x)) - fma(0.041666666666666664, x, 0.019444444444444445)), (-0.16666666666666666 * x)) - fma(-0.5, x, 0.16666666666666666))) - (1.0 + x)) / B;
	} else if (F <= -9e+22) {
		tmp = -1.0 / sin(B);
	} else if (F <= 1.8e+140) {
		tmp = -(x * (fma((B * B), (((B * B) * ((-0.0021164021164021165 * (B * B)) - 0.022222222222222223)) - 0.3333333333333333), 1.0) / B)) + ((F * 1.0) / (B * sqrt(fma(F, F, fma(2.0, x, 2.0)))));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.2e+99)
		tmp = Float64(Float64(Float64(Float64(B * B) * Float64(fma(Float64(B * B), Float64(fma(-0.16666666666666666, Float64(x * -0.3333333333333333), Float64(0.008333333333333333 * x)) - fma(0.041666666666666664, x, 0.019444444444444445)), Float64(-0.16666666666666666 * x)) - fma(-0.5, x, 0.16666666666666666))) - Float64(1.0 + x)) / B);
	elseif (F <= -9e+22)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 1.8e+140)
		tmp = Float64(Float64(-Float64(x * Float64(fma(Float64(B * B), Float64(Float64(Float64(B * B) * Float64(Float64(-0.0021164021164021165 * Float64(B * B)) - 0.022222222222222223)) - 0.3333333333333333), 1.0) / B))) + Float64(Float64(F * 1.0) / Float64(B * sqrt(fma(F, F, fma(2.0, x, 2.0))))));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.2e+99], N[(N[(N[(N[(B * B), $MachinePrecision] * N[(N[(N[(B * B), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(x * -0.3333333333333333), $MachinePrecision] + N[(0.008333333333333333 * x), $MachinePrecision]), $MachinePrecision] - N[(0.041666666666666664 * x + 0.019444444444444445), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * x), $MachinePrecision]), $MachinePrecision] - N[(-0.5 * x + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -9e+22], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.8e+140], N[((-N[(x * N[(N[(N[(B * B), $MachinePrecision] * N[(N[(N[(B * B), $MachinePrecision] * N[(N[(-0.0021164021164021165 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 0.022222222222222223), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]) + N[(N[(F * 1.0), $MachinePrecision] / N[(B * N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.2000000000000002e99

   1. Initial program 46.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right) \]

      3. div-add-rev N/A

         \[\leadsto -\frac{1 + x \cdot \cos B}{\sin B} \]

      4. lower-/.f64 N/A

         \[\leadsto -\frac{1 + x \cdot \cos B}{\sin B} \]

      5. lower-+.f64 N/A

         \[\leadsto -\frac{1 + x \cdot \cos B}{\sin B} \]

      6. *-commutative N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      7. lower-*.f64 N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      8. lower-cos.f64 N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      9. lift-sin.f64 99.7

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{-\frac{1 + \cos B \cdot x}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \frac{{B}^{2} \cdot \left(\left(\frac{-1}{6} \cdot x + {B}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) + \frac{1}{120} \cdot x\right) - \left(\frac{7}{360} + \frac{1}{24} \cdot x\right)\right)\right) - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{\color{blue}{B}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left(\left(\frac{-1}{6} \cdot x + {B}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) + \frac{1}{120} \cdot x\right) - \left(\frac{7}{360} + \frac{1}{24} \cdot x\right)\right)\right) - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{B} \]

   8. Applied rewrites 50.9%

      \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(-0.16666666666666666, x \cdot -0.3333333333333333, 0.008333333333333333 \cdot x\right) - \mathsf{fma}\left(0.041666666666666664, x, 0.019444444444444445\right), -0.16666666666666666 \cdot x\right) - \mathsf{fma}\left(-0.5, x, 0.16666666666666666\right)\right) - \left(1 + x\right)}{\color{blue}{B}} \]

   if -4.2000000000000002e99 < F < -8.9999999999999996e22

   1. Initial program 92.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right) \]

      3. div-add-rev N/A

         \[\leadsto -\frac{1 + x \cdot \cos B}{\sin B} \]

      4. lower-/.f64 N/A

         \[\leadsto -\frac{1 + x \cdot \cos B}{\sin B} \]

      5. lower-+.f64 N/A

         \[\leadsto -\frac{1 + x \cdot \cos B}{\sin B} \]

      6. *-commutative N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      7. lower-*.f64 N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      8. lower-cos.f64 N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      9. lift-sin.f64 99.8

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{-\frac{1 + \cos B \cdot x}{\sin B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\sin B} \]

      2. lift-sin.f64 60.4

         \[\leadsto \frac{-1}{\sin B} \]

   8. Applied rewrites 60.4%

      \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

   if -8.9999999999999996e22 < F < 1.8e140

   1. Initial program 97.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      4. inv-pow N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{-1}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{-1}} \]

      6. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\left(2 \cdot x + {F}^{2}\right) + 2\right)}^{-1}} \]

      7. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\left(2 \cdot x + {F}^{2}\right) + 2\right)}^{-1}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, {F}^{2}\right) + 2\right)}^{-1}} \]

      9. pow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      10. lift-*.f64 77.4

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

   5. Applied rewrites 77.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right)}{B}}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1 + {B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right)}{\color{blue}{B}}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      2. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right) + 1}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left({B}^{2}, {B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      4. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, {B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, {B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      6. lower--.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, {B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, {B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      10. lower--.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      12. unpow2 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      13. lower-*.f64 48.0

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(-0.0021164021164021165 \cdot \left(B \cdot B\right) - 0.022222222222222223\right) - 0.3333333333333333, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

   8. Applied rewrites 48.0%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(-0.0021164021164021165 \cdot \left(B \cdot B\right) - 0.022222222222222223\right) - 0.3333333333333333, 1\right)}{B}}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

       2. lift-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

       3. sqrt-div N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

       4. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{\color{blue}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

       5. lower-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\color{blue}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

       6. lower-sqrt.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

       7. pow2 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \left(2 \cdot x + F \cdot F\right)}} \]

       8. lower-+.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \left(2 \cdot x + F \cdot F\right)}} \]

       9. lift-fma.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}} \]

       10. lift-*.f64 47.9

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(-0.0021164021164021165 \cdot \left(B \cdot B\right) - 0.022222222222222223\right) - 0.3333333333333333, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}} \]

   11. Applied rewrites 47.9%

       \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(-0.0021164021164021165 \cdot \left(B \cdot B\right) - 0.022222222222222223\right) - 0.3333333333333333, 1\right)}{B}\right) + \color{blue}{\frac{F}{B} \cdot \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}}} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \color{blue}{\frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}}} \]

       2. lift-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{\color{blue}{1}}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}} \]

       3. lift-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\color{blue}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}}} \]

       4. lift-sqrt.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}} \]

       5. lift-+.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}} \]

       6. lift-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}} \]

       7. lift-fma.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \left(2 \cdot x + F \cdot F\right)}} \]

       8. frac-times N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{\color{blue}{B \cdot \sqrt{2 + \left(2 \cdot x + F \cdot F\right)}}} \]

       9. lower-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{\color{blue}{B \cdot \sqrt{2 + \left(2 \cdot x + F \cdot F\right)}}} \]

       10. lower-*.f64 N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{\color{blue}{B} \cdot \sqrt{2 + \left(2 \cdot x + F \cdot F\right)}} \]

       11. lower-*.f64 N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \color{blue}{\sqrt{2 + \left(2 \cdot x + F \cdot F\right)}}} \]

       12. lower-sqrt.f64 N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \sqrt{2 + \left(2 \cdot x + F \cdot F\right)}} \]

       13. +-commutative N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \sqrt{\left(2 \cdot x + F \cdot F\right) + 2}} \]

       14. pow2 N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \sqrt{\left(2 \cdot x + {F}^{2}\right) + 2}} \]

       15. associate-+r+ N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \sqrt{2 \cdot x + \left({F}^{2} + 2\right)}} \]

       16. pow2 N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} \]

       17. +-commutative N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}} \]

       18. pow2 N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \sqrt{\left({F}^{2} + 2\right) + 2 \cdot x}} \]

