VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.9% → 99.7%

Time: 10.4s

Alternatives: 26

Speedup: 1.6×

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: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -250000000:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 210000000:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{F}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -250000000.0)
   (/ (- (fma (cos B) x 1.0)) (sin B))
   (if (<= F 210000000.0)
     (fma (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F (sin B)) (/ (- x) (tan B)))
     (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -250000000.0) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else if (F <= 210000000.0) {
		tmp = fma(pow(fma(2.0, x, fma(F, F, 2.0)), -0.5), (F / sin(B)), (-x / tan(B)));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -250000000.0)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	elseif (F <= 210000000.0)
		tmp = fma((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5), Float64(F / sin(B)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -250000000.0], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 210000000.0], N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / 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[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.5e8

   1. Initial program 70.3%

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

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   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 -2.5e8 < F < 2.1e8

   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. *-commutative N/A

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

      5. lower-fma.f64 99.5

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

   4. Applied rewrites 99.6%

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

   if 2.1e8 < F

   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 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -250000000:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 210000000:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{F}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)}\\ t_1 := \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+282}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.022222222222222223, \mathsf{fma}\left(x \cdot 0.022222222222222223, -0.3333333333333333, x \cdot 0.009523809523809525\right) \cdot \left(B \cdot B\right)\right), B \cdot B, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right) - x}{B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (+
          (* x (/ -1.0 (tan B)))
          (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (/ -1.0 2.0)))))
        (t_1
         (-
          (* (sqrt (/ 1.0 (+ (fma 2.0 x (* F F)) 2.0))) (/ F B))
          (/ x (tan B)))))
   (if (<= t_0 -2e+20)
     t_1
     (if (<= t_0 10.0)
       (- (/ (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) (sin B)) (/ x B))
       (if (<= t_0 2e+282)
         t_1
         (+
          (/
           (-
            (*
             (fma
              (fma
               x
               0.022222222222222223
               (*
                (fma
                 (* x 0.022222222222222223)
                 -0.3333333333333333
                 (* x 0.009523809523809525))
                (* B B)))
              (* B B)
              (* 0.3333333333333333 x))
             (* B B))
            x)
           B)
          (/ 1.0 (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = (x * (-1.0 / tan(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), (-1.0 / 2.0)));
	double t_1 = (sqrt((1.0 / (fma(2.0, x, (F * F)) + 2.0))) * (F / B)) - (x / tan(B));
	double tmp;
	if (t_0 <= -2e+20) {
		tmp = t_1;
	} else if (t_0 <= 10.0) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) / sin(B)) - (x / B);
	} else if (t_0 <= 2e+282) {
		tmp = t_1;
	} else {
		tmp = (((fma(fma(x, 0.022222222222222223, (fma((x * 0.022222222222222223), -0.3333333333333333, (x * 0.009523809523809525)) * (B * B))), (B * B), (0.3333333333333333 * x)) * (B * B)) - x) / B) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = 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(-1.0 / 2.0))))
	t_1 = Float64(Float64(sqrt(Float64(1.0 / Float64(fma(2.0, x, Float64(F * F)) + 2.0))) * Float64(F / B)) - Float64(x / tan(B)))
	tmp = 0.0
	if (t_0 <= -2e+20)
		tmp = t_1;
	elseif (t_0 <= 10.0)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) / sin(B)) - Float64(x / B));
	elseif (t_0 <= 2e+282)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(Float64(fma(fma(x, 0.022222222222222223, Float64(fma(Float64(x * 0.022222222222222223), -0.3333333333333333, Float64(x * 0.009523809523809525)) * Float64(B * B))), Float64(B * B), Float64(0.3333333333333333 * x)) * Float64(B * B)) - x) / B) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], N[(-1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(1.0 / N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+20], t$95$1, If[LessEqual[t$95$0, 10.0], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+282], t$95$1, N[(N[(N[(N[(N[(N[(x * 0.022222222222222223 + N[(N[(N[(x * 0.022222222222222223), $MachinePrecision] * -0.3333333333333333 + N[(x * 0.009523809523809525), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + N[(0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < -2e20 or 10 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 2.00000000000000007e282

   1. Initial program 97.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 97.4

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

   4. Applied rewrites 97.6%

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

   6. Applied rewrites 99.0%

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

   7. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 97.5

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

   9. Applied rewrites 97.5%

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

   if -2e20 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 10

   1. Initial program 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 79.2

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

   4. Applied rewrites 79.2%

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

   6. Applied rewrites 79.3%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 60.6

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

   9. Applied rewrites 60.6%

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

   if 2.00000000000000007e282 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))))))

   1. Initial program 28.7%

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

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      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 66.7%

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

   6. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{{B}^{2} \cdot \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(\frac{-1}{9} \cdot x + \left(\frac{2}{15} \cdot x + {B}^{2} \cdot \left(\frac{-1}{3} \cdot \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right) + \left(\frac{-2}{45} \cdot x + \frac{17}{315} \cdot x\right)\right)\right)\right)\right) - x}{B}} + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6} \cdot F, B \cdot B, F\right)}{B} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{B}^{2} \cdot \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(\frac{-1}{9} \cdot x + \left(\frac{2}{15} \cdot x + {B}^{2} \cdot \left(\frac{-1}{3} \cdot \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right) + \left(\frac{-2}{45} \cdot x + \frac{17}{315} \cdot x\right)\right)\right)\right)\right) - x}{B}} + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6} \cdot F, B \cdot B, F\right)}{B} \]

   8. Applied rewrites 66.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.022222222222222223, \mathsf{fma}\left(x \cdot 0.022222222222222223, -0.3333333333333333, x \cdot 0.009523809523809525\right) \cdot \left(B \cdot B\right)\right), B \cdot B, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right) - x}{B}} + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right)}{B} \]

   9. Taylor expanded in F around inf

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \frac{1}{45}, \mathsf{fma}\left(x \cdot \frac{1}{45}, \frac{-1}{3}, x \cdot \frac{1}{105}\right) \cdot \left(B \cdot B\right)\right), B \cdot B, \frac{1}{3} \cdot x\right) \cdot \left(B \cdot B\right) - x}{B} + \color{blue}{\frac{1}{\sin B}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \frac{1}{45}, \mathsf{fma}\left(x \cdot \frac{1}{45}, \frac{-1}{3}, x \cdot \frac{1}{105}\right) \cdot \left(B \cdot B\right)\right), B \cdot B, \frac{1}{3} \cdot x\right) \cdot \left(B \cdot B\right) - x}{B} + \color{blue}{\frac{1}{\sin B}} \]

