VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.1% → 99.6%

Time: 12.0s

Alternatives: 23

Speedup: 1.4×

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: 77.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.4 \cdot 10^{+43}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 35000:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.4e+43)
   (/ (- (fma (cos B) x 1.0)) (sin B))
   (if (<= F 35000.0)
     (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B)) (/ (- x) (tan B)))
     (+ (* x (/ -1.0 (tan B))) (pow (sin B) -1.0)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4e+43) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else if (F <= 35000.0) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), (-x / tan(B)));
	} else {
		tmp = (x * (-1.0 / tan(B))) + pow(sin(B), -1.0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.4e+43)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	elseif (F <= 35000.0)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + (sin(B) ^ -1.0));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.4e+43], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 35000.0], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.40000000000000009e43

   1. Initial program 54.9%

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

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

      2. div-add-rev N/A

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

      3. distribute-neg-frac N/A

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

      4. lower-/.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -1.40000000000000009e43 < F < 35000

   1. Initial program 98.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

   if 35000 < F

   1. Initial program 52.8%

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

   3. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4 \cdot 10^{+43}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 35000:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\sin B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+50}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 35000:\\ \;\;\;\;\mathsf{fma}\left(F, {\left(\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5e+50)
   (/ (- (fma (cos B) x 1.0)) (sin B))
   (if (<= F 35000.0)
     (fma
      F
      (pow (* (sin B) (sqrt (fma 2.0 x (fma F F 2.0)))) -1.0)
      (/ (- x) (tan B)))
     (+ (* x (/ -1.0 (tan B))) (pow (sin B) -1.0)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5e+50) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else if (F <= 35000.0) {
		tmp = fma(F, pow((sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0)))), -1.0), (-x / tan(B)));
	} else {
		tmp = (x * (-1.0 / tan(B))) + pow(sin(B), -1.0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5e+50)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	elseif (F <= 35000.0)
		tmp = fma(F, (Float64(sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0)))) ^ -1.0), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + (sin(B) ^ -1.0));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5e+50], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 35000.0], N[(F * N[Power[N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5e50

   1. Initial program 54.0%

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

   3. Taylor expanded in 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

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

      2. div-add-rev N/A

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

      3. distribute-neg-frac N/A

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

      4. lower-/.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -5e50 < F < 35000

   1. Initial program 98.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. inv-pow N/A

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

      16. lower-/.f64 N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-+r+ N/A

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

      22. +-commutative N/A

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

   6. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.6

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-fma.f64 99.6

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

   8. Applied rewrites 99.6%

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

   if 35000 < F

   1. Initial program 52.8%

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

   3. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+50}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 35000:\\ \;\;\;\;\mathsf{fma}\left(F, {\left(\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\sin B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.00043:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.7:\\ \;\;\;\;\mathsf{fma}\left(F, {\left(\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.00043)
   (/ (- (fma (cos B) x 1.0)) (sin B))
   (if (<= F 1.7)
     (fma F (pow (* (sin B) (sqrt (fma 2.0 x 2.0))) -1.0) (/ (- x) (tan B)))
     (+ (* x (/ -1.0 (tan B))) (pow (sin B) -1.0)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.00043) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else if (F <= 1.7) {
		tmp = fma(F, pow((sin(B) * sqrt(fma(2.0, x, 2.0))), -1.0), (-x / tan(B)));
	} else {
		tmp = (x * (-1.0 / tan(B))) + pow(sin(B), -1.0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.00043)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	elseif (F <= 1.7)
		tmp = fma(F, (Float64(sin(B) * sqrt(fma(2.0, x, 2.0))) ^ -1.0), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + (sin(B) ^ -1.0));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.00043], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.7], N[(F * N[Power[N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 * x + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.29999999999999989e-4

   1. Initial program 58.0%

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

   3. Taylor expanded in 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

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

      2. div-add-rev N/A

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

      3. distribute-neg-frac N/A

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

      4. lower-/.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -4.29999999999999989e-4 < F < 1.69999999999999996

   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

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. inv-pow N/A

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

      16. lower-/.f64 N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-+r+ N/A

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

      22. +-commutative N/A

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

   6. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.6

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-fma.f64 99.6

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in F around 0

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

   11. Applied rewrites 99.6%

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

   if 1.69999999999999996 < F

   1. Initial program 53.3%

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

   3. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.2

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

   5. Applied rewrites 99.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00043:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.7:\\ \;\;\;\;\mathsf{fma}\left(F, {\left(\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, 2\right)}\right)}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\sin B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 92.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 5.7 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(F, {\left(\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B\right)}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 30000:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{\mathsf{fma}\left(B \cdot B, 0.3333333333333333, 1\right) \cdot B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.6e-17)
   (/ (- (fma (cos B) x 1.0)) (sin B))
   (if (<= F 5.7e-121)
     (fma
      F
      (pow (* (sqrt (+ (fma 2.0 x (* F F)) 2.0)) B) -1.0)
      (/ (- x) (tan B)))
     (if (<= F 30000.0)
       (-
        (/ (/ F (sin B)) (sqrt (fma x 2.0 (fma F F 2.0))))
        (/ x (* (fma (* B B) 0.3333333333333333 1.0) B)))
       (+ (* x (/ -1.0 (tan B))) (pow (sin B) -1.0))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.6e-17) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else if (F <= 5.7e-121) {
		tmp = fma(F, pow((sqrt((fma(2.0, x, (F * F)) + 2.0)) * B), -1.0), (-x / tan(B)));
	} else if (F <= 30000.0) {
		tmp = ((F / sin(B)) / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - (x / (fma((B * B), 0.3333333333333333, 1.0) * B));
	} else {
		tmp = (x * (-1.0 / tan(B))) + pow(sin(B), -1.0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.6e-17)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	elseif (F <= 5.7e-121)
		tmp = fma(F, (Float64(sqrt(Float64(fma(2.0, x, Float64(F * F)) + 2.0)) * B) ^ -1.0), Float64(Float64(-x) / tan(B)));
	elseif (F <= 30000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - Float64(x / Float64(fma(Float64(B * B), 0.3333333333333333, 1.0) * B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + (sin(B) ^ -1.0));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.6e-17], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.7e-121], N[(F * N[Power[N[(N[Sqrt[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * B), $MachinePrecision], -1.0], $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 30000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[(N[(B * B), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.59999999999999995e-17

   1. Initial program 59.5%

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

   3. Taylor expanded in 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

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

      2. div-add-rev N/A

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

      3. distribute-neg-frac N/A

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

      4. lower-/.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-sin.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if -3.59999999999999995e-17 < F < 5.70000000000000014e-121

   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

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. inv-pow N/A

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

      16. lower-/.f64 N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-+r+ N/A

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

      22. +-commutative N/A

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

   6. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.7

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-fma.f64 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in B around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 86.8

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

   11. Applied rewrites 86.8%

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

   if 5.70000000000000014e-121 < F < 3e4

   1. Initial program 99.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. inv-pow N/A

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

      16. lower-/.f64 N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-+r+ N/A

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

      22. +-commutative N/A

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 84.2

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

   9. Applied rewrites 84.2%

      \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\color{blue}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}}\right) \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l/ N/A

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

       4. lift-fma.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. lift-/.f64 84.4

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

       10. lower-fma.f64 N/A

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

       11. lower-+.f64 N/A

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

   11. Applied rewrites 84.5%

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

   if 3e4 < F

   1. Initial program 52.8%

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

   3. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 5.7 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(F, {\left(\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B\right)}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 30000:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{\mathsf{fma}\left(B \cdot B, 0.3333333333333333, 1\right) \cdot B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\sin B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\\ \mathbf{if}\;F \leq -7.5 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{\sin B \cdot F}, t\_0\right)\\ \mathbf{elif}\;F \leq 5.7 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(F, {\left(\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B\right)}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(F, {\left(\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-1}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{F}^{-1}}{\sin B}, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (* (fma 0.3333333333333333 (* B B) 1.0) B))))
   (if (<= F -7.5e+52)
     (fma F (/ -1.0 (* (sin B) F)) t_0)
     (if (<= F 5.7e-121)
       (fma
        F
        (pow (* (sqrt (+ (fma 2.0 x (* F F)) 2.0)) B) -1.0)
        (/ (- x) (tan B)))
       (if (<= F 7.2e+48)
         (fma F (pow (* (sin B) (sqrt (fma 2.0 x (fma F F 2.0)))) -1.0) t_0)
         (fma F (/ (pow F -1.0) (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / (fma(0.3333333333333333, (B * B), 1.0) * B);
	double tmp;
	if (F <= -7.5e+52) {
		tmp = fma(F, (-1.0 / (sin(B) * F)), t_0);
	} else if (F <= 5.7e-121) {
		tmp = fma(F, pow((sqrt((fma(2.0, x, (F * F)) + 2.0)) * B), -1.0), (-x / tan(B)));
	} else if (F <= 7.2e+48) {
		tmp = fma(F, pow((sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0)))), -1.0), t_0);
	} else {
		tmp = fma(F, (pow(F, -1.0) / sin(B)), t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / Float64(fma(0.3333333333333333, Float64(B * B), 1.0) * B))
	tmp = 0.0
	if (F <= -7.5e+52)
		tmp = fma(F, Float64(-1.0 / Float64(sin(B) * F)), t_0);
	elseif (F <= 5.7e-121)
		tmp = fma(F, (Float64(sqrt(Float64(fma(2.0, x, Float64(F * F)) + 2.0)) * B) ^ -1.0), Float64(Float64(-x) / tan(B)));
	elseif (F <= 7.2e+48)
		tmp = fma(F, (Float64(sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0)))) ^ -1.0), t_0);
	else
		tmp = fma(F, Float64((F ^ -1.0) / sin(B)), t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[(N[(0.3333333333333333 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -7.5e+52], N[(F * N[(-1.0 / N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, 5.7e-121], N[(F * N[Power[N[(N[Sqrt[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * B), $MachinePrecision], -1.0], $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.2e+48], N[(F * N[Power[N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] + t$95$0), $MachinePrecision], N[(F * N[(N[Power[F, -1.0], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7.49999999999999995e52