   13. Applied rewrites 50.1%

       \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(-0.0021164021164021165 \cdot \left(B \cdot B\right) - 0.022222222222222223\right) - 0.3333333333333333, 1\right)}{B}\right) + \frac{F \cdot 1}{\color{blue}{B \cdot \sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}} \]

   if 1.8e140 < F

   1. Initial program 36.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 28.6%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.0%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 57.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{+20}:\\ \;\;\;\;-\frac{1 + x}{\sin B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{+140}:\\ \;\;\;\;\left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(-0.0021164021164021165 \cdot \left(B \cdot B\right) - 0.022222222222222223\right) - 0.3333333333333333, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4e+20)
   (- (/ (+ 1.0 x) (sin B)))
   (if (<= F 1.8e+140)
     (+
      (-
       (*
        x
        (/
         (fma
          (* B B)
          (-
           (*
            (* B B)
            (- (* -0.0021164021164021165 (* B B)) 0.022222222222222223))
           0.3333333333333333)
          1.0)
         B)))
      (/ (* F 1.0) (* B (sqrt (fma F F (fma 2.0 x 2.0))))))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4e+20) {
		tmp = -((1.0 + x) / sin(B));
	} else if (F <= 1.8e+140) {
		tmp = -(x * (fma((B * B), (((B * B) * ((-0.0021164021164021165 * (B * B)) - 0.022222222222222223)) - 0.3333333333333333), 1.0) / B)) + ((F * 1.0) / (B * sqrt(fma(F, F, fma(2.0, x, 2.0)))));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4e+20)
		tmp = Float64(-Float64(Float64(1.0 + x) / sin(B)));
	elseif (F <= 1.8e+140)
		tmp = Float64(Float64(-Float64(x * Float64(fma(Float64(B * B), Float64(Float64(Float64(B * B) * Float64(Float64(-0.0021164021164021165 * Float64(B * B)) - 0.022222222222222223)) - 0.3333333333333333), 1.0) / B))) + Float64(Float64(F * 1.0) / Float64(B * sqrt(fma(F, F, fma(2.0, x, 2.0))))));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4e+20], (-N[(N[(1.0 + x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 1.8e+140], N[((-N[(x * N[(N[(N[(B * B), $MachinePrecision] * N[(N[(N[(B * B), $MachinePrecision] * N[(N[(-0.0021164021164021165 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 0.022222222222222223), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]) + N[(N[(F * 1.0), $MachinePrecision] / N[(B * N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4e20

   1. Initial program 57.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right) \]

      3. div-add-rev N/A

         \[\leadsto -\frac{1 + x \cdot \cos B}{\sin B} \]

      4. lower-/.f64 N/A

         \[\leadsto -\frac{1 + x \cdot \cos B}{\sin B} \]

      5. lower-+.f64 N/A

         \[\leadsto -\frac{1 + x \cdot \cos B}{\sin B} \]

      6. *-commutative N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      7. lower-*.f64 N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      8. lower-cos.f64 N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      9. lift-sin.f64 99.7

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{-\frac{1 + \cos B \cdot x}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto -\frac{1 + x}{\sin B} \]
   7. Step-by-step derivation

   8. Applied rewrites 77.9%

      \[\leadsto -\frac{1 + x}{\sin B} \]

   if -4e20 < F < 1.8e140

   1. Initial program 97.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      4. inv-pow N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{-1}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{-1}} \]

      6. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\left(2 \cdot x + {F}^{2}\right) + 2\right)}^{-1}} \]

      7. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\left(2 \cdot x + {F}^{2}\right) + 2\right)}^{-1}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, {F}^{2}\right) + 2\right)}^{-1}} \]

      9. pow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      10. lift-*.f64 77.4

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

   5. Applied rewrites 77.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right)}{B}}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1 + {B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right)}{\color{blue}{B}}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      2. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right) + 1}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left({B}^{2}, {B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      4. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, {B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, {B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      6. lower--.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, {B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, {B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      10. lower--.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      12. unpow2 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      13. lower-*.f64 47.9

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(-0.0021164021164021165 \cdot \left(B \cdot B\right) - 0.022222222222222223\right) - 0.3333333333333333, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

   8. Applied rewrites 47.9%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(-0.0021164021164021165 \cdot \left(B \cdot B\right) - 0.022222222222222223\right) - 0.3333333333333333, 1\right)}{B}}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

       2. lift-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

       3. sqrt-div N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

       4. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{\color{blue}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

       5. lower-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\color{blue}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

       6. lower-sqrt.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

       7. pow2 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \left(2 \cdot x + F \cdot F\right)}} \]

       8. lower-+.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \left(2 \cdot x + F \cdot F\right)}} \]

       9. lift-fma.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}} \]

       10. lift-*.f64 47.9

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(-0.0021164021164021165 \cdot \left(B \cdot B\right) - 0.022222222222222223\right) - 0.3333333333333333, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}} \]

   11. Applied rewrites 47.9%

       \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(-0.0021164021164021165 \cdot \left(B \cdot B\right) - 0.022222222222222223\right) - 0.3333333333333333, 1\right)}{B}\right) + \color{blue}{\frac{F}{B} \cdot \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}}} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \color{blue}{\frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}}} \]

       2. lift-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{\color{blue}{1}}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}} \]

       3. lift-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\color{blue}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}}} \]

       4. lift-sqrt.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}} \]

       5. lift-+.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}} \]

       6. lift-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}} \]

       7. lift-fma.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \left(2 \cdot x + F \cdot F\right)}} \]

       8. frac-times N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{\color{blue}{B \cdot \sqrt{2 + \left(2 \cdot x + F \cdot F\right)}}} \]

       9. lower-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{\color{blue}{B \cdot \sqrt{2 + \left(2 \cdot x + F \cdot F\right)}}} \]

       10. lower-*.f64 N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{\color{blue}{B} \cdot \sqrt{2 + \left(2 \cdot x + F \cdot F\right)}} \]

       11. lower-*.f64 N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \color{blue}{\sqrt{2 + \left(2 \cdot x + F \cdot F\right)}}} \]

       12. lower-sqrt.f64 N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \sqrt{2 + \left(2 \cdot x + F \cdot F\right)}} \]

       13. +-commutative N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \sqrt{\left(2 \cdot x + F \cdot F\right) + 2}} \]

       14. pow2 N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \sqrt{\left(2 \cdot x + {F}^{2}\right) + 2}} \]

       15. associate-+r+ N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \sqrt{2 \cdot x + \left({F}^{2} + 2\right)}} \]

       16. pow2 N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} \]

       17. +-commutative N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}} \]

       18. pow2 N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \sqrt{\left({F}^{2} + 2\right) + 2 \cdot x}} \]

   13. Applied rewrites 50.0%

       \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(-0.0021164021164021165 \cdot \left(B \cdot B\right) - 0.022222222222222223\right) - 0.3333333333333333, 1\right)}{B}\right) + \frac{F \cdot 1}{\color{blue}{B \cdot \sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}} \]

   if 1.8e140 < F

   1. Initial program 36.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 28.6%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.0%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 50.2% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.4 \cdot 10^{+89}:\\ \;\;\;\;\frac{\left(B \cdot B\right) \cdot \left(\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(-0.16666666666666666, x \cdot -0.3333333333333333, 0.008333333333333333 \cdot x\right) - \mathsf{fma}\left(0.041666666666666664, x, 0.019444444444444445\right), -0.16666666666666666 \cdot x\right) - \mathsf{fma}\left(-0.5, x, 0.16666666666666666\right)\right) - \left(1 + x\right)}{B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{+140}:\\ \;\;\;\;\left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(-0.0021164021164021165 \cdot \left(B \cdot B\right) - 0.022222222222222223\right) - 0.3333333333333333, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.4e+89)
   (/
    (-
     (*
      (* B B)
      (-
       (fma
        (* B B)
        (-
         (fma
          -0.16666666666666666
          (* x -0.3333333333333333)
          (* 0.008333333333333333 x))
         (fma 0.041666666666666664 x 0.019444444444444445))
        (* -0.16666666666666666 x))
       (fma -0.5 x 0.16666666666666666)))
     (+ 1.0 x))
    B)
   (if (<= F 1.8e+140)
     (+
      (-
       (*
        x
        (/
         (fma
          (* B B)
          (-
           (*
            (* B B)
            (- (* -0.0021164021164021165 (* B B)) 0.022222222222222223))
           0.3333333333333333)
          1.0)
         B)))
      (/ (* F 1.0) (* B (sqrt (fma F F (fma 2.0 x 2.0))))))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.4e+89) {
		tmp = (((B * B) * (fma((B * B), (fma(-0.16666666666666666, (x * -0.3333333333333333), (0.008333333333333333 * x)) - fma(0.041666666666666664, x, 0.019444444444444445)), (-0.16666666666666666 * x)) - fma(-0.5, x, 0.16666666666666666))) - (1.0 + x)) / B;
	} else if (F <= 1.8e+140) {
		tmp = -(x * (fma((B * B), (((B * B) * ((-0.0021164021164021165 * (B * B)) - 0.022222222222222223)) - 0.3333333333333333), 1.0) / B)) + ((F * 1.0) / (B * sqrt(fma(F, F, fma(2.0, x, 2.0)))));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.4e+89)
		tmp = Float64(Float64(Float64(Float64(B * B) * Float64(fma(Float64(B * B), Float64(fma(-0.16666666666666666, Float64(x * -0.3333333333333333), Float64(0.008333333333333333 * x)) - fma(0.041666666666666664, x, 0.019444444444444445)), Float64(-0.16666666666666666 * x)) - fma(-0.5, x, 0.16666666666666666))) - Float64(1.0 + x)) / B);
	elseif (F <= 1.8e+140)
		tmp = Float64(Float64(-Float64(x * Float64(fma(Float64(B * B), Float64(Float64(Float64(B * B) * Float64(Float64(-0.0021164021164021165 * Float64(B * B)) - 0.022222222222222223)) - 0.3333333333333333), 1.0) / B))) + Float64(Float64(F * 1.0) / Float64(B * sqrt(fma(F, F, fma(2.0, x, 2.0))))));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.4e+89], N[(N[(N[(N[(B * B), $MachinePrecision] * N[(N[(N[(B * B), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(x * -0.3333333333333333), $MachinePrecision] + N[(0.008333333333333333 * x), $MachinePrecision]), $MachinePrecision] - N[(0.041666666666666664 * x + 0.019444444444444445), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * x), $MachinePrecision]), $MachinePrecision] - N[(-0.5 * x + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.8e+140], N[((-N[(x * N[(N[(N[(B * B), $MachinePrecision] * N[(N[(N[(B * B), $MachinePrecision] * N[(N[(-0.0021164021164021165 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 0.022222222222222223), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]) + N[(N[(F * 1.0), $MachinePrecision] / N[(B * N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.4e89