       2. lower-sin.f64 82.9

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.022222222222222223, \mathsf{fma}\left(x \cdot 0.022222222222222223, -0.3333333333333333, x \cdot 0.009523809523809525\right) \cdot \left(B \cdot B\right)\right), B \cdot B, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right) - x}{B} + \frac{1}{\color{blue}{\sin B}} \]

   11. Applied rewrites 82.9%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.022222222222222223, \mathsf{fma}\left(x \cdot 0.022222222222222223, -0.3333333333333333, x \cdot 0.009523809523809525\right) \cdot \left(B \cdot B\right)\right), B \cdot B, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right) - x}{B} + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)} \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)} \leq 10:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)} \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.022222222222222223, \mathsf{fma}\left(x \cdot 0.022222222222222223, -0.3333333333333333, x \cdot 0.009523809523809525\right) \cdot \left(B \cdot B\right)\right), B \cdot B, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right) - x}{B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)}\\ t_1 := \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{+20}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 10:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+282}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (+
          (* x (/ -1.0 (tan B)))
          (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (/ -1.0 2.0)))))
        (t_1
         (-
          (* (sqrt (/ 1.0 (+ (fma 2.0 x (* F F)) 2.0))) (/ F B))
          (/ x (tan B)))))
   (if (<= t_0 -2e+20)
     t_1
     (if (<= t_0 10.0)
       (- (/ (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) (sin B)) (/ x B))
       (if (<= t_0 2e+282) t_1 (/ (- 1.0 x) B))))))

C

double code(double F, double B, double x) {
	double t_0 = (x * (-1.0 / tan(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), (-1.0 / 2.0)));
	double t_1 = (sqrt((1.0 / (fma(2.0, x, (F * F)) + 2.0))) * (F / B)) - (x / tan(B));
	double tmp;
	if (t_0 <= -2e+20) {
		tmp = t_1;
	} else if (t_0 <= 10.0) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) / sin(B)) - (x / B);
	} else if (t_0 <= 2e+282) {
		tmp = t_1;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-1.0 / 2.0))))
	t_1 = Float64(Float64(sqrt(Float64(1.0 / Float64(fma(2.0, x, Float64(F * F)) + 2.0))) * Float64(F / B)) - Float64(x / tan(B)))
	tmp = 0.0
	if (t_0 <= -2e+20)
		tmp = t_1;
	elseif (t_0 <= 10.0)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) / sin(B)) - Float64(x / B));
	elseif (t_0 <= 2e+282)
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], N[(-1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(1.0 / N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -2e+20], t$95$1, If[LessEqual[t$95$0, 10.0], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+282], t$95$1, N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < -2e20 or 10 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 2.00000000000000007e282

   1. Initial program 97.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 97.4

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

   4. Applied rewrites 97.6%

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

   6. Applied rewrites 99.0%

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

   7. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 97.5

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

   9. Applied rewrites 97.5%

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

   if -2e20 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 10

   1. Initial program 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 79.2

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

   4. Applied rewrites 79.2%

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

   6. Applied rewrites 79.3%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 60.6

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

   9. Applied rewrites 60.6%

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

   if 2.00000000000000007e282 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))))))

   1. Initial program 28.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 66.7

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

   5. Applied rewrites 66.7%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 82.9%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)} \leq -2 \cdot 10^{+20}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)} \leq 10:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)} \leq 2 \cdot 10^{+282}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -250000000:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 210000000:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}, \frac{F}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -250000000.0)
   (/ (- (fma (cos B) x 1.0)) (sin B))
   (if (<= F 210000000.0)
     (fma (sqrt (/ 1.0 (fma F F 2.0))) (/ F (sin B)) (/ (- x) (tan B)))
     (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -250000000.0) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else if (F <= 210000000.0) {
		tmp = fma(sqrt((1.0 / fma(F, F, 2.0))), (F / sin(B)), (-x / tan(B)));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -250000000.0)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	elseif (F <= 210000000.0)
		tmp = fma(sqrt(Float64(1.0 / fma(F, F, 2.0))), Float64(F / sin(B)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -250000000.0], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 210000000.0], N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / 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[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.5e8

   1. Initial program 70.3%

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

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   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 -2.5e8 < F < 2.1e8

   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. *-commutative N/A

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

      5. lower-fma.f64 99.5

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 99.6

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

   7. Applied rewrites 99.6%

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

   if 2.1e8 < F

   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 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -250000000:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 210000000:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}, \frac{F}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+30}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 50000000:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot \sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5e+30)
   (/ (- (fma (cos B) x 1.0)) (sin B))
   (if (<= F 50000000.0)
     (- (/ F (* (sqrt (fma 2.0 x (fma F F 2.0))) (sin B))) (/ x (tan B)))
     (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5e+30) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else if (F <= 50000000.0) {
		tmp = (F / (sqrt(fma(2.0, x, fma(F, F, 2.0))) * sin(B))) - (x / tan(B));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5e+30)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	elseif (F <= 50000000.0)
		tmp = Float64(Float64(F / Float64(sqrt(fma(2.0, x, fma(F, F, 2.0))) * sin(B))) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5e+30], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 50000000.0], N[(N[(F / N[(N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.9999999999999998e30

   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 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 -4.9999999999999998e30 < F < 5e7

   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. *-commutative N/A

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

      5. lower-fma.f64 99.4

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

   4. Applied rewrites 99.6%

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

   6. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lower-fma.f64 99.6

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

   8. Applied rewrites 99.6%

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

   if 5e7 < F

   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 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+30}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 50000000:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot \sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{F}{\sin B}, -\frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9e-5)
   (/ (- (fma (cos B) x 1.0)) (sin B))
   (if (<= F 3.5e-185)
     (- (* (sqrt (/ 1.0 (+ (fma 2.0 x (* F F)) 2.0))) (/ F B)) (/ x (tan B)))
     (if (<= F 2.3e-9)
       (fma (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F (sin B)) (- (/ x B)))
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9e-5) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else if (F <= 3.5e-185) {
		tmp = (sqrt((1.0 / (fma(2.0, x, (F * F)) + 2.0))) * (F / B)) - (x / tan(B));
	} else if (F <= 2.3e-9) {
		tmp = fma(pow(fma(2.0, x, fma(F, F, 2.0)), -0.5), (F / sin(B)), -(x / B));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9e-5)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	elseif (F <= 3.5e-185)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(fma(2.0, x, Float64(F * F)) + 2.0))) * Float64(F / B)) - Float64(x / tan(B)));
	elseif (F <= 2.3e-9)
		tmp = fma((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5), Float64(F / sin(B)), Float64(-Float64(x / B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9e-5], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.5e-185], N[(N[(N[Sqrt[N[(1.0 / N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.3e-9], N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] + (-N[(x / B), $MachinePrecision])), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -9.00000000000000057e-5