   1. Initial program 52.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 65.3%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 79.1

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

   10. Applied rewrites 79.1%

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

   if -7.49999999999999995e52 < F < 5.70000000000000014e-121

   1. Initial program 98.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. inv-pow N/A

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

      16. lower-/.f64 N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-+r+ N/A

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

      22. +-commutative N/A

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

   6. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.7

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-fma.f64 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in B around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 86.3

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

   11. Applied rewrites 86.3%

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

   if 5.70000000000000014e-121 < F < 7.19999999999999967e48

   1. Initial program 99.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. inv-pow N/A

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

      16. lower-/.f64 N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-+r+ N/A

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

      22. +-commutative N/A

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

   6. Applied rewrites 99.1%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.4

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-fma.f64 99.4

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in B around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. +-commutative N/A

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

       4. lower-fma.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 82.2

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

   11. Applied rewrites 82.2%

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

   if 7.19999999999999967e48 < F

   1. Initial program 48.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 64.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. inv-pow N/A

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

      16. lower-/.f64 N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-+r+ N/A

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

      22. +-commutative N/A

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

   6. Applied rewrites 64.3%

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

   7. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 48.9

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

   9. Applied rewrites 48.9%

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

   10. Taylor expanded in F around inf

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

       1. lower-/.f64 83.2

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

   12. Applied rewrites 83.2%

       \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{F}}}{\sin B}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.5 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{\sin B \cdot F}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right)\\ \mathbf{elif}\;F \leq 5.7 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(F, {\left(\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B\right)}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{+48}:\\ \;\;\;\;\mathsf{fma}\left(F, {\left(\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-1}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{F}^{-1}}{\sin B}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 5.7 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(F, {\left(\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B\right)}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{\mathsf{fma}\left(B \cdot B, 0.3333333333333333, 1\right) \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{F}^{-1}}{\sin B}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.6e-17)
   (/ (- (fma (cos B) x 1.0)) (sin B))
   (if (<= F 5.7e-121)
     (fma
      F
      (pow (* (sqrt (+ (fma 2.0 x (* F F)) 2.0)) B) -1.0)
      (/ (- x) (tan B)))
     (if (<= F 1.15e+49)
       (-
        (/ (/ F (sin B)) (sqrt (fma x 2.0 (fma F F 2.0))))
        (/ x (* (fma (* B B) 0.3333333333333333 1.0) B)))
       (fma
        F
        (/ (pow F -1.0) (sin B))
        (/ (- x) (* (fma 0.3333333333333333 (* B B) 1.0) B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.6e-17) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else if (F <= 5.7e-121) {
		tmp = fma(F, pow((sqrt((fma(2.0, x, (F * F)) + 2.0)) * B), -1.0), (-x / tan(B)));
	} else if (F <= 1.15e+49) {
		tmp = ((F / sin(B)) / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - (x / (fma((B * B), 0.3333333333333333, 1.0) * B));
	} else {
		tmp = fma(F, (pow(F, -1.0) / sin(B)), (-x / (fma(0.3333333333333333, (B * B), 1.0) * B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.6e-17)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	elseif (F <= 5.7e-121)
		tmp = fma(F, (Float64(sqrt(Float64(fma(2.0, x, Float64(F * F)) + 2.0)) * B) ^ -1.0), Float64(Float64(-x) / tan(B)));
	elseif (F <= 1.15e+49)
		tmp = Float64(Float64(Float64(F / sin(B)) / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - Float64(x / Float64(fma(Float64(B * B), 0.3333333333333333, 1.0) * B)));
	else
		tmp = fma(F, Float64((F ^ -1.0) / sin(B)), Float64(Float64(-x) / Float64(fma(0.3333333333333333, Float64(B * B), 1.0) * B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.6e-17], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.7e-121], N[(F * N[Power[N[(N[Sqrt[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * B), $MachinePrecision], -1.0], $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.15e+49], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[(N[(B * B), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(N[Power[F, -1.0], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[(N[(0.3333333333333333 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.59999999999999995e-17

   1. Initial program 59.5%

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

   3. Taylor expanded in 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

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

      2. div-add-rev N/A

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

      3. distribute-neg-frac N/A

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

      4. lower-/.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. +-commutative N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-sin.f64 98.2

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

   5. Applied rewrites 98.2%

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

   if -3.59999999999999995e-17 < F < 5.70000000000000014e-121

   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

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. inv-pow N/A

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

      16. lower-/.f64 N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-+r+ N/A

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

      22. +-commutative N/A

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

   6. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.7

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-fma.f64 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in B around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 86.8

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

   11. Applied rewrites 86.8%

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

   if 5.70000000000000014e-121 < F < 1.15000000000000001e49

   1. Initial program 99.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. inv-pow N/A

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

      16. lower-/.f64 N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-+r+ N/A

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

      22. +-commutative N/A

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 82.1

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

   9. Applied rewrites 82.1%

      \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\color{blue}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}}\right) \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l/ N/A

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

       4. lift-fma.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. lift-/.f64 82.3

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

       10. lower-fma.f64 N/A

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

       11. lower-+.f64 N/A

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

   11. Applied rewrites 82.4%

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

   if 1.15000000000000001e49 < F

   1. Initial program 47.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 63.8%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. inv-pow N/A

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

      16. lower-/.f64 N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-+r+ N/A

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

      22. +-commutative N/A

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

   6. Applied rewrites 63.8%

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

   7. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 48.4

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

   9. Applied rewrites 48.4%

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

   10. Taylor expanded in F around inf

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

       1. lower-/.f64 83.2

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

   12. Applied rewrites 83.2%

       \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{F}}}{\sin B}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 5.7 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(F, {\left(\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B\right)}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{\mathsf{fma}\left(B \cdot B, 0.3333333333333333, 1\right) \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{F}^{-1}}{\sin B}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\\ \mathbf{if}\;F \leq -7.5 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{\sin B \cdot F}, t\_0\right)\\ \mathbf{elif}\;F \leq 5.7 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(F, {\left(\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B\right)}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{\mathsf{fma}\left(B \cdot B, 0.3333333333333333, 1\right) \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{F}^{-1}}{\sin B}, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (* (fma 0.3333333333333333 (* B B) 1.0) B))))
   (if (<= F -7.5e+52)
     (fma F (/ -1.0 (* (sin B) F)) t_0)
     (if (<= F 5.7e-121)
       (fma
        F
        (pow (* (sqrt (+ (fma 2.0 x (* F F)) 2.0)) B) -1.0)
        (/ (- x) (tan B)))
       (if (<= F 1.15e+49)
         (-
          (/ (/ F (sin B)) (sqrt (fma x 2.0 (fma F F 2.0))))
          (/ x (* (fma (* B B) 0.3333333333333333 1.0) B)))
         (fma F (/ (pow F -1.0) (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / (fma(0.3333333333333333, (B * B), 1.0) * B);
	double tmp;
	if (F <= -7.5e+52) {
		tmp = fma(F, (-1.0 / (sin(B) * F)), t_0);
	} else if (F <= 5.7e-121) {
		tmp = fma(F, pow((sqrt((fma(2.0, x, (F * F)) + 2.0)) * B), -1.0), (-x / tan(B)));
	} else if (F <= 1.15e+49) {
		tmp = ((F / sin(B)) / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - (x / (fma((B * B), 0.3333333333333333, 1.0) * B));
	} else {
		tmp = fma(F, (pow(F, -1.0) / sin(B)), t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / Float64(fma(0.3333333333333333, Float64(B * B), 1.0) * B))
	tmp = 0.0
	if (F <= -7.5e+52)
		tmp = fma(F, Float64(-1.0 / Float64(sin(B) * F)), t_0);
	elseif (F <= 5.7e-121)
		tmp = fma(F, (Float64(sqrt(Float64(fma(2.0, x, Float64(F * F)) + 2.0)) * B) ^ -1.0), Float64(Float64(-x) / tan(B)));
	elseif (F <= 1.15e+49)
		tmp = Float64(Float64(Float64(F / sin(B)) / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - Float64(x / Float64(fma(Float64(B * B), 0.3333333333333333, 1.0) * B)));
	else
		tmp = fma(F, Float64((F ^ -1.0) / sin(B)), t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[(N[(0.3333333333333333 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -7.5e+52], N[(F * N[(-1.0 / N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, 5.7e-121], N[(F * N[Power[N[(N[Sqrt[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * B), $MachinePrecision], -1.0], $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.15e+49], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[(N[(B * B), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(N[Power[F, -1.0], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7.49999999999999995e52