   1. Initial program 47.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right) \]

      3. div-add-rev N/A

         \[\leadsto -\frac{1 + x \cdot \cos B}{\sin B} \]

      4. lower-/.f64 N/A

         \[\leadsto -\frac{1 + x \cdot \cos B}{\sin B} \]

      5. lower-+.f64 N/A

         \[\leadsto -\frac{1 + x \cdot \cos B}{\sin B} \]

      6. *-commutative N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      7. lower-*.f64 N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      8. lower-cos.f64 N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      9. lift-sin.f64 99.7

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{-\frac{1 + \cos B \cdot x}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \frac{{B}^{2} \cdot \left(\left(\frac{-1}{6} \cdot x + {B}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) + \frac{1}{120} \cdot x\right) - \left(\frac{7}{360} + \frac{1}{24} \cdot x\right)\right)\right) - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{\color{blue}{B}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left(\left(\frac{-1}{6} \cdot x + {B}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) + \frac{1}{120} \cdot x\right) - \left(\frac{7}{360} + \frac{1}{24} \cdot x\right)\right)\right) - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{B} \]

   8. Applied rewrites 50.7%

      \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(-0.16666666666666666, x \cdot -0.3333333333333333, 0.008333333333333333 \cdot x\right) - \mathsf{fma}\left(0.041666666666666664, x, 0.019444444444444445\right), -0.16666666666666666 \cdot x\right) - \mathsf{fma}\left(-0.5, x, 0.16666666666666666\right)\right) - \left(1 + x\right)}{\color{blue}{B}} \]

   if -4.4e89 < F < 1.8e140

   1. Initial program 97.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      4. inv-pow N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{-1}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{-1}} \]

      6. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\left(2 \cdot x + {F}^{2}\right) + 2\right)}^{-1}} \]

      7. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\left(2 \cdot x + {F}^{2}\right) + 2\right)}^{-1}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, {F}^{2}\right) + 2\right)}^{-1}} \]

      9. pow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      10. lift-*.f64 76.1

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

   5. Applied rewrites 76.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right)}{B}}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1 + {B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right)}{\color{blue}{B}}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      2. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right) + 1}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left({B}^{2}, {B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      4. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, {B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, {B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      6. lower--.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, {B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, {B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      10. lower--.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      12. unpow2 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      13. lower-*.f64 47.6

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(-0.0021164021164021165 \cdot \left(B \cdot B\right) - 0.022222222222222223\right) - 0.3333333333333333, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

   8. Applied rewrites 47.6%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(-0.0021164021164021165 \cdot \left(B \cdot B\right) - 0.022222222222222223\right) - 0.3333333333333333, 1\right)}{B}}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

       2. lift-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

       3. sqrt-div N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{\sqrt{1}}{\color{blue}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

       4. metadata-eval N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{\color{blue}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

       5. lower-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\color{blue}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

       6. lower-sqrt.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

       7. pow2 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \left(2 \cdot x + F \cdot F\right)}} \]

       8. lower-+.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \left(2 \cdot x + F \cdot F\right)}} \]

       9. lift-fma.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}} \]

       10. lift-*.f64 47.6

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(-0.0021164021164021165 \cdot \left(B \cdot B\right) - 0.022222222222222223\right) - 0.3333333333333333, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}} \]

   11. Applied rewrites 47.6%

       \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(-0.0021164021164021165 \cdot \left(B \cdot B\right) - 0.022222222222222223\right) - 0.3333333333333333, 1\right)}{B}\right) + \color{blue}{\frac{F}{B} \cdot \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}}} \]
   12. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \color{blue}{\frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}}} \]

       2. lift-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{\color{blue}{1}}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}} \]

       3. lift-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\color{blue}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}}} \]

       4. lift-sqrt.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}} \]

       5. lift-+.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}} \]

       6. lift-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}} \]

       7. lift-fma.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \left(2 \cdot x + F \cdot F\right)}} \]

       8. frac-times N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{\color{blue}{B \cdot \sqrt{2 + \left(2 \cdot x + F \cdot F\right)}}} \]

       9. lower-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{\color{blue}{B \cdot \sqrt{2 + \left(2 \cdot x + F \cdot F\right)}}} \]

       10. lower-*.f64 N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{\color{blue}{B} \cdot \sqrt{2 + \left(2 \cdot x + F \cdot F\right)}} \]

       11. lower-*.f64 N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \color{blue}{\sqrt{2 + \left(2 \cdot x + F \cdot F\right)}}} \]

       12. lower-sqrt.f64 N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \sqrt{2 + \left(2 \cdot x + F \cdot F\right)}} \]

       13. +-commutative N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \sqrt{\left(2 \cdot x + F \cdot F\right) + 2}} \]

       14. pow2 N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \sqrt{\left(2 \cdot x + {F}^{2}\right) + 2}} \]

       15. associate-+r+ N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \sqrt{2 \cdot x + \left({F}^{2} + 2\right)}} \]

       16. pow2 N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \sqrt{2 \cdot x + \left(F \cdot F + 2\right)}} \]

       17. +-commutative N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \sqrt{\left(F \cdot F + 2\right) + 2 \cdot x}} \]

       18. pow2 N/A

           \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F \cdot 1}{B \cdot \sqrt{\left({F}^{2} + 2\right) + 2 \cdot x}} \]

   13. Applied rewrites 50.1%

       \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(-0.0021164021164021165 \cdot \left(B \cdot B\right) - 0.022222222222222223\right) - 0.3333333333333333, 1\right)}{B}\right) + \frac{F \cdot 1}{\color{blue}{B \cdot \sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}} \]