   1. Initial program 72.3%

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

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   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 97.5

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

   5. Applied rewrites 97.5%

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

   if -9.00000000000000057e-5 < F < 3.4999999999999998e-185

   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. *-commutative N/A

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

      5. lower-fma.f64 99.5

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 88.6

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

   9. Applied rewrites 88.6%

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

   if 3.4999999999999998e-185 < F < 2.2999999999999999e-9

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 86.1

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

   5. Applied rewrites 86.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   7. Applied rewrites 86.1%

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

   if 2.2999999999999999e-9 < F

   1. Initial program 66.1%

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

   3. Taylor expanded in 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 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{F}{\sin B}, -\frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9e-5)
   (/ (- (fma (cos B) x 1.0)) (sin B))
   (if (<= F 3.5e-185)
     (- (* (sqrt (/ 1.0 (+ (fma 2.0 x (* F F)) 2.0))) (/ F B)) (/ x (tan B)))
     (if (<= F 2.3e-9)
       (- (/ (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) (sin B)) (/ x B))
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9e-5) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else if (F <= 3.5e-185) {
		tmp = (sqrt((1.0 / (fma(2.0, x, (F * F)) + 2.0))) * (F / B)) - (x / tan(B));
	} else if (F <= 2.3e-9) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) / sin(B)) - (x / B);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9e-5)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	elseif (F <= 3.5e-185)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(fma(2.0, x, Float64(F * F)) + 2.0))) * Float64(F / B)) - Float64(x / tan(B)));
	elseif (F <= 2.3e-9)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9e-5], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.5e-185], N[(N[(N[Sqrt[N[(1.0 / N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.3e-9], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -9.00000000000000057e-5

   1. Initial program 72.3%

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

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   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 97.5

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

   5. Applied rewrites 97.5%

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

   if -9.00000000000000057e-5 < F < 3.4999999999999998e-185

   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. *-commutative N/A

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

      5. lower-fma.f64 99.5

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 88.6

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

   9. Applied rewrites 88.6%

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

   if 3.4999999999999998e-185 < F < 2.2999999999999999e-9

   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. *-commutative N/A

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

      5. lower-fma.f64 99.5

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

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 86.0

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

   9. Applied rewrites 86.0%

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

   if 2.2999999999999999e-9 < F

   1. Initial program 66.1%

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

   3. Taylor expanded in 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 99.3

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

   5. Applied rewrites 99.3%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \cos B \cdot x, 1\right)}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9e-5)
   (/ (- (fma (cos B) x 1.0)) (sin B))
   (if (<= F 3.5e-185)
     (- (* (sqrt (/ 1.0 (+ (fma 2.0 x (* F F)) 2.0))) (/ F B)) (/ x (tan B)))
     (if (<= F 2.3e-9)
       (- (/ (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) (sin B)) (/ x B))
       (/ (fma -1.0 (* (cos B) x) 1.0) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9e-5) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else if (F <= 3.5e-185) {
		tmp = (sqrt((1.0 / (fma(2.0, x, (F * F)) + 2.0))) * (F / B)) - (x / tan(B));
	} else if (F <= 2.3e-9) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) / sin(B)) - (x / 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 <= -9e-5)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	elseif (F <= 3.5e-185)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(fma(2.0, x, Float64(F * F)) + 2.0))) * Float64(F / B)) - Float64(x / tan(B)));
	elseif (F <= 2.3e-9)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) / sin(B)) - Float64(x / 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, -9e-5], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.5e-185], N[(N[(N[Sqrt[N[(1.0 / N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.3e-9], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $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 < -9.00000000000000057e-5

   1. Initial program 72.3%

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

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   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 97.5

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

   5. Applied rewrites 97.5%

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

   if -9.00000000000000057e-5 < F < 3.4999999999999998e-185

   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. *-commutative N/A

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

      5. lower-fma.f64 99.5

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 88.6

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

   9. Applied rewrites 88.6%

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

   if 3.4999999999999998e-185 < F < 2.2999999999999999e-9

   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. *-commutative N/A

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

      5. lower-fma.f64 99.5

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

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 99.3%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 86.0

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

   9. Applied rewrites 86.0%

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

   if 2.2999999999999999e-9 < F

   1. Initial program 66.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 66.1

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

   4. Applied rewrites 66.2%

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

   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 99.2

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

   7. Applied rewrites 99.2%

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

4. Final simplification 94.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \cos B \cdot x, 1\right)}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 77.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+24} \lor \neg \left(x \leq 3.6 \cdot 10^{-6}\right):\\ \;\;\;\;\left(-x\right) \cdot \frac{\cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -6.5e+24) (not (<= x 3.6e-6)))
   (* (- x) (/ (cos B) (sin B)))
   (- (/ (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) (sin B)) (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -6.5e+24) || !(x <= 3.6e-6)) {
		tmp = -x * (cos(B) / sin(B));
	} else {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) / sin(B)) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -6.5e+24) || !(x <= 3.6e-6))
		tmp = Float64(Float64(-x) * Float64(cos(B) / sin(B)));
	else
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) / sin(B)) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -6.5e+24], N[Not[LessEqual[x, 3.6e-6]], $MachinePrecision]], N[((-x) * N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.4999999999999996e24 or 3.59999999999999984e-6 < x