   1. Initial program 52.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 65.3%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 79.1

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

   10. Applied rewrites 79.1%

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

   if -7.49999999999999995e52 < F < 5.70000000000000014e-121

   1. Initial program 98.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. inv-pow N/A

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

      16. lower-/.f64 N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-+r+ N/A

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

      22. +-commutative N/A

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

   6. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.7

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-fma.f64 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in B around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 86.3

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

   11. Applied rewrites 86.3%

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

   if 5.70000000000000014e-121 < F < 1.15000000000000001e49

   1. Initial program 99.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. inv-pow N/A

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

      16. lower-/.f64 N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-+r+ N/A

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

      22. +-commutative N/A

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 82.1

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

   9. Applied rewrites 82.1%

      \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\color{blue}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}}\right) \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l/ N/A

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

       4. lift-fma.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-fma.f64 N/A

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

       7. *-commutative N/A

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

       8. lift-*.f64 N/A

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

       9. lift-/.f64 82.3

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

       10. lower-fma.f64 N/A

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

       11. lower-+.f64 N/A

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

   11. Applied rewrites 82.4%

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

   if 1.15000000000000001e49 < F

   1. Initial program 47.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 63.8%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+l+ N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. inv-pow N/A

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

      16. lower-/.f64 N/A

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

      17. lift-fma.f64 N/A

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

      18. lift-*.f64 N/A

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

      19. lift-fma.f64 N/A

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

      20. lift-*.f64 N/A

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

      21. associate-+r+ N/A

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

      22. +-commutative N/A

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

   6. Applied rewrites 63.8%

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

   7. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 48.4

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

   9. Applied rewrites 48.4%

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

   10. Taylor expanded in F around inf

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

       1. lower-/.f64 83.2

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

   12. Applied rewrites 83.2%

       \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{F}}}{\sin B}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.5 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{\sin B \cdot F}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right)\\ \mathbf{elif}\;F \leq 5.7 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(F, {\left(\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B\right)}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{+49}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{\mathsf{fma}\left(B \cdot B, 0.3333333333333333, 1\right) \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{F}^{-1}}{\sin B}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-114}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{\left(\mathsf{fma}\left(-0.16666666666666666, B \cdot B, 1\right) \cdot B\right) \cdot F}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{F}^{-1}}{\sin B}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -1e-114)
   (fma
    F
    (/ -1.0 (* (* (fma -0.16666666666666666 (* B B) 1.0) B) F))
    (/ (- x) (tan B)))
   (if (<= x 8.5e-114)
     (* (sqrt (pow (fma F F 2.0) -1.0)) (/ F (sin B)))
     (if (<= x 1.15e-58)
       (fma
        F
        (/ (pow F -1.0) (sin B))
        (/ (- x) (* (fma 0.3333333333333333 (* B B) 1.0) B)))
       (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -1e-114) {
		tmp = fma(F, (-1.0 / ((fma(-0.16666666666666666, (B * B), 1.0) * B) * F)), (-x / tan(B)));
	} else if (x <= 8.5e-114) {
		tmp = sqrt(pow(fma(F, F, 2.0), -1.0)) * (F / sin(B));
	} else if (x <= 1.15e-58) {
		tmp = fma(F, (pow(F, -1.0) / sin(B)), (-x / (fma(0.3333333333333333, (B * B), 1.0) * B)));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -1e-114)
		tmp = fma(F, Float64(-1.0 / Float64(Float64(fma(-0.16666666666666666, Float64(B * B), 1.0) * B) * F)), Float64(Float64(-x) / tan(B)));
	elseif (x <= 8.5e-114)
		tmp = Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * Float64(F / sin(B)));
	elseif (x <= 1.15e-58)
		tmp = fma(F, Float64((F ^ -1.0) / sin(B)), Float64(Float64(-x) / Float64(fma(0.3333333333333333, Float64(B * B), 1.0) * B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -1e-114], N[(F * N[(-1.0 / N[(N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.5e-114], N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.15e-58], N[(F * N[(N[Power[F, -1.0], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[(N[(0.3333333333333333 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.0000000000000001e-114

   1. Initial program 74.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 84.1%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sin.f64 52.3

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

   7. Applied rewrites 52.3%

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

   8. Taylor expanded in B around 0

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

   10. Applied rewrites 64.8%

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

   if -1.0000000000000001e-114 < x < 8.5000000000000006e-114

   1. Initial program 78.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}} \cdot \frac{F}{\sin B} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}} \cdot \frac{F}{\sin B} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}} \cdot \frac{F}{\sin B} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}} \cdot \frac{F}{\sin B} \]

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 62.3

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

   5. Applied rewrites 62.3%

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

   if 8.5000000000000006e-114 < x < 1.1499999999999999e-58

   1. Initial program 26.5%

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

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}, -x \cdot \frac{1}{\tan B}\right)} \]

   4. Applied rewrites 34.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{\left(\frac{-1}{2}\right)}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      3. sqrt-pow2 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{{\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-1}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      4. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{2 \cdot x + \mathsf{fma}\left(F, F, 2\right)}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{2 \cdot x} + \mathsf{fma}\left(F, F, 2\right)}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{\mathsf{fma}\left(F, F, 2\right) + 2 \cdot x}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{\left(F \cdot F + 2\right)} + 2 \cdot x}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\left(\color{blue}{F \cdot F} + 2\right) + 2 \cdot x}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      9. associate-+l+ N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{F \cdot F + \left(2 + 2 \cdot x\right)}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{F \cdot F + \color{blue}{\left(2 \cdot x + 2\right)}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{F \cdot F} + \left(2 \cdot x + 2\right)}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{F \cdot F + \left(\color{blue}{2 \cdot x} + 2\right)}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      13. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{F \cdot F + \color{blue}{\mathsf{fma}\left(2, x, 2\right)}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      14. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      15. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      17. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\color{blue}{F \cdot F + \mathsf{fma}\left(2, x, 2\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\color{blue}{F \cdot F} + \mathsf{fma}\left(2, x, 2\right)}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      19. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{F \cdot F + \color{blue}{\left(2 \cdot x + 2\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{F \cdot F + \left(\color{blue}{2 \cdot x} + 2\right)}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      21. associate-+r+ N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\color{blue}{\left(F \cdot F + 2 \cdot x\right) + 2}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      22. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\color{blue}{\left(2 \cdot x + F \cdot F\right)} + 2}}}{\sin B}, \frac{-x}{\tan B}\right) \]

   6. Applied rewrites 34.2%

      \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

   7. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\color{blue}{B \cdot \left(1 + \frac{1}{3} \cdot {B}^{2}\right)}}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\color{blue}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot B}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\color{blue}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot B}}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\color{blue}{\left(\frac{1}{3} \cdot {B}^{2} + 1\right)} \cdot B}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\color{blue}{\mathsf{fma}\left(\frac{1}{3}, {B}^{2}, 1\right)} \cdot B}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\mathsf{fma}\left(\frac{1}{3}, \color{blue}{B \cdot B}, 1\right) \cdot B}\right) \]

      6. lower-*.f64 19.0

         \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, \color{blue}{B \cdot B}, 1\right) \cdot B}\right) \]

   9. Applied rewrites 19.0%

      \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\color{blue}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}}\right) \]

   10. Taylor expanded in F around inf

       \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{F}}}{\sin B}, \frac{-x}{\mathsf{fma}\left(\frac{1}{3}, B \cdot B, 1\right) \cdot B}\right) \]
   11. Step-by-step derivation

       1. lower-/.f64 67.2

          \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{F}}}{\sin B}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right) \]

   12. Applied rewrites 67.2%

       \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{F}}}{\sin B}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right) \]

   if 1.1499999999999999e-58 < x

   1. Initial program 79.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      2. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      3. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      4. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 65.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 67.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B} \]
   10. Step-by-step derivation