   if 1.8e140 < F

   1. Initial program 36.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 28.6%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.0%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 50.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.96 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(B \cdot B\right) \cdot \left(\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(-0.16666666666666666, x \cdot -0.3333333333333333, 0.008333333333333333 \cdot x\right) - \mathsf{fma}\left(0.041666666666666664, x, 0.019444444444444445\right), -0.16666666666666666 \cdot x\right) - \mathsf{fma}\left(-0.5, x, 0.16666666666666666\right)\right) - \left(1 + x\right)}{B}\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.96e+99)
   (/
    (-
     (*
      (* B B)
      (-
       (fma
        (* B B)
        (-
         (fma
          -0.16666666666666666
          (* x -0.3333333333333333)
          (* 0.008333333333333333 x))
         (fma 0.041666666666666664 x 0.019444444444444445))
        (* -0.16666666666666666 x))
       (fma -0.5 x 0.16666666666666666)))
     (+ 1.0 x))
    B)
   (if (<= F 1.95e-8)
     (/ (- (* (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0)))) F) x) B)
     (/ (- (fma -0.5 (/ (fma 2.0 x 2.0) (* F F)) 1.0) x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.96e+99) {
		tmp = (((B * B) * (fma((B * B), (fma(-0.16666666666666666, (x * -0.3333333333333333), (0.008333333333333333 * x)) - fma(0.041666666666666664, x, 0.019444444444444445)), (-0.16666666666666666 * x)) - fma(-0.5, x, 0.16666666666666666))) - (1.0 + x)) / B;
	} else if (F <= 1.95e-8) {
		tmp = ((sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B;
	} else {
		tmp = (fma(-0.5, (fma(2.0, x, 2.0) / (F * F)), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.96e+99)
		tmp = Float64(Float64(Float64(Float64(B * B) * Float64(fma(Float64(B * B), Float64(fma(-0.16666666666666666, Float64(x * -0.3333333333333333), Float64(0.008333333333333333 * x)) - fma(0.041666666666666664, x, 0.019444444444444445)), Float64(-0.16666666666666666 * x)) - fma(-0.5, x, 0.16666666666666666))) - Float64(1.0 + x)) / B);
	elseif (F <= 1.95e-8)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B);
	else
		tmp = Float64(Float64(fma(-0.5, Float64(fma(2.0, x, 2.0) / Float64(F * F)), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.96e+99], N[(N[(N[(N[(B * B), $MachinePrecision] * N[(N[(N[(B * B), $MachinePrecision] * N[(N[(-0.16666666666666666 * N[(x * -0.3333333333333333), $MachinePrecision] + N[(0.008333333333333333 * x), $MachinePrecision]), $MachinePrecision] - N[(0.041666666666666664 * x + 0.019444444444444445), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * x), $MachinePrecision]), $MachinePrecision] - N[(-0.5 * x + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.95e-8], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(-0.5 * N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.95999999999999992e99

   1. Initial program 46.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right) \]

      3. div-add-rev N/A

         \[\leadsto -\frac{1 + x \cdot \cos B}{\sin B} \]

      4. lower-/.f64 N/A

         \[\leadsto -\frac{1 + x \cdot \cos B}{\sin B} \]

      5. lower-+.f64 N/A

         \[\leadsto -\frac{1 + x \cdot \cos B}{\sin B} \]

      6. *-commutative N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      7. lower-*.f64 N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      8. lower-cos.f64 N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      9. lift-sin.f64 99.7

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{-\frac{1 + \cos B \cdot x}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \frac{{B}^{2} \cdot \left(\left(\frac{-1}{6} \cdot x + {B}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) + \frac{1}{120} \cdot x\right) - \left(\frac{7}{360} + \frac{1}{24} \cdot x\right)\right)\right) - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{\color{blue}{B}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left(\left(\frac{-1}{6} \cdot x + {B}^{2} \cdot \left(\left(\frac{-1}{6} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) + \frac{1}{120} \cdot x\right) - \left(\frac{7}{360} + \frac{1}{24} \cdot x\right)\right)\right) - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{B} \]

   8. Applied rewrites 50.9%

      \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(-0.16666666666666666, x \cdot -0.3333333333333333, 0.008333333333333333 \cdot x\right) - \mathsf{fma}\left(0.041666666666666664, x, 0.019444444444444445\right), -0.16666666666666666 \cdot x\right) - \mathsf{fma}\left(-0.5, x, 0.16666666666666666\right)\right) - \left(1 + x\right)}{\color{blue}{B}} \]

   if -1.95999999999999992e99 < F < 1.94999999999999992e-8

   1. Initial program 98.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{{\left(\left(2 \cdot x + F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B} \]

      5. unpow-1 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\left(2 \cdot x + F \cdot F\right) + 2}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\left(2 \cdot x + F \cdot F\right) + 2}} \cdot F - x}{B} \]

      7. pow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\left(2 \cdot x + {F}^{2}\right) + 2}} \cdot F - x}{B} \]

      8. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 \cdot x + \left({F}^{2} + 2\right)}} \cdot F - x}{B} \]

      9. pow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 \cdot x + \left(F \cdot F + 2\right)}} \cdot F - x}{B} \]

      10. lift-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{2 \cdot x + \mathsf{fma}\left(F, F, 2\right)}} \cdot F - x}{B} \]

      11. lift-fma.f64 50.7

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B} \]

   7. Applied rewrites 50.7%

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B} \]

   if 1.94999999999999992e-8 < F

   1. Initial program 59.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 38.0%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) - x}{B} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} + 1\right) - x}{B} \]

      2. div-add N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \left(\frac{2}{{F}^{2}} + \frac{2 \cdot x}{{F}^{2}}\right) + 1\right) - x}{B} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \left(\frac{2 \cdot 1}{{F}^{2}} + \frac{2 \cdot x}{{F}^{2}}\right) + 1\right) - x}{B} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \left(2 \cdot \frac{1}{{F}^{2}} + \frac{2 \cdot x}{{F}^{2}}\right) + 1\right) - x}{B} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \left(2 \cdot \frac{1}{{F}^{2}} + 2 \cdot \frac{x}{{F}^{2}}\right) + 1\right) - x}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \left(2 \cdot \frac{x}{{F}^{2}} + 2 \cdot \frac{1}{{F}^{2}}\right) + 1\right) - x}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, 2 \cdot \frac{x}{{F}^{2}} + 2 \cdot \frac{1}{{F}^{2}}, 1\right) - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, 2 \cdot \frac{1}{{F}^{2}} + 2 \cdot \frac{x}{{F}^{2}}, 1\right) - x}{B} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2 \cdot 1}{{F}^{2}} + 2 \cdot \frac{x}{{F}^{2}}, 1\right) - x}{B} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2}{{F}^{2}} + 2 \cdot \frac{x}{{F}^{2}}, 1\right) - x}{B} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2}{{F}^{2}} + \frac{2 \cdot x}{{F}^{2}}, 1\right) - x}{B} \]

      12. div-add N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{2}}, 1\right) - x}{B} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{2}}, 1\right) - x}{B} \]

      14. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2 \cdot x + 2}{{F}^{2}}, 1\right) - x}{B} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{{F}^{2}}, 1\right) - x}{B} \]

      16. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}{B} \]

      17. lift-*.f64 49.4

          \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}{B} \]

   8. Applied rewrites 49.4%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 50.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.75 \cdot 10^{+99}:\\ \;\;\;\;\left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(-0.0021164021164021165 \cdot \left(B \cdot B\right) - 0.022222222222222223\right) - 0.3333333333333333, 1\right)}{B}\right) + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.75e+99)
   (+
    (-
     (*
      x
      (/
       (fma
        (* B B)
        (-
         (*
          (* B B)
          (- (* -0.0021164021164021165 (* B B)) 0.022222222222222223))
         0.3333333333333333)
        1.0)
       B)))
    (/ -1.0 B))
   (if (<= F 1.95e-8)
     (/ (- (* (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0)))) F) x) B)
     (/ (- (fma -0.5 (/ (fma 2.0 x 2.0) (* F F)) 1.0) x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.75e+99) {
		tmp = -(x * (fma((B * B), (((B * B) * ((-0.0021164021164021165 * (B * B)) - 0.022222222222222223)) - 0.3333333333333333), 1.0) / B)) + (-1.0 / B);
	} else if (F <= 1.95e-8) {
		tmp = ((sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B;
	} else {
		tmp = (fma(-0.5, (fma(2.0, x, 2.0) / (F * F)), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.75e+99)
		tmp = Float64(Float64(-Float64(x * Float64(fma(Float64(B * B), Float64(Float64(Float64(B * B) * Float64(Float64(-0.0021164021164021165 * Float64(B * B)) - 0.022222222222222223)) - 0.3333333333333333), 1.0) / B))) + Float64(-1.0 / B));
	elseif (F <= 1.95e-8)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B);
	else
		tmp = Float64(Float64(fma(-0.5, Float64(fma(2.0, x, 2.0) / Float64(F * F)), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.75e+99], N[((-N[(x * N[(N[(N[(B * B), $MachinePrecision] * N[(N[(N[(B * B), $MachinePrecision] * N[(N[(-0.0021164021164021165 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 0.022222222222222223), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.95e-8], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(-0.5 * N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.7499999999999999e99

   1. Initial program 46.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      4. inv-pow N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{-1}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{-1}} \]

      6. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\left(2 \cdot x + {F}^{2}\right) + 2\right)}^{-1}} \]

      7. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\left(2 \cdot x + {F}^{2}\right) + 2\right)}^{-1}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, {F}^{2}\right) + 2\right)}^{-1}} \]

      9. pow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      10. lift-*.f64 39.2