   1. Initial program 87.0%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-sin.f64 98.6

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

   5. Applied rewrites 98.6%

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

   if -6.4999999999999996e24 < x < 3.59999999999999984e-6

   1. Initial program 78.0%

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

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

      5. lower-fma.f64 78.0

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

   4. Applied rewrites 78.0%

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

   6. Applied rewrites 80.1%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 69.7

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

   9. Applied rewrites 69.7%

      \[\leadsto \frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \color{blue}{\frac{x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+24} \lor \neg \left(x \leq 3.6 \cdot 10^{-6}\right):\\ \;\;\;\;\left(-x\right) \cdot \frac{\cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+24}:\\ \;\;\;\;\left(-x\right) \cdot \frac{\cos B}{\sin B}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos B \cdot x}{-\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -6.5e+24)
   (* (- x) (/ (cos B) (sin B)))
   (if (<= x 3.6e-6)
     (- (/ (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) (sin B)) (/ x B))
     (/ (* (cos B) x) (- (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -6.5e+24) {
		tmp = -x * (cos(B) / sin(B));
	} else if (x <= 3.6e-6) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) / sin(B)) - (x / B);
	} else {
		tmp = (cos(B) * x) / -sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -6.5e+24)
		tmp = Float64(Float64(-x) * Float64(cos(B) / sin(B)));
	elseif (x <= 3.6e-6)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(cos(B) * x) / Float64(-sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -6.5e+24], N[((-x) * N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.6e-6], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.4999999999999996e24

   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 F around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-sin.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if -6.4999999999999996e24 < x < 3.59999999999999984e-6

   1. Initial program 78.0%

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

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

      5. lower-fma.f64 78.0

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

   4. Applied rewrites 78.0%

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

   6. Applied rewrites 80.1%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 69.7

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

   9. Applied rewrites 69.7%

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

   if 3.59999999999999984e-6 < x

   1. Initial program 92.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 98.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 98.4

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 98.4

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

      14. lift-neg.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval 98.4

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

   4. Applied rewrites 98.4%

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

   5. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   6. 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. lower-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 96.3

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

   7. Applied rewrites 96.3%

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

   8. Taylor expanded in x around inf

      \[\leadsto -\frac{x \cdot \cos B}{\sin B} \]
   9. Step-by-step derivation

   10. Applied rewrites 98.5%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{+24}:\\ \;\;\;\;\left(-x\right) \cdot \frac{\cos B}{\sin B}\\ \mathbf{elif}\;x \leq 3.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos B \cdot x}{-\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9e-5)
   (/ (- (fma (cos B) x 1.0)) (sin B))
   (if (<= F 3.5e-185)
     (- (* (sqrt (/ 1.0 (+ (fma 2.0 x (* F F)) 2.0))) (/ F B)) (/ x (tan B)))
     (if (<= F 2.5e+154)
       (- (/ (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) (sin B)) (/ x B))
       (+
        (* x (/ -1.0 (tan B)))
        (/ (fma (* B B) 0.16666666666666666 1.0) B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9e-5) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else if (F <= 3.5e-185) {
		tmp = (sqrt((1.0 / (fma(2.0, x, (F * F)) + 2.0))) * (F / B)) - (x / tan(B));
	} else if (F <= 2.5e+154) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) / sin(B)) - (x / B);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (fma((B * B), 0.16666666666666666, 1.0) / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9e-5)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	elseif (F <= 3.5e-185)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(fma(2.0, x, Float64(F * F)) + 2.0))) * Float64(F / B)) - Float64(x / tan(B)));
	elseif (F <= 2.5e+154)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(fma(Float64(B * B), 0.16666666666666666, 1.0) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9e-5], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.5e-185], N[(N[(N[Sqrt[N[(1.0 / N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.5e+154], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(B * B), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -9.00000000000000057e-5

   1. Initial program 72.3%

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

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   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 97.5

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

   5. Applied rewrites 97.5%

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

   if -9.00000000000000057e-5 < F < 3.4999999999999998e-185

   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. *-commutative N/A

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

      5. lower-fma.f64 99.5

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

   4. Applied rewrites 99.7%

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

   6. Applied rewrites 99.7%

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

   7. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 88.6

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

   9. Applied rewrites 88.6%

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

   if 3.4999999999999998e-185 < F < 2.50000000000000002e154

   1. Initial program 94.0%

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

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

      5. lower-fma.f64 94.1

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

   4. Applied rewrites 94.1%

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

   6. Applied rewrites 98.3%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 81.4

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

   9. Applied rewrites 81.4%

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

   if 2.50000000000000002e154 < F

   1. Initial program 37.9%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      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 23.9%

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

   6. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1 + \frac{1}{6} \cdot {B}^{2}}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 61.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-185}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+35}:\\ \;\;\;\;\frac{1 + x}{-\sin B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1e+35)
   (/ (+ 1.0 x) (- (sin B)))
   (if (<= F 2.5e+154)
     (- (/ (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) (sin B)) (/ x B))
     (+ (* x (/ -1.0 (tan B))) (/ (fma (* B B) 0.16666666666666666 1.0) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e+35) {
		tmp = (1.0 + x) / -sin(B);
	} else if (F <= 2.5e+154) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) / sin(B)) - (x / B);
	} else {
		tmp = (x * (-1.0 / tan(B))) + (fma((B * B), 0.16666666666666666, 1.0) / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1e+35)
		tmp = Float64(Float64(1.0 + x) / Float64(-sin(B)));
	elseif (F <= 2.5e+154)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(fma(Float64(B * B), 0.16666666666666666, 1.0) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1e+35], N[(N[(1.0 + x), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 2.5e+154], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(B * B), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.9999999999999997e34

   1. Initial program 66.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 76.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 76.4

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 76.4

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

      14. lift-neg.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval 76.4

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

   4. Applied rewrites 76.4%

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

   5. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   6. 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. lower-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 99.7

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in B around 0

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

   10. Applied rewrites 76.8%

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

   if -9.9999999999999997e34 < F < 2.50000000000000002e154

   1. Initial program 97.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 97.1

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

   4. Applied rewrites 97.2%

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

   6. Applied rewrites 99.1%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 76.4

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

   9. Applied rewrites 76.4%

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

   if 2.50000000000000002e154 < F

   1. Initial program 37.9%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      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 23.9%