   11. Applied rewrites 87.0%

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-114}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{\left(\mathsf{fma}\left(-0.16666666666666666, B \cdot B, 1\right) \cdot B\right) \cdot F}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-114}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{F}^{-1}}{\sin B}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 5.7 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(F, {\left(\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B\right)}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 30000:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{\mathsf{fma}\left(B \cdot B, 0.3333333333333333, 1\right) \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \cos B \cdot x, 1\right)}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.6e-17)
   (/ (- (fma (cos B) x 1.0)) (sin B))
   (if (<= F 5.7e-121)
     (fma
      F
      (pow (* (sqrt (+ (fma 2.0 x (* F F)) 2.0)) B) -1.0)
      (/ (- x) (tan B)))
     (if (<= F 30000.0)
       (-
        (/ (/ F (sin B)) (sqrt (fma x 2.0 (fma F F 2.0))))
        (/ x (* (fma (* B B) 0.3333333333333333 1.0) B)))
       (/ (fma -1.0 (* (cos B) x) 1.0) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.6e-17) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else if (F <= 5.7e-121) {
		tmp = fma(F, pow((sqrt((fma(2.0, x, (F * F)) + 2.0)) * B), -1.0), (-x / tan(B)));
	} else if (F <= 30000.0) {
		tmp = ((F / sin(B)) / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - (x / (fma((B * B), 0.3333333333333333, 1.0) * B));
	} else {
		tmp = fma(-1.0, (cos(B) * x), 1.0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.6e-17)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	elseif (F <= 5.7e-121)
		tmp = fma(F, (Float64(sqrt(Float64(fma(2.0, x, Float64(F * F)) + 2.0)) * B) ^ -1.0), Float64(Float64(-x) / tan(B)));
	elseif (F <= 30000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - Float64(x / Float64(fma(Float64(B * B), 0.3333333333333333, 1.0) * B)));
	else
		tmp = Float64(fma(-1.0, Float64(cos(B) * x), 1.0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.6e-17], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.7e-121], N[(F * N[Power[N[(N[Sqrt[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * B), $MachinePrecision], -1.0], $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 30000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[(N[(B * B), $MachinePrecision] * 0.3333333333333333 + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.59999999999999995e-17

   1. Initial program 59.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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

         \[\leadsto \color{blue}{\mathsf{neg}\left(\left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)\right)} \]

      2. div-add-rev N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{\frac{1 + x \cdot \cos B}{\sin B}}\right) \]

      3. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(1 + x \cdot \cos B\right)\right)}{\sin B}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(\left(1 + x \cdot \cos B\right)\right)}{\sin B}} \]

      5. lower-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{-\left(1 + x \cdot \cos B\right)}}{\sin B} \]

      6. +-commutative N/A

         \[\leadsto \frac{-\color{blue}{\left(x \cdot \cos B + 1\right)}}{\sin B} \]

      7. *-commutative N/A

         \[\leadsto \frac{-\left(\color{blue}{\cos B \cdot x} + 1\right)}{\sin B} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{-\color{blue}{\mathsf{fma}\left(\cos B, x, 1\right)}}{\sin B} \]

      9. lower-cos.f64 N/A

         \[\leadsto \frac{-\mathsf{fma}\left(\color{blue}{\cos B}, x, 1\right)}{\sin B} \]

      10. lower-sin.f64 98.2

          \[\leadsto \frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\color{blue}{\sin B}} \]

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}} \]

   if -3.59999999999999995e-17 < F < 5.70000000000000014e-121

   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

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}, -x \cdot \frac{1}{\tan B}\right)} \]

   4. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{\left(\frac{-1}{2}\right)}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      3. sqrt-pow2 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{{\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-1}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      4. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{2 \cdot x + \mathsf{fma}\left(F, F, 2\right)}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{2 \cdot x} + \mathsf{fma}\left(F, F, 2\right)}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{\mathsf{fma}\left(F, F, 2\right) + 2 \cdot x}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{\left(F \cdot F + 2\right)} + 2 \cdot x}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\left(\color{blue}{F \cdot F} + 2\right) + 2 \cdot x}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      9. associate-+l+ N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{F \cdot F + \left(2 + 2 \cdot x\right)}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{F \cdot F + \color{blue}{\left(2 \cdot x + 2\right)}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{F \cdot F} + \left(2 \cdot x + 2\right)}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{F \cdot F + \left(\color{blue}{2 \cdot x} + 2\right)}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      13. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{F \cdot F + \color{blue}{\mathsf{fma}\left(2, x, 2\right)}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      14. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      15. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      17. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\color{blue}{F \cdot F + \mathsf{fma}\left(2, x, 2\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\color{blue}{F \cdot F} + \mathsf{fma}\left(2, x, 2\right)}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      19. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{F \cdot F + \color{blue}{\left(2 \cdot x + 2\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{F \cdot F + \left(\color{blue}{2 \cdot x} + 2\right)}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      21. associate-+r+ N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\color{blue}{\left(F \cdot F + 2 \cdot x\right) + 2}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      22. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\color{blue}{\left(2 \cdot x + F \cdot F\right)} + 2}}}{\sin B}, \frac{-x}{\tan B}\right) \]

   6. Applied rewrites 99.7%

      \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}}, \frac{-x}{\tan B}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      3. associate-/l/ N/A

         \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot \sin B}}, \frac{-x}{\tan B}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot \sin B}}, \frac{-x}{\tan B}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{1}{\color{blue}{\sin B \cdot \sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}, \frac{-x}{\tan B}\right) \]

      6. lower-*.f64 99.7

         \[\leadsto \mathsf{fma}\left(F, \frac{1}{\color{blue}{\sin B \cdot \sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}, \frac{-x}{\tan B}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{1}{\sin B \cdot \sqrt{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F + 2}\right)}}, \frac{-x}{\tan B}\right) \]

      8. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{1}{\sin B \cdot \sqrt{\color{blue}{x \cdot 2 + \left(F \cdot F + 2\right)}}}, \frac{-x}{\tan B}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{1}{\sin B \cdot \sqrt{\color{blue}{2 \cdot x} + \left(F \cdot F + 2\right)}}, \frac{-x}{\tan B}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\sin B \cdot \sqrt{\color{blue}{\mathsf{fma}\left(2, x, F \cdot F + 2\right)}}}, \frac{-x}{\tan B}\right) \]

      11. lift-fma.f64 99.7

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \frac{-x}{\tan B}\right) \]

   8. Applied rewrites 99.7%

      \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{1}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}, \frac{-x}{\tan B}\right) \]

   9. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(F, \frac{1}{\color{blue}{B \cdot \sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \frac{-x}{\tan B}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\color{blue}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)} \cdot B}}, \frac{-x}{\tan B}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\color{blue}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)} \cdot B}}, \frac{-x}{\tan B}\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\color{blue}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot B}, \frac{-x}{\tan B}\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\sqrt{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}} \cdot B}, \frac{-x}{\tan B}\right) \]

       5. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\sqrt{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}} \cdot B}, \frac{-x}{\tan B}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(2, x, {F}^{2}\right)} + 2} \cdot B}, \frac{-x}{\tan B}\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\sqrt{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2} \cdot B}, \frac{-x}{\tan B}\right) \]

       8. lower-*.f64 86.8

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\sqrt{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2} \cdot B}, \frac{-x}{\tan B}\right) \]

   11. Applied rewrites 86.8%

       \[\leadsto \mathsf{fma}\left(F, \frac{1}{\color{blue}{\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B}}, \frac{-x}{\tan B}\right) \]

   if 5.70000000000000014e-121 < F < 3e4

   1. Initial program 99.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}, -x \cdot \frac{1}{\tan B}\right)} \]

   4. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{\left(\frac{-1}{2}\right)}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      3. sqrt-pow2 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{{\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-1}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      4. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{2 \cdot x + \mathsf{fma}\left(F, F, 2\right)}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{2 \cdot x} + \mathsf{fma}\left(F, F, 2\right)}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{\mathsf{fma}\left(F, F, 2\right) + 2 \cdot x}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{\left(F \cdot F + 2\right)} + 2 \cdot x}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\left(\color{blue}{F \cdot F} + 2\right) + 2 \cdot x}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      9. associate-+l+ N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{F \cdot F + \left(2 + 2 \cdot x\right)}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{F \cdot F + \color{blue}{\left(2 \cdot x + 2\right)}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{F \cdot F} + \left(2 \cdot x + 2\right)}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{F \cdot F + \left(\color{blue}{2 \cdot x} + 2\right)}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      13. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{F \cdot F + \color{blue}{\mathsf{fma}\left(2, x, 2\right)}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      14. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      15. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      17. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\color{blue}{F \cdot F + \mathsf{fma}\left(2, x, 2\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\color{blue}{F \cdot F} + \mathsf{fma}\left(2, x, 2\right)}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      19. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{F \cdot F + \color{blue}{\left(2 \cdot x + 2\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{F \cdot F + \left(\color{blue}{2 \cdot x} + 2\right)}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      21. associate-+r+ N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\color{blue}{\left(F \cdot F + 2 \cdot x\right) + 2}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      22. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\color{blue}{\left(2 \cdot x + F \cdot F\right)} + 2}}}{\sin B}, \frac{-x}{\tan B}\right) \]

   6. Applied rewrites 99.1%

      \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

   7. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\color{blue}{B \cdot \left(1 + \frac{1}{3} \cdot {B}^{2}\right)}}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\color{blue}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot B}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\color{blue}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot B}}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\color{blue}{\left(\frac{1}{3} \cdot {B}^{2} + 1\right)} \cdot B}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\color{blue}{\mathsf{fma}\left(\frac{1}{3}, {B}^{2}, 1\right)} \cdot B}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\mathsf{fma}\left(\frac{1}{3}, \color{blue}{B \cdot B}, 1\right) \cdot B}\right) \]