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

   5. Applied rewrites 39.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right)}{B}}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1 + {B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right)}{\color{blue}{B}}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      2. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{{B}^{2} \cdot \left({B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}\right) + 1}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left({B}^{2}, {B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      4. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, {B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, {B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      6. lower--.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, {B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, {B}^{2} \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      10. lower--.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      11. lower-*.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot {B}^{2} - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      12. unpow2 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

      13. lower-*.f64 15.4

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(-0.0021164021164021165 \cdot \left(B \cdot B\right) - 0.022222222222222223\right) - 0.3333333333333333, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

   8. Applied rewrites 15.4%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(-0.0021164021164021165 \cdot \left(B \cdot B\right) - 0.022222222222222223\right) - 0.3333333333333333, 1\right)}{B}}\right) + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \]

   9. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(\frac{-2}{945} \cdot \left(B \cdot B\right) - \frac{1}{45}\right) - \frac{1}{3}, 1\right)}{B}\right) + \frac{-1}{\color{blue}{B}} \]
   10. Step-by-step derivation

       1. lower-/.f64 50.7

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(-0.0021164021164021165 \cdot \left(B \cdot B\right) - 0.022222222222222223\right) - 0.3333333333333333, 1\right)}{B}\right) + \frac{-1}{B} \]

   11. Applied rewrites 50.7%

       \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(B \cdot B, \left(B \cdot B\right) \cdot \left(-0.0021164021164021165 \cdot \left(B \cdot B\right) - 0.022222222222222223\right) - 0.3333333333333333, 1\right)}{B}\right) + \frac{-1}{\color{blue}{B}} \]

   if -1.7499999999999999e99 < F < 1.94999999999999992e-8

   1. Initial program 98.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{{\left(\left(2 \cdot x + F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B} \]

      5. unpow-1 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\left(2 \cdot x + F \cdot F\right) + 2}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\left(2 \cdot x + F \cdot F\right) + 2}} \cdot F - x}{B} \]

      7. pow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\left(2 \cdot x + {F}^{2}\right) + 2}} \cdot F - x}{B} \]

      8. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 \cdot x + \left({F}^{2} + 2\right)}} \cdot F - x}{B} \]

      9. pow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 \cdot x + \left(F \cdot F + 2\right)}} \cdot F - x}{B} \]

      10. lift-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{2 \cdot x + \mathsf{fma}\left(F, F, 2\right)}} \cdot F - x}{B} \]

      11. lift-fma.f64 50.7

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B} \]

   7. Applied rewrites 50.7%

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B} \]

   if 1.94999999999999992e-8 < F

   1. Initial program 59.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 38.0%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) - x}{B} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} + 1\right) - x}{B} \]

      2. div-add N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \left(\frac{2}{{F}^{2}} + \frac{2 \cdot x}{{F}^{2}}\right) + 1\right) - x}{B} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \left(\frac{2 \cdot 1}{{F}^{2}} + \frac{2 \cdot x}{{F}^{2}}\right) + 1\right) - x}{B} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \left(2 \cdot \frac{1}{{F}^{2}} + \frac{2 \cdot x}{{F}^{2}}\right) + 1\right) - x}{B} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \left(2 \cdot \frac{1}{{F}^{2}} + 2 \cdot \frac{x}{{F}^{2}}\right) + 1\right) - x}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \left(2 \cdot \frac{x}{{F}^{2}} + 2 \cdot \frac{1}{{F}^{2}}\right) + 1\right) - x}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, 2 \cdot \frac{x}{{F}^{2}} + 2 \cdot \frac{1}{{F}^{2}}, 1\right) - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, 2 \cdot \frac{1}{{F}^{2}} + 2 \cdot \frac{x}{{F}^{2}}, 1\right) - x}{B} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2 \cdot 1}{{F}^{2}} + 2 \cdot \frac{x}{{F}^{2}}, 1\right) - x}{B} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2}{{F}^{2}} + 2 \cdot \frac{x}{{F}^{2}}, 1\right) - x}{B} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2}{{F}^{2}} + \frac{2 \cdot x}{{F}^{2}}, 1\right) - x}{B} \]

      12. div-add N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{2}}, 1\right) - x}{B} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{2}}, 1\right) - x}{B} \]

      14. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2 \cdot x + 2}{{F}^{2}}, 1\right) - x}{B} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{{F}^{2}}, 1\right) - x}{B} \]

      16. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}{B} \]

      17. lift-*.f64 49.4

          \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}{B} \]

   8. Applied rewrites 49.4%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 50.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.96 \cdot 10^{+99}:\\ \;\;\;\;\frac{\left(B \cdot B\right) \cdot \left(-0.16666666666666666 \cdot x - \mathsf{fma}\left(-0.5, x, 0.16666666666666666\right)\right) - \left(1 + x\right)}{B}\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.96e+99)
   (/
    (-
     (*
      (* B B)
      (- (* -0.16666666666666666 x) (fma -0.5 x 0.16666666666666666)))
     (+ 1.0 x))
    B)
   (if (<= F 1.95e-8)
     (/ (- (* (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0)))) F) x) B)
     (/ (- (fma -0.5 (/ (fma 2.0 x 2.0) (* F F)) 1.0) x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.96e+99) {
		tmp = (((B * B) * ((-0.16666666666666666 * x) - fma(-0.5, x, 0.16666666666666666))) - (1.0 + x)) / B;
	} else if (F <= 1.95e-8) {
		tmp = ((sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B;
	} else {
		tmp = (fma(-0.5, (fma(2.0, x, 2.0) / (F * F)), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.96e+99)
		tmp = Float64(Float64(Float64(Float64(B * B) * Float64(Float64(-0.16666666666666666 * x) - fma(-0.5, x, 0.16666666666666666))) - Float64(1.0 + x)) / B);
	elseif (F <= 1.95e-8)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B);
	else
		tmp = Float64(Float64(fma(-0.5, Float64(fma(2.0, x, 2.0) / Float64(F * F)), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.96e+99], N[(N[(N[(N[(B * B), $MachinePrecision] * N[(N[(-0.16666666666666666 * x), $MachinePrecision] - N[(-0.5 * x + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.95e-8], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(-0.5 * N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.95999999999999992e99

   1. Initial program 46.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right) \]

      3. div-add-rev N/A

         \[\leadsto -\frac{1 + x \cdot \cos B}{\sin B} \]

      4. lower-/.f64 N/A

         \[\leadsto -\frac{1 + x \cdot \cos B}{\sin B} \]

      5. lower-+.f64 N/A

         \[\leadsto -\frac{1 + x \cdot \cos B}{\sin B} \]

      6. *-commutative N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      7. lower-*.f64 N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      8. lower-cos.f64 N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      9. lift-sin.f64 99.7

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

   5. Applied rewrites 99.7%

      \[\leadsto \color{blue}{-\frac{1 + \cos B \cdot x}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \frac{{B}^{2} \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{\color{blue}{B}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{B} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{B} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{B} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{B} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{B} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{-1}{2} \cdot x + \frac{1}{6}\right)\right) - \left(1 + x\right)}{B} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\frac{-1}{6} \cdot x - \mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{6}\right)\right) - \left(1 + x\right)}{B} \]

      10. lower-+.f64 51.1

          \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(-0.16666666666666666 \cdot x - \mathsf{fma}\left(-0.5, x, 0.16666666666666666\right)\right) - \left(1 + x\right)}{B} \]

   8. Applied rewrites 51.1%

      \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(-0.16666666666666666 \cdot x - \mathsf{fma}\left(-0.5, x, 0.16666666666666666\right)\right) - \left(1 + x\right)}{\color{blue}{B}} \]

   if -1.95999999999999992e99 < F < 1.94999999999999992e-8

   1. Initial program 98.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]
   6. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{{\left(\left(2 \cdot x + F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B} \]

      5. unpow-1 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\left(2 \cdot x + F \cdot F\right) + 2}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\left(2 \cdot x + F \cdot F\right) + 2}} \cdot F - x}{B} \]

      7. pow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\left(2 \cdot x + {F}^{2}\right) + 2}} \cdot F - x}{B} \]

      8. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 \cdot x + \left({F}^{2} + 2\right)}} \cdot F - x}{B} \]

      9. pow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 \cdot x + \left(F \cdot F + 2\right)}} \cdot F - x}{B} \]

      10. lift-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{2 \cdot x + \mathsf{fma}\left(F, F, 2\right)}} \cdot F - x}{B} \]

      11. lift-fma.f64 50.7

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B} \]