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

   6. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1 + \frac{1}{6} \cdot {B}^{2}}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 61.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+35}:\\ \;\;\;\;\frac{1 + x}{-\sin B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}}\\ \mathbf{if}\;F \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + x}{-\sin B}\\ \mathbf{elif}\;F \leq 4.25 \cdot 10^{-162}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, F, \mathsf{fma}\left(0.16666666666666666 \cdot F, t\_0, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right) - x\right)}{B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (+ (fma 2.0 x (* F F)) 2.0)))))
   (if (<= F -9e-5)
     (/ (+ 1.0 x) (- (sin B)))
     (if (<= F 4.25e-162)
       (/
        (fma
         t_0
         F
         (-
          (*
           (fma (* 0.16666666666666666 F) t_0 (* 0.3333333333333333 x))
           (* B B))
          x))
        B)
       (if (<= F 2.2e-9)
         (* (sqrt (/ 1.0 (fma F F 2.0))) (/ F (sin B)))
         (+
          (* x (/ -1.0 (tan B)))
          (/ (fma (* B B) 0.16666666666666666 1.0) B)))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt((1.0 / (fma(2.0, x, (F * F)) + 2.0)));
	double tmp;
	if (F <= -9e-5) {
		tmp = (1.0 + x) / -sin(B);
	} else if (F <= 4.25e-162) {
		tmp = fma(t_0, F, ((fma((0.16666666666666666 * F), t_0, (0.3333333333333333 * x)) * (B * B)) - x)) / B;
	} else if (F <= 2.2e-9) {
		tmp = sqrt((1.0 / fma(F, F, 2.0))) * (F / sin(B));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (fma((B * B), 0.16666666666666666, 1.0) / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = sqrt(Float64(1.0 / Float64(fma(2.0, x, Float64(F * F)) + 2.0)))
	tmp = 0.0
	if (F <= -9e-5)
		tmp = Float64(Float64(1.0 + x) / Float64(-sin(B)));
	elseif (F <= 4.25e-162)
		tmp = Float64(fma(t_0, F, Float64(Float64(fma(Float64(0.16666666666666666 * F), t_0, Float64(0.3333333333333333 * x)) * Float64(B * B)) - x)) / B);
	elseif (F <= 2.2e-9)
		tmp = Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) * Float64(F / sin(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(fma(Float64(B * B), 0.16666666666666666, 1.0) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[F, -9e-5], N[(N[(1.0 + x), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 4.25e-162], N[(N[(t$95$0 * F + N[(N[(N[(N[(0.16666666666666666 * F), $MachinePrecision] * t$95$0 + N[(0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.2e-9], N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(B * B), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -9.00000000000000057e-5

   1. Initial program 72.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 80.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 80.4

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 80.4

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

      14. lift-neg.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval 80.4

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

   4. Applied rewrites 80.4%

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

   5. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   6. 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. lower-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 97.5

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

   7. Applied rewrites 97.5%

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

   8. Taylor expanded in B around 0

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

   10. Applied rewrites 75.4%

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

   if -9.00000000000000057e-5 < F < 4.24999999999999977e-162

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.6

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 99.6

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 99.6

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

      14. lift-neg.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]

   7. Applied rewrites 62.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}}, F, \mathsf{fma}\left(0.16666666666666666 \cdot F, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}}, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right) - x\right)}{B}} \]

   if 4.24999999999999977e-162 < F < 2.1999999999999998e-9

   1. Initial program 99.5%

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

   3. Taylor expanded in 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 70.8

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

   5. Applied rewrites 70.8%

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

   if 2.1999999999999998e-9 < F

   1. Initial program 66.1%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      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 45.8%

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

   6. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1 + \frac{1}{6} \cdot {B}^{2}}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 62.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + x}{-\sin B}\\ \mathbf{elif}\;F \leq 4.25 \cdot 10^{-162}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}}, F, \mathsf{fma}\left(0.16666666666666666 \cdot F, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}}, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right) - x\right)}{B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{-9}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.1% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 1.25:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.022222222222222223, \mathsf{fma}\left(x \cdot 0.022222222222222223, -0.3333333333333333, x \cdot 0.009523809523809525\right) \cdot \left(B \cdot B\right)\right), B \cdot B, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right) - x}{B} + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 1.25)
   (+
    (/
     (-
      (*
       (fma
        (fma
         x
         0.022222222222222223
         (*
          (fma
           (* x 0.022222222222222223)
           -0.3333333333333333
           (* x 0.009523809523809525))
          (* B B)))
        (* B B)
        (* 0.3333333333333333 x))
       (* B B))
      x)
     B)
    (/
     (*
      (sqrt (/ 1.0 (fma F F (fma 2.0 x 2.0))))
      (fma (* 0.16666666666666666 F) (* B B) F))
     B))
   (* (sqrt (/ 1.0 (fma F F 2.0))) (/ F (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 1.25) {
		tmp = (((fma(fma(x, 0.022222222222222223, (fma((x * 0.022222222222222223), -0.3333333333333333, (x * 0.009523809523809525)) * (B * B))), (B * B), (0.3333333333333333 * x)) * (B * B)) - x) / B) + ((sqrt((1.0 / fma(F, F, fma(2.0, x, 2.0)))) * fma((0.16666666666666666 * F), (B * B), F)) / B);
	} else {
		tmp = sqrt((1.0 / fma(F, F, 2.0))) * (F / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 1.25)
		tmp = Float64(Float64(Float64(Float64(fma(fma(x, 0.022222222222222223, Float64(fma(Float64(x * 0.022222222222222223), -0.3333333333333333, Float64(x * 0.009523809523809525)) * Float64(B * B))), Float64(B * B), Float64(0.3333333333333333 * x)) * Float64(B * B)) - x) / B) + Float64(Float64(sqrt(Float64(1.0 / fma(F, F, fma(2.0, x, 2.0)))) * fma(Float64(0.16666666666666666 * F), Float64(B * B), F)) / B));
	else
		tmp = Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) * Float64(F / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 1.25], N[(N[(N[(N[(N[(N[(x * 0.022222222222222223 + N[(N[(N[(x * 0.022222222222222223), $MachinePrecision] * -0.3333333333333333 + N[(x * 0.009523809523809525), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + N[(0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 / N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(0.16666666666666666 * F), $MachinePrecision] * N[(B * B), $MachinePrecision] + F), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.25

   1. Initial program 79.6%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      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 64.9%

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

   6. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{{B}^{2} \cdot \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(\frac{-1}{9} \cdot x + \left(\frac{2}{15} \cdot x + {B}^{2} \cdot \left(\frac{-1}{3} \cdot \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right) + \left(\frac{-2}{45} \cdot x + \frac{17}{315} \cdot x\right)\right)\right)\right)\right) - x}{B}} + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6} \cdot F, B \cdot B, F\right)}{B} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{{B}^{2} \cdot \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(\frac{-1}{9} \cdot x + \left(\frac{2}{15} \cdot x + {B}^{2} \cdot \left(\frac{-1}{3} \cdot \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right) + \left(\frac{-2}{45} \cdot x + \frac{17}{315} \cdot x\right)\right)\right)\right)\right) - x}{B}} + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\frac{1}{6} \cdot F, B \cdot B, F\right)}{B} \]