      6. lower-*.f64 84.2

         \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, \color{blue}{B \cdot B}, 1\right) \cdot B}\right) \]

   9. Applied rewrites 84.2%

      \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\color{blue}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}}\right) \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}}, \frac{-x}{\mathsf{fma}\left(\frac{1}{3}, B \cdot B, 1\right) \cdot B}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}}{\sin B}, \frac{-x}{\mathsf{fma}\left(\frac{1}{3}, B \cdot B, 1\right) \cdot B}\right) \]

       3. associate-/l/ N/A

          \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot \sin B}}, \frac{-x}{\mathsf{fma}\left(\frac{1}{3}, B \cdot B, 1\right) \cdot B}\right) \]

       4. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\sqrt{\color{blue}{x \cdot 2 + \mathsf{fma}\left(F, F, 2\right)}} \cdot \sin B}, \frac{-x}{\mathsf{fma}\left(\frac{1}{3}, B \cdot B, 1\right) \cdot B}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\sqrt{\color{blue}{2 \cdot x} + \mathsf{fma}\left(F, F, 2\right)} \cdot \sin B}, \frac{-x}{\mathsf{fma}\left(\frac{1}{3}, B \cdot B, 1\right) \cdot B}\right) \]

       6. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot \sin B}, \frac{-x}{\mathsf{fma}\left(\frac{1}{3}, B \cdot B, 1\right) \cdot B}\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\color{blue}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}, \frac{-x}{\mathsf{fma}\left(\frac{1}{3}, B \cdot B, 1\right) \cdot B}\right) \]

       8. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\color{blue}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}, \frac{-x}{\mathsf{fma}\left(\frac{1}{3}, B \cdot B, 1\right) \cdot B}\right) \]

       9. lift-/.f64 84.4

          \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{1}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right) \]

       10. lower-fma.f64 N/A

           \[\leadsto \color{blue}{F \cdot \frac{1}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} + \frac{-x}{\mathsf{fma}\left(\frac{1}{3}, B \cdot B, 1\right) \cdot B}} \]

       11. lower-+.f64 N/A

           \[\leadsto \color{blue}{F \cdot \frac{1}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} + \frac{-x}{\mathsf{fma}\left(\frac{1}{3}, B \cdot B, 1\right) \cdot B}} \]

   11. Applied rewrites 84.5%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} + \frac{-x}{\mathsf{fma}\left(B \cdot B, 0.3333333333333333, 1\right) \cdot B}} \]

   if 3e4 < F

   1. Initial program 52.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}, -x \cdot \frac{1}{\tan B}\right)} \]

   4. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)} \]

   5. Taylor expanded in F around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B} + \frac{1}{\sin B}} \]
   6. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot \cos B\right)}{\sin B}} + \frac{1}{\sin B} \]

      2. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot \cos B\right) + 1}{\sin B}} \]

      3. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1 \cdot \left(x \cdot \cos B\right) + 1}{\sin B}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1, x \cdot \cos B, 1\right)}}{\sin B} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, \color{blue}{\cos B \cdot x}, 1\right)}{\sin B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, \color{blue}{\cos B \cdot x}, 1\right)}{\sin B} \]

      7. lower-cos.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, \color{blue}{\cos B} \cdot x, 1\right)}{\sin B} \]

      8. lower-sin.f64 99.7

         \[\leadsto \frac{\mathsf{fma}\left(-1, \cos B \cdot x, 1\right)}{\color{blue}{\sin B}} \]

   7. Applied rewrites 99.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, \cos B \cdot x, 1\right)}{\sin B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.6 \cdot 10^{-17}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 5.7 \cdot 10^{-121}:\\ \;\;\;\;\mathsf{fma}\left(F, {\left(\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B\right)}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 30000:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{\mathsf{fma}\left(B \cdot B, 0.3333333333333333, 1\right) \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \cos B \cdot x, 1\right)}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\\ \mathbf{if}\;F \leq -7.5 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{\sin B \cdot F}, t\_0\right)\\ \mathbf{elif}\;F \leq 10000000:\\ \;\;\;\;\mathsf{fma}\left(F, {\left(\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B\right)}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{F}^{-1}}{\sin B}, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (* (fma 0.3333333333333333 (* B B) 1.0) B))))
   (if (<= F -7.5e+52)
     (fma F (/ -1.0 (* (sin B) F)) t_0)
     (if (<= F 10000000.0)
       (fma
        F
        (pow (* (sqrt (+ (fma 2.0 x (* F F)) 2.0)) B) -1.0)
        (/ (- x) (tan B)))
       (fma F (/ (pow F -1.0) (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -x / (fma(0.3333333333333333, (B * B), 1.0) * B);
	double tmp;
	if (F <= -7.5e+52) {
		tmp = fma(F, (-1.0 / (sin(B) * F)), t_0);
	} else if (F <= 10000000.0) {
		tmp = fma(F, pow((sqrt((fma(2.0, x, (F * F)) + 2.0)) * B), -1.0), (-x / tan(B)));
	} else {
		tmp = fma(F, (pow(F, -1.0) / sin(B)), t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / Float64(fma(0.3333333333333333, Float64(B * B), 1.0) * B))
	tmp = 0.0
	if (F <= -7.5e+52)
		tmp = fma(F, Float64(-1.0 / Float64(sin(B) * F)), t_0);
	elseif (F <= 10000000.0)
		tmp = fma(F, (Float64(sqrt(Float64(fma(2.0, x, Float64(F * F)) + 2.0)) * B) ^ -1.0), Float64(Float64(-x) / tan(B)));
	else
		tmp = fma(F, Float64((F ^ -1.0) / sin(B)), t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[(N[(0.3333333333333333 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -7.5e+52], N[(F * N[(-1.0 / N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, 10000000.0], N[(F * N[Power[N[(N[Sqrt[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * B), $MachinePrecision], -1.0], $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(N[Power[F, -1.0], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.49999999999999995e52

   1. Initial program 52.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}, -x \cdot \frac{1}{\tan B}\right)} \]

   4. Applied rewrites 65.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{-1}{F \cdot \sin B}}, \frac{-x}{\tan B}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{-1}{F \cdot \sin B}}, \frac{-x}{\tan B}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{\color{blue}{\sin B \cdot F}}, \frac{-x}{\tan B}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{\color{blue}{\sin B \cdot F}}, \frac{-x}{\tan B}\right) \]

      4. lower-sin.f64 99.6

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{\color{blue}{\sin B} \cdot F}, \frac{-x}{\tan B}\right) \]

   7. Applied rewrites 99.6%

      \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{-1}{\sin B \cdot F}}, \frac{-x}{\tan B}\right) \]

   8. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(F, \frac{-1}{\sin B \cdot F}, \frac{-x}{\color{blue}{B \cdot \left(1 + \frac{1}{3} \cdot {B}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{\sin B \cdot F}, \frac{-x}{\color{blue}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot B}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{\sin B \cdot F}, \frac{-x}{\color{blue}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot B}}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{\sin B \cdot F}, \frac{-x}{\color{blue}{\left(\frac{1}{3} \cdot {B}^{2} + 1\right)} \cdot B}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{\sin B \cdot F}, \frac{-x}{\color{blue}{\mathsf{fma}\left(\frac{1}{3}, {B}^{2}, 1\right)} \cdot B}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{\sin B \cdot F}, \frac{-x}{\mathsf{fma}\left(\frac{1}{3}, \color{blue}{B \cdot B}, 1\right) \cdot B}\right) \]

      6. lower-*.f64 79.1

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{\sin B \cdot F}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, \color{blue}{B \cdot B}, 1\right) \cdot B}\right) \]

   10. Applied rewrites 79.1%

       \[\leadsto \mathsf{fma}\left(F, \frac{-1}{\sin B \cdot F}, \frac{-x}{\color{blue}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}}\right) \]

   if -7.49999999999999995e52 < F < 1e7

   1. Initial program 98.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}, -x \cdot \frac{1}{\tan B}\right)} \]

   4. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{\left(\frac{-1}{2}\right)}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      3. sqrt-pow2 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{{\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-1}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      4. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{2 \cdot x + \mathsf{fma}\left(F, F, 2\right)}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{2 \cdot x} + \mathsf{fma}\left(F, F, 2\right)}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{\mathsf{fma}\left(F, F, 2\right) + 2 \cdot x}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{\left(F \cdot F + 2\right)} + 2 \cdot x}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\left(\color{blue}{F \cdot F} + 2\right) + 2 \cdot x}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      9. associate-+l+ N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{F \cdot F + \left(2 + 2 \cdot x\right)}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{F \cdot F + \color{blue}{\left(2 \cdot x + 2\right)}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{F \cdot F} + \left(2 \cdot x + 2\right)}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{F \cdot F + \left(\color{blue}{2 \cdot x} + 2\right)}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      13. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{F \cdot F + \color{blue}{\mathsf{fma}\left(2, x, 2\right)}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      14. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      15. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      17. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\color{blue}{F \cdot F + \mathsf{fma}\left(2, x, 2\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\color{blue}{F \cdot F} + \mathsf{fma}\left(2, x, 2\right)}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      19. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{F \cdot F + \color{blue}{\left(2 \cdot x + 2\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{F \cdot F + \left(\color{blue}{2 \cdot x} + 2\right)}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      21. associate-+r+ N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\color{blue}{\left(F \cdot F + 2 \cdot x\right) + 2}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      22. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\color{blue}{\left(2 \cdot x + F \cdot F\right)} + 2}}}{\sin B}, \frac{-x}{\tan B}\right) \]