   7. Applied rewrites 50.7%

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B} \]

   if 1.94999999999999992e-8 < F

   1. Initial program 59.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 38.0%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) - x}{B} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} + 1\right) - x}{B} \]

      2. div-add N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \left(\frac{2}{{F}^{2}} + \frac{2 \cdot x}{{F}^{2}}\right) + 1\right) - x}{B} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \left(\frac{2 \cdot 1}{{F}^{2}} + \frac{2 \cdot x}{{F}^{2}}\right) + 1\right) - x}{B} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \left(2 \cdot \frac{1}{{F}^{2}} + \frac{2 \cdot x}{{F}^{2}}\right) + 1\right) - x}{B} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \left(2 \cdot \frac{1}{{F}^{2}} + 2 \cdot \frac{x}{{F}^{2}}\right) + 1\right) - x}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \left(2 \cdot \frac{x}{{F}^{2}} + 2 \cdot \frac{1}{{F}^{2}}\right) + 1\right) - x}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, 2 \cdot \frac{x}{{F}^{2}} + 2 \cdot \frac{1}{{F}^{2}}, 1\right) - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, 2 \cdot \frac{1}{{F}^{2}} + 2 \cdot \frac{x}{{F}^{2}}, 1\right) - x}{B} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2 \cdot 1}{{F}^{2}} + 2 \cdot \frac{x}{{F}^{2}}, 1\right) - x}{B} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2}{{F}^{2}} + 2 \cdot \frac{x}{{F}^{2}}, 1\right) - x}{B} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2}{{F}^{2}} + \frac{2 \cdot x}{{F}^{2}}, 1\right) - x}{B} \]

      12. div-add N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{2}}, 1\right) - x}{B} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{2}}, 1\right) - x}{B} \]

      14. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2 \cdot x + 2}{{F}^{2}}, 1\right) - x}{B} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{{F}^{2}}, 1\right) - x}{B} \]

      16. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}{B} \]

      17. lift-*.f64 49.4

          \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}{B} \]

   8. Applied rewrites 49.4%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 50.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.8:\\ \;\;\;\;\mathsf{fma}\left(0.5, \frac{2}{B \cdot \left(F \cdot F\right)}, -\frac{1 + x}{B}\right)\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{0.5} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.8)
   (fma 0.5 (/ 2.0 (* B (* F F))) (- (/ (+ 1.0 x) B)))
   (if (<= F 1.95e-8)
     (/ (- (* (sqrt 0.5) F) x) B)
     (/ (- (fma -0.5 (/ (fma 2.0 x 2.0) (* F F)) 1.0) x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.8) {
		tmp = fma(0.5, (2.0 / (B * (F * F))), -((1.0 + x) / B));
	} else if (F <= 1.95e-8) {
		tmp = ((sqrt(0.5) * F) - x) / B;
	} else {
		tmp = (fma(-0.5, (fma(2.0, x, 2.0) / (F * F)), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.8)
		tmp = fma(0.5, Float64(2.0 / Float64(B * Float64(F * F))), Float64(-Float64(Float64(1.0 + x) / B)));
	elseif (F <= 1.95e-8)
		tmp = Float64(Float64(Float64(sqrt(0.5) * F) - x) / B);
	else
		tmp = Float64(Float64(fma(-0.5, Float64(fma(2.0, x, 2.0) / Float64(F * F)), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.8], N[(0.5 * N[(2.0 / N[(B * N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + (-N[(N[(1.0 + x), $MachinePrecision] / B), $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 1.95e-8], N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(-0.5 * N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.79999999999999982

   1. Initial program 59.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 37.8%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto -1 \cdot \frac{1 + x}{B} + \color{blue}{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{B \cdot {F}^{2}}} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto -1 \cdot \frac{1 + x}{B} + \frac{1}{2} \cdot \frac{2 + 2 \cdot x}{\color{blue}{B} \cdot {F}^{2}} \]

      2. +-commutative N/A

         \[\leadsto \frac{1}{2} \cdot \frac{2 + 2 \cdot x}{B \cdot {F}^{2}} + -1 \cdot \color{blue}{\frac{1 + x}{B}} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{2 + 2 \cdot x}{\color{blue}{B \cdot {F}^{2}}}, -1 \cdot \frac{1 + x}{B}\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{2 + 2 \cdot x}{\color{blue}{B} \cdot {F}^{2}}, -1 \cdot \frac{1 + x}{B}\right) \]

      5. associate-/r* N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\frac{2 + 2 \cdot x}{B}}{{F}^{\color{blue}{2}}}, -1 \cdot \frac{1 + x}{B}\right) \]

      6. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\frac{2 + 2 \cdot x}{B}}{{F}^{\color{blue}{2}}}, -1 \cdot \frac{1 + x}{B}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\frac{2 + 2 \cdot x}{B}}{{F}^{2}}, -1 \cdot \frac{1 + x}{B}\right) \]

      8. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\frac{2 \cdot x + 2}{B}}{{F}^{2}}, -1 \cdot \frac{1 + x}{B}\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\frac{\mathsf{fma}\left(2, x, 2\right)}{B}}{{F}^{2}}, -1 \cdot \frac{1 + x}{B}\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\frac{\mathsf{fma}\left(2, x, 2\right)}{B}}{F \cdot F}, -1 \cdot \frac{1 + x}{B}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\frac{\mathsf{fma}\left(2, x, 2\right)}{B}}{F \cdot F}, -1 \cdot \frac{1 + x}{B}\right) \]

      12. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\frac{\mathsf{fma}\left(2, x, 2\right)}{B}}{F \cdot F}, \mathsf{neg}\left(\frac{1 + x}{B}\right)\right) \]

      13. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\frac{\mathsf{fma}\left(2, x, 2\right)}{B}}{F \cdot F}, -\frac{1 + x}{B}\right) \]

      14. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\frac{\mathsf{fma}\left(2, x, 2\right)}{B}}{F \cdot F}, -\frac{1 + x}{B}\right) \]

      15. lower-+.f64 39.7

          \[\leadsto \mathsf{fma}\left(0.5, \frac{\frac{\mathsf{fma}\left(2, x, 2\right)}{B}}{F \cdot F}, -\frac{1 + x}{B}\right) \]

   8. Applied rewrites 39.7%

      \[\leadsto \mathsf{fma}\left(0.5, \color{blue}{\frac{\frac{\mathsf{fma}\left(2, x, 2\right)}{B}}{F \cdot F}}, -\frac{1 + x}{B}\right) \]

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{2}{B \cdot \color{blue}{{F}^{2}}}, -\frac{1 + x}{B}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{2}{B \cdot {F}^{\color{blue}{2}}}, -\frac{1 + x}{B}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{2}{B \cdot {F}^{2}}, -\frac{1 + x}{B}\right) \]

       3. pow2 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{2}{B \cdot \left(F \cdot F\right)}, -\frac{1 + x}{B}\right) \]

       4. lift-*.f64 50.8

          \[\leadsto \mathsf{fma}\left(0.5, \frac{2}{B \cdot \left(F \cdot F\right)}, -\frac{1 + x}{B}\right) \]

   11. Applied rewrites 50.8%

       \[\leadsto \mathsf{fma}\left(0.5, \frac{2}{B \cdot \color{blue}{\left(F \cdot F\right)}}, -\frac{1 + x}{B}\right) \]

   if -6.79999999999999982 < F < 1.94999999999999992e-8

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot F - x}{B} \]
   7. Step-by-step derivation

      1. inv-pow N/A

         \[\leadsto \frac{\sqrt{{\left(2 + 2 \cdot x\right)}^{-1}} \cdot F - x}{B} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\sqrt{{\left(2 + 2 \cdot x\right)}^{-1}} \cdot F - x}{B} \]

      3. +-commutative N/A

         \[\leadsto \frac{\sqrt{{\left(2 \cdot x + 2\right)}^{-1}} \cdot F - x}{B} \]

      4. lower-fma.f64 50.5

         \[\leadsto \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}} \cdot F - x}{B} \]

   8. Applied rewrites 50.5%

      \[\leadsto \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}} \cdot F - x}{B} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\sqrt{\frac{1}{2}} \cdot F - x}{B} \]
   10. Step-by-step derivation

   11. Applied rewrites 50.5%

       \[\leadsto \frac{\sqrt{0.5} \cdot F - x}{B} \]

   if 1.94999999999999992e-8 < F

   1. Initial program 59.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 38.0%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) - x}{B} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} + 1\right) - x}{B} \]