   8. Applied rewrites 61.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.022222222222222223, \mathsf{fma}\left(x \cdot 0.022222222222222223, -0.3333333333333333, x \cdot 0.009523809523809525\right) \cdot \left(B \cdot B\right)\right), B \cdot B, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right) - x}{B}} + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right)}{B} \]

   if 1.25 < B

   1. Initial program 88.2%

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

   3. Taylor expanded in 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 32.2

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

   5. Applied rewrites 32.2%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.25:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, 0.022222222222222223, \mathsf{fma}\left(x \cdot 0.022222222222222223, -0.3333333333333333, x \cdot 0.009523809523809525\right) \cdot \left(B \cdot B\right)\right), B \cdot B, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right) - x}{B} + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 58.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + x}{-\sin B}\\ \mathbf{elif}\;F \leq 185000000:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9e-5)
   (/ (+ 1.0 x) (- (sin B)))
   (if (<= F 185000000.0)
     (+
      (- (/ x B))
      (/
       (*
        (sqrt (/ 1.0 (fma F F (fma 2.0 x 2.0))))
        (fma (* 0.16666666666666666 F) (* B B) F))
       B))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9e-5) {
		tmp = (1.0 + x) / -sin(B);
	} else if (F <= 185000000.0) {
		tmp = -(x / B) + ((sqrt((1.0 / fma(F, F, fma(2.0, x, 2.0)))) * fma((0.16666666666666666 * F), (B * B), F)) / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9e-5)
		tmp = Float64(Float64(1.0 + x) / Float64(-sin(B)));
	elseif (F <= 185000000.0)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(sqrt(Float64(1.0 / fma(F, F, fma(2.0, x, 2.0)))) * fma(Float64(0.16666666666666666 * F), Float64(B * B), F)) / B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9e-5], N[(N[(1.0 + x), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 185000000.0], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[Sqrt[N[(1.0 / N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(0.16666666666666666 * F), $MachinePrecision] * N[(B * B), $MachinePrecision] + F), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.00000000000000057e-5

   1. Initial program 72.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 80.4

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 80.4

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 80.4

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

      14. lift-neg.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval 80.4

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

   4. Applied rewrites 80.4%

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

   5. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   6. 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. lower-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 97.5

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

   7. Applied rewrites 97.5%

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

   8. Taylor expanded in B around 0

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

   10. Applied rewrites 75.4%

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

   if -9.00000000000000057e-5 < F < 1.85e8

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      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 65.4%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 58.3

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

   8. Applied rewrites 58.3%

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

   if 1.85e8 < F

   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 39.3

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

   5. Applied rewrites 39.3%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 52.6%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{1 + x}{-\sin B}\\ \mathbf{elif}\;F \leq 185000000:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 51.2% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\frac{0.5}{F} \cdot \frac{\mathsf{fma}\left(2, x, 2\right)}{F} - 1\right) - x}{B}\\ \mathbf{elif}\;F \leq 185000000:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9e-5)
   (/ (- (- (* (/ 0.5 F) (/ (fma 2.0 x 2.0) F)) 1.0) x) B)
   (if (<= F 185000000.0)
     (+
      (- (/ x B))
      (/
       (*
        (sqrt (/ 1.0 (fma F F (fma 2.0 x 2.0))))
        (fma (* 0.16666666666666666 F) (* B B) F))
       B))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9e-5) {
		tmp = ((((0.5 / F) * (fma(2.0, x, 2.0) / F)) - 1.0) - x) / B;
	} else if (F <= 185000000.0) {
		tmp = -(x / B) + ((sqrt((1.0 / fma(F, F, fma(2.0, x, 2.0)))) * fma((0.16666666666666666 * F), (B * B), F)) / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9e-5)
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 / F) * Float64(fma(2.0, x, 2.0) / F)) - 1.0) - x) / B);
	elseif (F <= 185000000.0)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(sqrt(Float64(1.0 / fma(F, F, fma(2.0, x, 2.0)))) * fma(Float64(0.16666666666666666 * F), Float64(B * B), F)) / B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9e-5], N[(N[(N[(N[(N[(0.5 / F), $MachinePrecision] * N[(N[(2.0 * x + 2.0), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 185000000.0], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[Sqrt[N[(1.0 / N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(0.16666666666666666 * F), $MachinePrecision] * N[(B * B), $MachinePrecision] + F), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.00000000000000057e-5

   1. Initial program 72.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 41.8

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

   5. Applied rewrites 41.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 52.2%

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

   if -9.00000000000000057e-5 < F < 1.85e8

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      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 65.4%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 58.3

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

   8. Applied rewrites 58.3%

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

   if 1.85e8 < F

   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 39.3

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

   5. Applied rewrites 39.3%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 52.6%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\frac{0.5}{F} \cdot \frac{\mathsf{fma}\left(2, x, 2\right)}{F} - 1\right) - x}{B}\\ \mathbf{elif}\;F \leq 185000000:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 44.6% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot F}{B}\\ \mathbf{if}\;F \leq -1350000:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -3.5 \cdot 10^{-35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{-152}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 10800:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (* (sqrt (/ 1.0 (fma F F 2.0))) F) B)))
   (if (<= F -1350000.0)
     (/ (- -1.0 x) B)
     (if (<= F -3.5e-35)
       t_0
       (if (<= F 1.5e-152)
         (/ (- x) B)
         (if (<= F 10800.0) t_0 (/ (- 1.0 x) B)))))))

C

double code(double F, double B, double x) {
	double t_0 = (sqrt((1.0 / fma(F, F, 2.0))) * F) / B;
	double tmp;
	if (F <= -1350000.0) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -3.5e-35) {
		tmp = t_0;
	} else if (F <= 1.5e-152) {
		tmp = -x / B;
	} else if (F <= 10800.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) * F) / B)
	tmp = 0.0
	if (F <= -1350000.0)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -3.5e-35)
		tmp = t_0;
	elseif (F <= 1.5e-152)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 10800.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[F, -1350000.0], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -3.5e-35], t$95$0, If[LessEqual[F, 1.5e-152], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 10800.0], t$95$0, N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.35e6