   6. Applied rewrites 99.5%

      \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]
   7. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}}, \frac{-x}{\tan B}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      3. associate-/l/ N/A

         \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot \sin B}}, \frac{-x}{\tan B}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot \sin B}}, \frac{-x}{\tan B}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{1}{\color{blue}{\sin B \cdot \sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}, \frac{-x}{\tan B}\right) \]

      6. lower-*.f64 99.6

         \[\leadsto \mathsf{fma}\left(F, \frac{1}{\color{blue}{\sin B \cdot \sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}, \frac{-x}{\tan B}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{1}{\sin B \cdot \sqrt{\mathsf{fma}\left(x, 2, \color{blue}{F \cdot F + 2}\right)}}, \frac{-x}{\tan B}\right) \]

      8. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{1}{\sin B \cdot \sqrt{\color{blue}{x \cdot 2 + \left(F \cdot F + 2\right)}}}, \frac{-x}{\tan B}\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{1}{\sin B \cdot \sqrt{\color{blue}{2 \cdot x} + \left(F \cdot F + 2\right)}}, \frac{-x}{\tan B}\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\sin B \cdot \sqrt{\color{blue}{\mathsf{fma}\left(2, x, F \cdot F + 2\right)}}}, \frac{-x}{\tan B}\right) \]

      11. lift-fma.f64 99.6

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}}, \frac{-x}{\tan B}\right) \]

   8. Applied rewrites 99.6%

      \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{1}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}, \frac{-x}{\tan B}\right) \]

   9. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(F, \frac{1}{\color{blue}{B \cdot \sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}}}, \frac{-x}{\tan B}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\color{blue}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)} \cdot B}}, \frac{-x}{\tan B}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\color{blue}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)} \cdot B}}, \frac{-x}{\tan B}\right) \]

       3. lower-sqrt.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\color{blue}{\sqrt{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot B}, \frac{-x}{\tan B}\right) \]

       4. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\sqrt{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}} \cdot B}, \frac{-x}{\tan B}\right) \]

       5. lower-+.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\sqrt{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}} \cdot B}, \frac{-x}{\tan B}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\sqrt{\color{blue}{\mathsf{fma}\left(2, x, {F}^{2}\right)} + 2} \cdot B}, \frac{-x}{\tan B}\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\sqrt{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2} \cdot B}, \frac{-x}{\tan B}\right) \]

       8. lower-*.f64 81.9

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\sqrt{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2} \cdot B}, \frac{-x}{\tan B}\right) \]

   11. Applied rewrites 81.9%

       \[\leadsto \mathsf{fma}\left(F, \frac{1}{\color{blue}{\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B}}, \frac{-x}{\tan B}\right) \]

   if 1e7 < F

   1. Initial program 52.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}, -x \cdot \frac{1}{\tan B}\right)} \]

   4. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      2. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{\left(\frac{-1}{2}\right)}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      3. sqrt-pow2 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{{\left(\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\right)}^{-1}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      4. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{2 \cdot x + \mathsf{fma}\left(F, F, 2\right)}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{2 \cdot x} + \mathsf{fma}\left(F, F, 2\right)}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{\mathsf{fma}\left(F, F, 2\right) + 2 \cdot x}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      7. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{\left(F \cdot F + 2\right)} + 2 \cdot x}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\left(\color{blue}{F \cdot F} + 2\right) + 2 \cdot x}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      9. associate-+l+ N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{F \cdot F + \left(2 + 2 \cdot x\right)}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{F \cdot F + \color{blue}{\left(2 \cdot x + 2\right)}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{F \cdot F} + \left(2 \cdot x + 2\right)}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      12. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{F \cdot F + \left(\color{blue}{2 \cdot x} + 2\right)}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      13. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{F \cdot F + \color{blue}{\mathsf{fma}\left(2, x, 2\right)}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      14. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\sqrt{\color{blue}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}\right)}^{-1}}{\sin B}, \frac{-x}{\tan B}\right) \]

      15. inv-pow N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      17. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\color{blue}{F \cdot F + \mathsf{fma}\left(2, x, 2\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\color{blue}{F \cdot F} + \mathsf{fma}\left(2, x, 2\right)}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      19. lift-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{F \cdot F + \color{blue}{\left(2 \cdot x + 2\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      20. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{F \cdot F + \left(\color{blue}{2 \cdot x} + 2\right)}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      21. associate-+r+ N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\color{blue}{\left(F \cdot F + 2 \cdot x\right) + 2}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

      22. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\color{blue}{\left(2 \cdot x + F \cdot F\right)} + 2}}}{\sin B}, \frac{-x}{\tan B}\right) \]

   6. Applied rewrites 66.9%

      \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}}{\sin B}, \frac{-x}{\tan B}\right) \]

   7. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\color{blue}{B \cdot \left(1 + \frac{1}{3} \cdot {B}^{2}\right)}}\right) \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\color{blue}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot B}}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\color{blue}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right) \cdot B}}\right) \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\color{blue}{\left(\frac{1}{3} \cdot {B}^{2} + 1\right)} \cdot B}\right) \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\color{blue}{\mathsf{fma}\left(\frac{1}{3}, {B}^{2}, 1\right)} \cdot B}\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\mathsf{fma}\left(\frac{1}{3}, \color{blue}{B \cdot B}, 1\right) \cdot B}\right) \]

      6. lower-*.f64 51.4

         \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, \color{blue}{B \cdot B}, 1\right) \cdot B}\right) \]

   9. Applied rewrites 51.4%

      \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, \frac{-x}{\color{blue}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}}\right) \]

   10. Taylor expanded in F around inf

       \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{F}}}{\sin B}, \frac{-x}{\mathsf{fma}\left(\frac{1}{3}, B \cdot B, 1\right) \cdot B}\right) \]
   11. Step-by-step derivation

       1. lower-/.f64 83.2

          \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{F}}}{\sin B}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right) \]

   12. Applied rewrites 83.2%

       \[\leadsto \mathsf{fma}\left(F, \frac{\color{blue}{\frac{1}{F}}}{\sin B}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.5 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{\sin B \cdot F}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right)\\ \mathbf{elif}\;F \leq 10000000:\\ \;\;\;\;\mathsf{fma}\left(F, {\left(\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B\right)}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{F}^{-1}}{\sin B}, \frac{-x}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{if}\;x \leq -620000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-102}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))))
   (if (<= x -620000000000.0)
     t_0
     (if (<= x -7.2e-145)
       (/ (- (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
       (if (<= x 3.6e-102)
         (* (sqrt (pow (fma F F 2.0) -1.0)) (/ 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 <= -620000000000.0) {
		tmp = t_0;
	} else if (x <= -7.2e-145) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else if (x <= 3.6e-102) {
		tmp = sqrt(pow(fma(F, F, 2.0), -1.0)) * (F / sin(B));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B))
	tmp = 0.0
	if (x <= -620000000000.0)
		tmp = t_0;
	elseif (x <= -7.2e-145)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	elseif (x <= 3.6e-102)
		tmp = Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * 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, -620000000000.0], t$95$0, If[LessEqual[x, -7.2e-145], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[x, 3.6e-102], N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.2e11 or 3.6e-102 < x

   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

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      2. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      3. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      4. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 55.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B} \]
   10. Step-by-step derivation

   11. Applied rewrites 85.9%

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B} \]

   if -6.2e11 < x < -7.2000000000000001e-145

   1. Initial program 70.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 41.6

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 41.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]
   6. Step-by-step derivation

   7. Applied rewrites 41.7%

      \[\leadsto \frac{\frac{F \cdot 1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B} \]

   if -7.2000000000000001e-145 < x < 3.6e-102

   1. Initial program 75.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}} \cdot \frac{F}{\sin B} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}} \cdot \frac{F}{\sin B} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}} \cdot \frac{F}{\sin B} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}} \cdot \frac{F}{\sin B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}} \cdot \frac{F}{\sin B} \]

      8. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      9. lower-sin.f64 62.4

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\color{blue}{\sin B}} \]