      2. div-add N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \left(\frac{2}{{F}^{2}} + \frac{2 \cdot x}{{F}^{2}}\right) + 1\right) - x}{B} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \left(\frac{2 \cdot 1}{{F}^{2}} + \frac{2 \cdot x}{{F}^{2}}\right) + 1\right) - x}{B} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \left(2 \cdot \frac{1}{{F}^{2}} + \frac{2 \cdot x}{{F}^{2}}\right) + 1\right) - x}{B} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \left(2 \cdot \frac{1}{{F}^{2}} + 2 \cdot \frac{x}{{F}^{2}}\right) + 1\right) - x}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \left(2 \cdot \frac{x}{{F}^{2}} + 2 \cdot \frac{1}{{F}^{2}}\right) + 1\right) - x}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, 2 \cdot \frac{x}{{F}^{2}} + 2 \cdot \frac{1}{{F}^{2}}, 1\right) - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, 2 \cdot \frac{1}{{F}^{2}} + 2 \cdot \frac{x}{{F}^{2}}, 1\right) - x}{B} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2 \cdot 1}{{F}^{2}} + 2 \cdot \frac{x}{{F}^{2}}, 1\right) - x}{B} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2}{{F}^{2}} + 2 \cdot \frac{x}{{F}^{2}}, 1\right) - x}{B} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2}{{F}^{2}} + \frac{2 \cdot x}{{F}^{2}}, 1\right) - x}{B} \]

      12. div-add N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{2}}, 1\right) - x}{B} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{2}}, 1\right) - x}{B} \]

      14. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2 \cdot x + 2}{{F}^{2}}, 1\right) - x}{B} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{{F}^{2}}, 1\right) - x}{B} \]

      16. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}{B} \]

      17. lift-*.f64 49.4

          \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}{B} \]

   8. Applied rewrites 49.4%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 50.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.8:\\ \;\;\;\;\frac{\left(B \cdot B\right) \cdot \left(-0.16666666666666666 \cdot x - \mathsf{fma}\left(-0.5, x, 0.16666666666666666\right)\right) - \left(1 + x\right)}{B}\\ \mathbf{elif}\;F \leq 1.95 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{0.5} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.8)
   (/
    (-
     (*
      (* B B)
      (- (* -0.16666666666666666 x) (fma -0.5 x 0.16666666666666666)))
     (+ 1.0 x))
    B)
   (if (<= F 1.95e-8)
     (/ (- (* (sqrt 0.5) F) x) B)
     (/ (- (fma -0.5 (/ (fma 2.0 x 2.0) (* F F)) 1.0) x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.8) {
		tmp = (((B * B) * ((-0.16666666666666666 * x) - fma(-0.5, x, 0.16666666666666666))) - (1.0 + x)) / B;
	} else if (F <= 1.95e-8) {
		tmp = ((sqrt(0.5) * F) - x) / B;
	} else {
		tmp = (fma(-0.5, (fma(2.0, x, 2.0) / (F * F)), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.8)
		tmp = Float64(Float64(Float64(Float64(B * B) * Float64(Float64(-0.16666666666666666 * x) - fma(-0.5, x, 0.16666666666666666))) - Float64(1.0 + x)) / B);
	elseif (F <= 1.95e-8)
		tmp = Float64(Float64(Float64(sqrt(0.5) * F) - x) / B);
	else
		tmp = Float64(Float64(fma(-0.5, Float64(fma(2.0, x, 2.0) / Float64(F * F)), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.8], N[(N[(N[(N[(B * B), $MachinePrecision] * N[(N[(-0.16666666666666666 * x), $MachinePrecision] - N[(-0.5 * x + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.95e-8], N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(-0.5 * N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.79999999999999982

   1. Initial program 59.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right) \]

      3. div-add-rev N/A

         \[\leadsto -\frac{1 + x \cdot \cos B}{\sin B} \]

      4. lower-/.f64 N/A

         \[\leadsto -\frac{1 + x \cdot \cos B}{\sin B} \]

      5. lower-+.f64 N/A

         \[\leadsto -\frac{1 + x \cdot \cos B}{\sin B} \]

      6. *-commutative N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      7. lower-*.f64 N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      8. lower-cos.f64 N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      9. lift-sin.f64 99.3

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{-\frac{1 + \cos B \cdot x}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \frac{{B}^{2} \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{\color{blue}{B}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{B} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{B} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{B} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{B} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{B} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{-1}{2} \cdot x + \frac{1}{6}\right)\right) - \left(1 + x\right)}{B} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\frac{-1}{6} \cdot x - \mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{6}\right)\right) - \left(1 + x\right)}{B} \]

      10. lower-+.f64 50.7

          \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(-0.16666666666666666 \cdot x - \mathsf{fma}\left(-0.5, x, 0.16666666666666666\right)\right) - \left(1 + x\right)}{B} \]

   8. Applied rewrites 50.7%

      \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(-0.16666666666666666 \cdot x - \mathsf{fma}\left(-0.5, x, 0.16666666666666666\right)\right) - \left(1 + x\right)}{\color{blue}{B}} \]

   if -6.79999999999999982 < F < 1.94999999999999992e-8

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot F - x}{B} \]
   7. Step-by-step derivation

      1. inv-pow N/A

         \[\leadsto \frac{\sqrt{{\left(2 + 2 \cdot x\right)}^{-1}} \cdot F - x}{B} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\sqrt{{\left(2 + 2 \cdot x\right)}^{-1}} \cdot F - x}{B} \]

      3. +-commutative N/A

         \[\leadsto \frac{\sqrt{{\left(2 \cdot x + 2\right)}^{-1}} \cdot F - x}{B} \]

      4. lower-fma.f64 50.5

         \[\leadsto \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}} \cdot F - x}{B} \]

   8. Applied rewrites 50.5%

      \[\leadsto \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}} \cdot F - x}{B} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\sqrt{\frac{1}{2}} \cdot F - x}{B} \]
   10. Step-by-step derivation

   11. Applied rewrites 50.5%

       \[\leadsto \frac{\sqrt{0.5} \cdot F - x}{B} \]

   if 1.94999999999999992e-8 < F

   1. Initial program 59.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 38.0%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{\left(1 + \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}\right) - x}{B} \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} + 1\right) - x}{B} \]

      2. div-add N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \left(\frac{2}{{F}^{2}} + \frac{2 \cdot x}{{F}^{2}}\right) + 1\right) - x}{B} \]

      3. metadata-eval N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \left(\frac{2 \cdot 1}{{F}^{2}} + \frac{2 \cdot x}{{F}^{2}}\right) + 1\right) - x}{B} \]

      4. associate-*r/ N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \left(2 \cdot \frac{1}{{F}^{2}} + \frac{2 \cdot x}{{F}^{2}}\right) + 1\right) - x}{B} \]

      5. associate-*r/ N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \left(2 \cdot \frac{1}{{F}^{2}} + 2 \cdot \frac{x}{{F}^{2}}\right) + 1\right) - x}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\left(\frac{-1}{2} \cdot \left(2 \cdot \frac{x}{{F}^{2}} + 2 \cdot \frac{1}{{F}^{2}}\right) + 1\right) - x}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, 2 \cdot \frac{x}{{F}^{2}} + 2 \cdot \frac{1}{{F}^{2}}, 1\right) - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, 2 \cdot \frac{1}{{F}^{2}} + 2 \cdot \frac{x}{{F}^{2}}, 1\right) - x}{B} \]

      9. associate-*r/ N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2 \cdot 1}{{F}^{2}} + 2 \cdot \frac{x}{{F}^{2}}, 1\right) - x}{B} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2}{{F}^{2}} + 2 \cdot \frac{x}{{F}^{2}}, 1\right) - x}{B} \]

      11. associate-*r/ N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2}{{F}^{2}} + \frac{2 \cdot x}{{F}^{2}}, 1\right) - x}{B} \]

      12. div-add N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{2}}, 1\right) - x}{B} \]

      13. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2 + 2 \cdot x}{{F}^{2}}, 1\right) - x}{B} \]

      14. +-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{2 \cdot x + 2}{{F}^{2}}, 1\right) - x}{B} \]

      15. lower-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{{F}^{2}}, 1\right) - x}{B} \]

      16. pow2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\frac{-1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}{B} \]

      17. lift-*.f64 49.4

          \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}{B} \]