   1. Initial program 70.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 42.7

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

   5. Applied rewrites 42.7%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 53.6%

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

   if -1.35e6 < F < -3.49999999999999996e-35 or 1.5e-152 < F < 10800

   1. Initial program 99.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 52.0

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

   5. Applied rewrites 52.0%

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

   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 43.5%

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

   if -3.49999999999999996e-35 < F < 1.5e-152

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      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 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 47.7%

      \[\leadsto \frac{-x}{B} \]

   if 10800 < F

   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 39.3

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

   5. Applied rewrites 39.3%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 52.6%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1350000:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -3.5 \cdot 10^{-35}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot F}{B}\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{-152}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 10800:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 44.6% accurate, 5.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{B}\\ \mathbf{if}\;F \leq -1350000:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -3.5 \cdot 10^{-35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{-152}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 10800:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (sqrt (/ 1.0 (fma F F 2.0))) (/ F B))))
   (if (<= F -1350000.0)
     (/ (- -1.0 x) B)
     (if (<= F -3.5e-35)
       t_0
       (if (<= F 1.5e-152)
         (/ (- x) B)
         (if (<= F 10800.0) t_0 (/ (- 1.0 x) B)))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt((1.0 / fma(F, F, 2.0))) * (F / B);
	double tmp;
	if (F <= -1350000.0) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -3.5e-35) {
		tmp = t_0;
	} else if (F <= 1.5e-152) {
		tmp = -x / B;
	} else if (F <= 10800.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) * Float64(F / B))
	tmp = 0.0
	if (F <= -1350000.0)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -3.5e-35)
		tmp = t_0;
	elseif (F <= 1.5e-152)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 10800.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1350000.0], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -3.5e-35], t$95$0, If[LessEqual[F, 1.5e-152], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 10800.0], t$95$0, N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.35e6

   1. Initial program 70.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 42.7

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

   5. Applied rewrites 42.7%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 53.6%

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

   if -1.35e6 < F < -3.49999999999999996e-35 or 1.5e-152 < F < 10800

   1. Initial program 99.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 52.0

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

   5. Applied rewrites 52.0%

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

   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 43.4%

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

   if -3.49999999999999996e-35 < F < 1.5e-152

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      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 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 47.7%

      \[\leadsto \frac{-x}{B} \]

   if 10800 < F

   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 39.3

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

   5. Applied rewrites 39.3%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 52.6%

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

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1350000:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -3.5 \cdot 10^{-35}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{-152}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 10800:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 51.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\frac{0.5}{F} \cdot \frac{\mathsf{fma}\left(2, x, 2\right)}{F} - 1\right) - x}{B}\\ \mathbf{elif}\;F \leq 10000:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9e-5)
   (/ (- (- (* (/ 0.5 F) (/ (fma 2.0 x 2.0) F)) 1.0) x) B)
   (if (<= F 10000.0)
     (/ (- (* (sqrt (/ 1.0 (fma F F (fma 2.0 x 2.0)))) F) x) B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9e-5) {
		tmp = ((((0.5 / F) * (fma(2.0, x, 2.0) / F)) - 1.0) - x) / B;
	} else if (F <= 10000.0) {
		tmp = ((sqrt((1.0 / fma(F, F, fma(2.0, x, 2.0)))) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9e-5)
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 / F) * Float64(fma(2.0, x, 2.0) / F)) - 1.0) - x) / B);
	elseif (F <= 10000.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(F, F, fma(2.0, x, 2.0)))) * F) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9e-5], N[(N[(N[(N[(N[(0.5 / F), $MachinePrecision] * N[(N[(2.0 * x + 2.0), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 10000.0], N[(N[(N[(N[Sqrt[N[(1.0 / N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]), $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 < -9.00000000000000057e-5

   1. Initial program 72.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 41.8

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

   5. Applied rewrites 41.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 52.2%

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

   if -9.00000000000000057e-5 < F < 1e4

   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 58.0

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

   5. Applied rewrites 58.0%

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

   if 1e4 < F

   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 39.3

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

   5. Applied rewrites 39.3%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 52.6%

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

4. Final simplification 54.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\frac{0.5}{F} \cdot \frac{\mathsf{fma}\left(2, x, 2\right)}{F} - 1\right) - x}{B}\\ \mathbf{elif}\;F \leq 10000:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 51.1% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(2, x, 2\right)}{F}\\ \mathbf{if}\;F \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\frac{0.5}{F} \cdot t\_0 - 1\right) - x}{B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.5}{F}, t\_0, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (fma 2.0 x 2.0) F)))
   (if (<= F -9e-5)
     (/ (- (- (* (/ 0.5 F) t_0) 1.0) x) B)
     (if (<= F 2.3e-9)
       (/ (- (* (sqrt (/ 1.0 (fma 2.0 x 2.0))) F) x) B)
       (/ (- (fma (/ -0.5 F) t_0 1.0) x) B)))))

C

double code(double F, double B, double x) {
	double t_0 = fma(2.0, x, 2.0) / F;
	double tmp;
	if (F <= -9e-5) {
		tmp = ((((0.5 / F) * t_0) - 1.0) - x) / B;
	} else if (F <= 2.3e-9) {
		tmp = ((sqrt((1.0 / fma(2.0, x, 2.0))) * F) - x) / B;
	} else {
		tmp = (fma((-0.5 / F), t_0, 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(fma(2.0, x, 2.0) / F)
	tmp = 0.0
	if (F <= -9e-5)
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 / F) * t_0) - 1.0) - x) / B);
	elseif (F <= 2.3e-9)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, 2.0))) * F) - x) / B);
	else
		tmp = Float64(Float64(fma(Float64(-0.5 / F), t_0, 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(2.0 * x + 2.0), $MachinePrecision] / F), $MachinePrecision]}, If[LessEqual[F, -9e-5], N[(N[(N[(N[(N[(0.5 / F), $MachinePrecision] * t$95$0), $MachinePrecision] - 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.3e-9], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(-0.5 / F), $MachinePrecision] * t$95$0 + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.00000000000000057e-5

   1. Initial program 72.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 41.8

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

   5. Applied rewrites 41.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 52.2%

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

   if -9.00000000000000057e-5 < F < 2.2999999999999999e-9

   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 58.6

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

   5. Applied rewrites 58.6%

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

   6. Taylor expanded in F around 0

      \[\leadsto \frac{\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot F - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 58.3%