   5. Applied rewrites 62.4%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -620000000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;x \leq -7.2 \cdot 10^{-145}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-102}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-114}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{\left(\mathsf{fma}\left(-0.16666666666666666, B \cdot B, 1\right) \cdot B\right) \cdot F}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-102}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -1e-114)
   (fma
    F
    (/ -1.0 (* (* (fma -0.16666666666666666 (* B B) 1.0) B) F))
    (/ (- x) (tan B)))
   (if (<= x 3.6e-102)
     (* (sqrt (pow (fma F F 2.0) -1.0)) (/ F (sin B)))
     (+ (* x (/ -1.0 (tan B))) (/ -1.0 B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -1e-114) {
		tmp = fma(F, (-1.0 / ((fma(-0.16666666666666666, (B * B), 1.0) * B) * F)), (-x / tan(B)));
	} else if (x <= 3.6e-102) {
		tmp = sqrt(pow(fma(F, F, 2.0), -1.0)) * (F / sin(B));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -1e-114)
		tmp = fma(F, Float64(-1.0 / Float64(Float64(fma(-0.16666666666666666, Float64(B * B), 1.0) * B) * F)), Float64(Float64(-x) / tan(B)));
	elseif (x <= 3.6e-102)
		tmp = Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * Float64(F / sin(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -1e-114], N[(F * N[(-1.0 / N[(N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.6e-102], N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.0000000000000001e-114

   1. Initial program 74.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. associate-/l* N/A

         \[\leadsto \color{blue}{F \cdot \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}, -x \cdot \frac{1}{\tan B}\right)} \]

   4. Applied rewrites 84.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{-1}{F \cdot \sin B}}, \frac{-x}{\tan B}\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{-1}{F \cdot \sin B}}, \frac{-x}{\tan B}\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{\color{blue}{\sin B \cdot F}}, \frac{-x}{\tan B}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{\color{blue}{\sin B \cdot F}}, \frac{-x}{\tan B}\right) \]

      4. lower-sin.f64 52.3

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{\color{blue}{\sin B} \cdot F}, \frac{-x}{\tan B}\right) \]

   7. Applied rewrites 52.3%

      \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{-1}{\sin B \cdot F}}, \frac{-x}{\tan B}\right) \]

   8. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(F, \frac{-1}{\left(B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{2}\right)\right) \cdot F}, \frac{-x}{\tan B}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 64.8%

       \[\leadsto \mathsf{fma}\left(F, \frac{-1}{\left(\mathsf{fma}\left(-0.16666666666666666, B \cdot B, 1\right) \cdot B\right) \cdot F}, \frac{-x}{\tan B}\right) \]

   if -1.0000000000000001e-114 < x < 3.6e-102

   1. Initial program 76.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}} \cdot \frac{F}{\sin B} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}} \cdot \frac{F}{\sin B} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}} \cdot \frac{F}{\sin B} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}} \cdot \frac{F}{\sin B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}} \cdot \frac{F}{\sin B} \]

      8. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      9. lower-sin.f64 60.9

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\color{blue}{\sin B}} \]

   5. Applied rewrites 60.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}} \]

   if 3.6e-102 < x

   1. Initial program 74.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      2. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      3. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      4. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 60.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 61.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B} \]
   10. Step-by-step derivation

   11. Applied rewrites 82.0%

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-114}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{\left(\mathsf{fma}\left(-0.16666666666666666, B \cdot B, 1\right) \cdot B\right) \cdot F}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-102}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{-72}:\\ \;\;\;\;t\_0 + \frac{F}{B} \cdot {F}^{-1}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-102}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= x -2.9e-72)
     (+ t_0 (* (/ F B) (pow F -1.0)))
     (if (<= x 3.6e-102)
       (* (sqrt (pow (fma F F 2.0) -1.0)) (/ F (sin B)))
       (+ t_0 (/ -1.0 B))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (x <= -2.9e-72) {
		tmp = t_0 + ((F / B) * pow(F, -1.0));
	} else if (x <= 3.6e-102) {
		tmp = sqrt(pow(fma(F, F, 2.0), -1.0)) * (F / sin(B));
	} else {
		tmp = t_0 + (-1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (x <= -2.9e-72)
		tmp = Float64(t_0 + Float64(Float64(F / B) * (F ^ -1.0)));
	elseif (x <= 3.6e-102)
		tmp = Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * Float64(F / sin(B)));
	else
		tmp = Float64(t_0 + Float64(-1.0 / 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]}, If[LessEqual[x, -2.9e-72], N[(t$95$0 + N[(N[(F / B), $MachinePrecision] * N[Power[F, -1.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.6e-102], N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.89999999999999998e-72

   1. Initial program 75.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{F}} \]
   4. Step-by-step derivation

      1. lower-/.f64 66.1

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{F}} \]

   5. Applied rewrites 66.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{F}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot \frac{1}{F} \]
   7. Step-by-step derivation

      1. lower-/.f64 71.1

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot \frac{1}{F} \]

   8. Applied rewrites 71.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot \frac{1}{F} \]

   if -2.89999999999999998e-72 < x < 3.6e-102

   1. Initial program 75.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}} \cdot \frac{F}{\sin B} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}} \cdot \frac{F}{\sin B} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}} \cdot \frac{F}{\sin B} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}} \cdot \frac{F}{\sin B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}} \cdot \frac{F}{\sin B} \]

      8. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      9. lower-sin.f64 57.9

         \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\color{blue}{\sin B}} \]

   5. Applied rewrites 57.9%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}} \]

   if 3.6e-102 < x

   1. Initial program 74.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      2. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      3. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      4. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 60.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 61.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B} \]
   10. Step-by-step derivation

   11. Applied rewrites 82.0%

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-72}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot {F}^{-1}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-102}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.85 \cdot 10^{+86}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq 2800:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.85e+86)
   (+ (- (/ x B)) (/ (- (* -0.16666666666666666 (* B B)) 1.0) B))
   (if (<= F 2800.0)
     (/ (- (* (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0)) F) x) B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.85e+86) {
		tmp = -(x / B) + (((-0.16666666666666666 * (B * B)) - 1.0) / B);
	} else if (F <= 2800.0) {
		tmp = ((sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.85e+86)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(Float64(-0.16666666666666666 * Float64(B * B)) - 1.0) / B));
	elseif (F <= 2800.0)
		tmp = Float64(Float64(Float64(sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0)) * F) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.85e+86], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2800.0], N[(N[(N[(N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $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 < -4.85e86

   1. Initial program 48.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      2. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      3. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      4. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 32.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 63.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\frac{-1}{6} \cdot \left(B \cdot B\right) - 1}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 51.8

          \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   11. Applied rewrites 51.8%

       \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   if -4.85e86 < F < 2800

   1. Initial program 98.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 47.4

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 47.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   if 2800 < F

   1. Initial program 53.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 35.1

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 35.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.3%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.85 \cdot 10^{+86}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq 2800:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.9 \cdot 10^{-63}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{-114}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 0.000145:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.9e-63)
   (+ (- (/ x B)) (/ (- (* -0.16666666666666666 (* B B)) 1.0) B))
   (if (<= F 5.4e-114)
     (/ (- x) B)
     (if (<= F 0.000145)
       (/ (* (sqrt (pow (fma F F 2.0) -1.0)) F) B)
       (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.9e-63) {
		tmp = -(x / B) + (((-0.16666666666666666 * (B * B)) - 1.0) / B);
	} else if (F <= 5.4e-114) {
		tmp = -x / B;
	} else if (F <= 0.000145) {
		tmp = (sqrt(pow(fma(F, F, 2.0), -1.0)) * F) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.9e-63)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(Float64(-0.16666666666666666 * Float64(B * B)) - 1.0) / B));
	elseif (F <= 5.4e-114)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 0.000145)
		tmp = Float64(Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * F) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.9e-63], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.4e-114], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 0.000145], N[(N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.90000000000000009e-63

   1. Initial program 63.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      2. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      3. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      4. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 43.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\frac{-1}{6} \cdot \left(B \cdot B\right) - 1}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 48.9

          \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   11. Applied rewrites 48.9%

       \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   if -1.90000000000000009e-63 < F < 5.3999999999999999e-114

   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

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 42.4

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 42.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 38.2%

      \[\leadsto \frac{-x}{B} \]

   if 5.3999999999999999e-114 < F < 1.45e-4

   1. Initial program 99.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 55.2

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 55.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 38.6%

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot F}{B} \]

   if 1.45e-4 < F

   1. Initial program 53.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 35.9

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 35.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.3%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.9 \cdot 10^{-63}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{-114}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 0.000145:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 43.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.9 \cdot 10^{-63}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{-114}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 0.000145:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.9e-63)
   (+ (- (/ x B)) (/ (- (* -0.16666666666666666 (* B B)) 1.0) B))
   (if (<= F 5.4e-114)
     (/ (- x) B)
     (if (<= F 0.000145)
       (* (sqrt (pow (fma F F 2.0) -1.0)) (/ F B))
       (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.9e-63) {
		tmp = -(x / B) + (((-0.16666666666666666 * (B * B)) - 1.0) / B);
	} else if (F <= 5.4e-114) {
		tmp = -x / B;
	} else if (F <= 0.000145) {
		tmp = sqrt(pow(fma(F, F, 2.0), -1.0)) * (F / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.9e-63)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(Float64(-0.16666666666666666 * Float64(B * B)) - 1.0) / B));
	elseif (F <= 5.4e-114)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 0.000145)
		tmp = Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * Float64(F / B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.9e-63], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.4e-114], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 0.000145], N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.90000000000000009e-63

   1. Initial program 63.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      2. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      3. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      4. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 43.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\frac{-1}{6} \cdot \left(B \cdot B\right) - 1}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 48.9