   8. Applied rewrites 49.4%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 50.2% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.8:\\ \;\;\;\;\frac{\left(B \cdot B\right) \cdot \left(-0.16666666666666666 \cdot x - \mathsf{fma}\left(-0.5, x, 0.16666666666666666\right)\right) - \left(1 + x\right)}{B}\\ \mathbf{elif}\;F \leq 7500000000000:\\ \;\;\;\;\frac{\sqrt{0.5} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.8)
   (/
    (-
     (*
      (* B B)
      (- (* -0.16666666666666666 x) (fma -0.5 x 0.16666666666666666)))
     (+ 1.0 x))
    B)
   (if (<= F 7500000000000.0) (/ (- (* (sqrt 0.5) F) x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.8) {
		tmp = (((B * B) * ((-0.16666666666666666 * x) - fma(-0.5, x, 0.16666666666666666))) - (1.0 + x)) / B;
	} else if (F <= 7500000000000.0) {
		tmp = ((sqrt(0.5) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.8)
		tmp = Float64(Float64(Float64(Float64(B * B) * Float64(Float64(-0.16666666666666666 * x) - fma(-0.5, x, 0.16666666666666666))) - Float64(1.0 + x)) / B);
	elseif (F <= 7500000000000.0)
		tmp = Float64(Float64(Float64(sqrt(0.5) * F) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.8], N[(N[(N[(N[(B * B), $MachinePrecision] * N[(N[(-0.16666666666666666 * x), $MachinePrecision] - N[(-0.5 * x + 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 7500000000000.0], N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.79999999999999982

   1. Initial program 59.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right) \]

      3. div-add-rev N/A

         \[\leadsto -\frac{1 + x \cdot \cos B}{\sin B} \]

      4. lower-/.f64 N/A

         \[\leadsto -\frac{1 + x \cdot \cos B}{\sin B} \]

      5. lower-+.f64 N/A

         \[\leadsto -\frac{1 + x \cdot \cos B}{\sin B} \]

      6. *-commutative N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      7. lower-*.f64 N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      8. lower-cos.f64 N/A

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

      9. lift-sin.f64 99.3

         \[\leadsto -\frac{1 + \cos B \cdot x}{\sin B} \]

   5. Applied rewrites 99.3%

      \[\leadsto \color{blue}{-\frac{1 + \cos B \cdot x}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \frac{{B}^{2} \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{\color{blue}{B}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{B} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{{B}^{2} \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{B} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{B} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{B} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{B} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{1}{6} + \frac{-1}{2} \cdot x\right)\right) - \left(1 + x\right)}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\frac{-1}{6} \cdot x - \left(\frac{-1}{2} \cdot x + \frac{1}{6}\right)\right) - \left(1 + x\right)}{B} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(\frac{-1}{6} \cdot x - \mathsf{fma}\left(\frac{-1}{2}, x, \frac{1}{6}\right)\right) - \left(1 + x\right)}{B} \]

      10. lower-+.f64 50.7

          \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(-0.16666666666666666 \cdot x - \mathsf{fma}\left(-0.5, x, 0.16666666666666666\right)\right) - \left(1 + x\right)}{B} \]

   8. Applied rewrites 50.7%

      \[\leadsto \frac{\left(B \cdot B\right) \cdot \left(-0.16666666666666666 \cdot x - \mathsf{fma}\left(-0.5, x, 0.16666666666666666\right)\right) - \left(1 + x\right)}{\color{blue}{B}} \]

   if -6.79999999999999982 < F < 7.5e12

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 50.5%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot F - x}{B} \]
   7. Step-by-step derivation

      1. inv-pow N/A

         \[\leadsto \frac{\sqrt{{\left(2 + 2 \cdot x\right)}^{-1}} \cdot F - x}{B} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\sqrt{{\left(2 + 2 \cdot x\right)}^{-1}} \cdot F - x}{B} \]

      3. +-commutative N/A

         \[\leadsto \frac{\sqrt{{\left(2 \cdot x + 2\right)}^{-1}} \cdot F - x}{B} \]

      4. lower-fma.f64 49.7

         \[\leadsto \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}} \cdot F - x}{B} \]

   8. Applied rewrites 49.7%

      \[\leadsto \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}} \cdot F - x}{B} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\sqrt{\frac{1}{2}} \cdot F - x}{B} \]
   10. Step-by-step derivation

   11. Applied rewrites 49.6%

       \[\leadsto \frac{\sqrt{0.5} \cdot F - x}{B} \]

   if 7.5e12 < F

   1. Initial program 57.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 37.7%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.5%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 49.6% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.05 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 7500000000000:\\ \;\;\;\;\frac{\sqrt{0.5} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.05e-29)
   (/ (- -1.0 x) B)
   (if (<= F 7500000000000.0) (/ (- (* (sqrt 0.5) F) x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.05e-29) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 7500000000000.0) {
		tmp = ((sqrt(0.5) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.05d-29)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 7500000000000.0d0) then
        tmp = ((sqrt(0.5d0) * f) - x) / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.05e-29) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 7500000000000.0) {
		tmp = ((Math.sqrt(0.5) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.05e-29:
		tmp = (-1.0 - x) / B
	elif F <= 7500000000000.0:
		tmp = ((math.sqrt(0.5) * F) - x) / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.05e-29)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 7500000000000.0)
		tmp = Float64(Float64(Float64(sqrt(0.5) * F) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.05e-29)
		tmp = (-1.0 - x) / B;
	elseif (F <= 7500000000000.0)
		tmp = ((sqrt(0.5) * F) - x) / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.05e-29], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 7500000000000.0], N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.04999999999999995e-29

   1. Initial program 62.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 38.7%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.6%

      \[\leadsto \frac{-1 - x}{B} \]

   if -1.04999999999999995e-29 < F < 7.5e12

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot F - x}{B} \]
   7. Step-by-step derivation

      1. inv-pow N/A

         \[\leadsto \frac{\sqrt{{\left(2 + 2 \cdot x\right)}^{-1}} \cdot F - x}{B} \]

      2. lower-pow.f64 N/A

         \[\leadsto \frac{\sqrt{{\left(2 + 2 \cdot x\right)}^{-1}} \cdot F - x}{B} \]

      3. +-commutative N/A

         \[\leadsto \frac{\sqrt{{\left(2 \cdot x + 2\right)}^{-1}} \cdot F - x}{B} \]

      4. lower-fma.f64 49.9

         \[\leadsto \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}} \cdot F - x}{B} \]

   8. Applied rewrites 49.9%

      \[\leadsto \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}} \cdot F - x}{B} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\sqrt{\frac{1}{2}} \cdot F - x}{B} \]
   10. Step-by-step derivation

   11. Applied rewrites 49.8%

       \[\leadsto \frac{\sqrt{0.5} \cdot F - x}{B} \]

   if 7.5e12 < F

   1. Initial program 57.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 37.7%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.5%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 43.4% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.2e-75)
   (/ (- -1.0 x) B)
   (if (<= F 4.2e-93) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.2e-75) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.2e-93) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-4.2d-75)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 4.2d-93) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.2e-75) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.2e-93) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.2e-75:
		tmp = (-1.0 - x) / B
	elif F <= 4.2e-93:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.2e-75)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 4.2e-93)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -4.2e-75)
		tmp = (-1.0 - x) / B;
	elseif (F <= 4.2e-93)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.2e-75], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4.2e-93], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.2000000000000002e-75

   1. Initial program 66.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.8%

      \[\leadsto \frac{-1 - x}{B} \]

   if -4.2000000000000002e-75 < F < 4.2000000000000002e-93

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 50.6%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

      2. lower-neg.f64 39.2

         \[\leadsto \frac{-x}{B} \]

   8. Applied rewrites 39.2%

      \[\leadsto \frac{-x}{B} \]

   if 4.2000000000000002e-93 < F

   1. Initial program 66.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 40.7%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.0%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 36.7% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.2e-75) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.2e-75) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-4.2d-75)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.2e-75) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.2e-75:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.2e-75)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -4.2e-75)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.2e-75], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -4.2000000000000002e-75

   1. Initial program 66.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 39.5%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.8%

      \[\leadsto \frac{-1 - x}{B} \]

   if -4.2000000000000002e-75 < F

   1. Initial program 82.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 45.5%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

      2. lower-neg.f64 32.1

         \[\leadsto \frac{-x}{B} \]

   8. Applied rewrites 32.1%

      \[\leadsto \frac{-x}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 29.6% accurate, 26.3× speedup

Math

\[\begin{array}{l} \\ \frac{-x}{B} \end{array} \]

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 77.0%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

5. Applied rewrites 43.5%

   \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

6. Taylor expanded in F around 0

   \[\leadsto \frac{-1 \cdot x}{B} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

   2. lower-neg.f64 29.6

      \[\leadsto \frac{-x}{B} \]

8. Applied rewrites 29.6%

   \[\leadsto \frac{-x}{B} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025091 
(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))))))