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

   if 2.2999999999999999e-9 < F

   1. Initial program 66.1%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      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 39.3

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

   5. Applied rewrites 39.3%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 52.0%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-0.5}{F}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F}, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\frac{0.5}{F} \cdot \frac{\mathsf{fma}\left(2, x, 2\right)}{F} - 1\right) - x}{B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.5}{F}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F}, 1\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 51.2% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.9:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.5}{F}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F}, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.9)
   (/ (- -1.0 x) B)
   (if (<= F 2.3e-9)
     (/ (- (* (sqrt (/ 1.0 (fma 2.0 x 2.0))) F) x) B)
     (/ (- (fma (/ -0.5 F) (/ (fma 2.0 x 2.0) F) 1.0) x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.9) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.3e-9) {
		tmp = ((sqrt((1.0 / fma(2.0, x, 2.0))) * F) - x) / B;
	} else {
		tmp = (fma((-0.5 / F), (fma(2.0, x, 2.0) / F), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.9)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.3e-9)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, 2.0))) * F) - x) / B);
	else
		tmp = Float64(Float64(fma(Float64(-0.5 / F), Float64(fma(2.0, x, 2.0) / F), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.9], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.3e-9], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(-0.5 / F), $MachinePrecision] * N[(N[(2.0 * x + 2.0), $MachinePrecision] / F), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.8999999999999999

   1. Initial program 71.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 42.3

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

   5. Applied rewrites 42.3%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 52.6%

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

   if -1.8999999999999999 < F < 2.2999999999999999e-9

   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 58.0

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

   5. Applied rewrites 58.0%

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

   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 57.8%

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

   if 2.2999999999999999e-9 < F

   1. Initial program 66.1%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      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 39.3

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

   5. Applied rewrites 39.3%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 52.0%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-0.5}{F}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F}, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.9:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.5}{F}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F}, 1\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 51.2% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.9:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.9)
   (/ (- -1.0 x) B)
   (if (<= F 2.3e-9)
     (/ (- (* (sqrt (/ 1.0 (fma 2.0 x 2.0))) F) x) B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.9) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.3e-9) {
		tmp = ((sqrt((1.0 / fma(2.0, x, 2.0))) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.9)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.3e-9)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, 2.0))) * F) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.9], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.3e-9], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + 2.0), $MachinePrecision]), $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 < -1.8999999999999999

   1. Initial program 71.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 42.3

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

   5. Applied rewrites 42.3%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 52.6%

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

   if -1.8999999999999999 < F < 2.2999999999999999e-9

   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 58.0

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

   5. Applied rewrites 58.0%

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

   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 57.8%

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

   if 2.2999999999999999e-9 < F

   1. Initial program 66.1%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      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 39.3

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

   5. Applied rewrites 39.3%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 51.5%

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

4. Final simplification 54.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.9:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 43.7% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.6 \cdot 10^{-59}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{-114}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(-0.3333333333333333 \cdot x, B \cdot B, x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.6e-59)
   (/ (- -1.0 x) B)
   (if (<= F 2.1e-114)
     (/ (- (fma (* -0.3333333333333333 x) (* B B) x)) B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.6e-59) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.1e-114) {
		tmp = -fma((-0.3333333333333333 * x), (B * B), x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.6e-59)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.1e-114)
		tmp = Float64(Float64(-fma(Float64(-0.3333333333333333 * x), Float64(B * B), x)) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.6e-59], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.1e-114], N[((-N[(N[(-0.3333333333333333 * x), $MachinePrecision] * N[(B * B), $MachinePrecision] + x), $MachinePrecision]) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.6e-59

   1. Initial program 75.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 44.6

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

   5. Applied rewrites 44.6%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 48.3%

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

   if -1.6e-59 < F < 2.09999999999999993e-114

   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. *-commutative N/A

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

      5. lower-fma.f64 99.5

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

      3. distribute-lft-neg-in N/A

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-sin.f64 71.9

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

   7. Applied rewrites 71.9%

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

   8. Taylor expanded in B around 0

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

   10. Applied rewrites 46.2%

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

   if 2.09999999999999993e-114 < F

   1. Initial program 72.3%

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

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

   8. Applied rewrites 45.2%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.6 \cdot 10^{-59}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{-114}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(-0.3333333333333333 \cdot x, B \cdot B, x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 44.1% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{-50}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.5e-65)
   (/ (- -1.0 x) B)
   (if (<= F 5.6e-50) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.5e-65) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 5.6e-50) {
		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 <= (-1.5d-65)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 5.6d-50) 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 <= -1.5e-65) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 5.6e-50) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.5e-65:
		tmp = (-1.0 - x) / B
	elif F <= 5.6e-50:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.5e-65)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 5.6e-50)
		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 <= -1.5e-65)
		tmp = (-1.0 - x) / B;
	elseif (F <= 5.6e-50)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.5e-65], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 5.6e-50], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.49999999999999999e-65

   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 44.1

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

   5. Applied rewrites 44.1%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 47.8%

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

   if -1.49999999999999999e-65 < F < 5.5999999999999996e-50

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      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 57.1

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

   5. Applied rewrites 57.1%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 43.9%

      \[\leadsto \frac{-x}{B} \]

   if 5.5999999999999996e-50 < F

   1. Initial program 68.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 42.1

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

   5. Applied rewrites 42.1%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 47.9%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{-50}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 36.9% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.5e-65) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.5e-65) {
		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 <= (-1.5d-65)) 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 <= -1.5e-65) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.5e-65:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.5e-65)
		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 <= -1.5e-65)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.5e-65], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.49999999999999999e-65

   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 44.1

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

   5. Applied rewrites 44.1%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 47.8%

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

   if -1.49999999999999999e-65 < F

   1. Initial program 84.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 49.8

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

   5. Applied rewrites 49.8%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 33.3%

      \[\leadsto \frac{-x}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 38.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{-65}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 30.0% 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 81.6%

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

3. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

   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 47.9

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

5. Applied rewrites 47.9%

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

6. Taylor expanded in F around 0

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

8. Applied rewrites 29.3%

   \[\leadsto \frac{-x}{B} \]

9. Final simplification 29.3%

   \[\leadsto \frac{-x}{B} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025015 
(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))))))