          \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   11. Applied rewrites 48.9%

       \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   if -1.90000000000000009e-63 < F < 5.3999999999999999e-114

   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

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 42.4

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 42.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 38.2%

      \[\leadsto \frac{-x}{B} \]

   if 5.3999999999999999e-114 < F < 1.45e-4

   1. Initial program 99.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 55.2

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 55.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 38.5%

      \[\leadsto \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \color{blue}{\frac{F}{B}} \]

   if 1.45e-4 < F

   1. Initial program 53.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 35.9

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 35.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.3%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 44.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.9 \cdot 10^{-63}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{-114}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 0.000145:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 50.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.00043:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq 2900:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.00043)
   (+ (- (/ x B)) (/ (- (* -0.16666666666666666 (* B B)) 1.0) B))
   (if (<= F 2900.0)
     (/ (- (* (sqrt (pow (fma 2.0 x 2.0) -1.0)) F) x) B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.00043) {
		tmp = -(x / B) + (((-0.16666666666666666 * (B * B)) - 1.0) / B);
	} else if (F <= 2900.0) {
		tmp = ((sqrt(pow(fma(2.0, x, 2.0), -1.0)) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.00043)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(Float64(-0.16666666666666666 * Float64(B * B)) - 1.0) / B));
	elseif (F <= 2900.0)
		tmp = Float64(Float64(Float64(sqrt((fma(2.0, x, 2.0) ^ -1.0)) * F) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.00043], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2900.0], N[(N[(N[(N[Sqrt[N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $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 < -4.29999999999999989e-4

   1. Initial program 58.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      2. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      3. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      4. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 40.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 63.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\frac{-1}{6} \cdot \left(B \cdot B\right) - 1}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 51.8

          \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   11. Applied rewrites 51.8%

       \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   if -4.29999999999999989e-4 < F < 2900

   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

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 46.4

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 46.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   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

   8. Applied rewrites 46.4%

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B} \]

   if 2900 < F

   1. Initial program 52.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 35.5

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 35.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.8%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00043:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq 2900:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 55.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 8e-8)
   (/ (- (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
   (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 8e-8) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 8e-8)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 8e-8], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $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 < 8.0000000000000002e-8

   1. Initial program 72.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in 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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 54.8

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]
   6. Step-by-step derivation

   7. Applied rewrites 54.9%

      \[\leadsto \frac{\frac{F \cdot 1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B} \]

   if 8.0000000000000002e-8 < B

   1. Initial program 82.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      2. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      3. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      4. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 9.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 15.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B} \]
   10. Step-by-step derivation

   11. Applied rewrites 47.6%

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8 \cdot 10^{-8}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 43.5% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.9 \cdot 10^{-63}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq 0.0014:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.9e-63)
   (+ (- (/ x B)) (/ (- (* -0.16666666666666666 (* B B)) 1.0) B))
   (if (<= F 0.0014) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.9e-63) {
		tmp = -(x / B) + (((-0.16666666666666666 * (B * B)) - 1.0) / B);
	} else if (F <= 0.0014) {
		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 <= (-1.9d-63)) then
        tmp = -(x / b) + ((((-0.16666666666666666d0) * (b * b)) - 1.0d0) / b)
    else if (f <= 0.0014d0) 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 <= -1.9e-63) {
		tmp = -(x / B) + (((-0.16666666666666666 * (B * B)) - 1.0) / B);
	} else if (F <= 0.0014) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.9e-63:
		tmp = -(x / B) + (((-0.16666666666666666 * (B * B)) - 1.0) / B)
	elif F <= 0.0014:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.9e-63)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(Float64(-0.16666666666666666 * Float64(B * B)) - 1.0) / B));
	elseif (F <= 0.0014)
		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 <= -1.9e-63)
		tmp = -(x / B) + (((-0.16666666666666666 * (B * B)) - 1.0) / B);
	elseif (F <= 0.0014)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.9e-63], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0014], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.90000000000000009e-63

   1. Initial program 63.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\frac{1}{6} \cdot \left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left(\left({B}^{2} \cdot F\right) \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      2. associate-*l* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{\left({B}^{2} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} \cdot \frac{1}{6} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      3. associate-*r* N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{{B}^{2} \cdot \left(\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) \cdot \frac{1}{6}\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      4. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{{B}^{2} \cdot \color{blue}{\left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)} + F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right)\right)}{B}} \]

   5. Applied rewrites 43.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right)}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\frac{-1}{6} \cdot \left(B \cdot B\right) - 1}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 48.9

          \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   11. Applied rewrites 48.9%

       \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   if -1.90000000000000009e-63 < F < 0.00139999999999999999

   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

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 46.4

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 46.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 33.4%

      \[\leadsto \frac{-x}{B} \]

   if 0.00139999999999999999 < F

   1. Initial program 53.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 35.1

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 35.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.3%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 42.7% accurate, 8.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7 \cdot 10^{-96}:\\ \;\;\;\;\frac{\frac{\left(-1 + x\right) \cdot \left(-1 - x\right)}{-1 + x}}{B}\\ \mathbf{elif}\;F \leq 0.0014:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7e-96)
   (/ (/ (* (+ -1.0 x) (- -1.0 x)) (+ -1.0 x)) B)
   (if (<= F 0.0014) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7e-96) {
		tmp = (((-1.0 + x) * (-1.0 - x)) / (-1.0 + x)) / B;
	} else if (F <= 0.0014) {
		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 <= (-7d-96)) then
        tmp = ((((-1.0d0) + x) * ((-1.0d0) - x)) / ((-1.0d0) + x)) / b
    else if (f <= 0.0014d0) 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 <= -7e-96) {
		tmp = (((-1.0 + x) * (-1.0 - x)) / (-1.0 + x)) / B;
	} else if (F <= 0.0014) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -7e-96:
		tmp = (((-1.0 + x) * (-1.0 - x)) / (-1.0 + x)) / B
	elif F <= 0.0014:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7e-96)
		tmp = Float64(Float64(Float64(Float64(-1.0 + x) * Float64(-1.0 - x)) / Float64(-1.0 + x)) / B);
	elseif (F <= 0.0014)
		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 <= -7e-96)
		tmp = (((-1.0 + x) * (-1.0 - x)) / (-1.0 + x)) / B;
	elseif (F <= 0.0014)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7e-96], N[(N[(N[(N[(-1.0 + x), $MachinePrecision] * N[(-1.0 - x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 0.0014], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.9999999999999998e-96

   1. Initial program 64.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 38.2

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 38.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 46.9%

      \[\leadsto \frac{-1 - x}{B} \]
   9. Step-by-step derivation

   10. Applied rewrites 46.9%

       \[\leadsto \frac{\frac{\left(-1 + x\right) \cdot \left(-1 - x\right)}{-1 + x}}{B} \]

   if -6.9999999999999998e-96 < F < 0.00139999999999999999

   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

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 47.6

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 47.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 34.2%

      \[\leadsto \frac{-x}{B} \]

   if 0.00139999999999999999 < F

   1. Initial program 53.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 35.1

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 35.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.3%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 43.1% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.15 \cdot 10^{-138}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 0.0014:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.15e-138)
   (/ (- -1.0 x) B)
   (if (<= F 0.0014) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.15e-138) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 0.0014) {
		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 <= (-1.15d-138)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 0.0014d0) 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 <= -1.15e-138) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 0.0014) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.15e-138:
		tmp = (-1.0 - x) / B
	elif F <= 0.0014:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.15e-138)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 0.0014)
		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 <= -1.15e-138)
		tmp = (-1.0 - x) / B;
	elseif (F <= 0.0014)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.15e-138], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 0.0014], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.14999999999999995e-138

   1. Initial program 68.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 37.6

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 37.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.2%

      \[\leadsto \frac{-1 - x}{B} \]

   if -1.14999999999999995e-138 < F < 0.00139999999999999999

   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

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 48.7

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 48.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 35.3%

      \[\leadsto \frac{-x}{B} \]

   if 0.00139999999999999999 < F

   1. Initial program 53.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 35.1

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 35.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.3%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 35.9% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.15 \cdot 10^{-138}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.15e-138) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.15e-138) {
		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 <= (-1.15d-138)) 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 <= -1.15e-138) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.15e-138:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.15e-138)
		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 <= -1.15e-138)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.15e-138], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.14999999999999995e-138

   1. Initial program 68.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 37.6

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 37.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 44.2%

      \[\leadsto \frac{-1 - x}{B} \]

   if -1.14999999999999995e-138 < F

   1. Initial program 78.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 42.5

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 42.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 31.0%

      \[\leadsto \frac{-x}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 29.3% 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 75.4%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
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

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

   7. associate-+r+ N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

   8. +-commutative N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

   9. unpow2 N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

   11. +-commutative N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

   12. lower-fma.f64 41.1

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

5. Applied rewrites 41.1%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

6. Taylor expanded in F around 0

   \[\leadsto \frac{-1 \cdot x}{B} \]
7. Step-by-step derivation

8. Applied rewrites 28.7%

   \[\leadsto \frac{-x}{B} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024351 
(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))))))