VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.0% → 99.1%

Time: 4.8s

Alternatives: 26

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -155000000000:\\ \;\;\;\;-\frac{1 + \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -155000000000.0)
   (- (/ (+ 1.0 (* (cos B) x)) (sin B)))
   (if (<= F 2e-9)
     (+
      (- (/ (* x 1.0) (tan B)))
      (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
     (+ (- (* x (/ 1.0 (tan B)))) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -155000000000.0) {
		tmp = -((1.0 + (cos(B) * x)) / sin(B));
	} else if (F <= 2e-9) {
		tmp = -((x * 1.0) / tan(B)) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = -(x * (1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-155000000000.0d0)) then
        tmp = -((1.0d0 + (cos(b) * x)) / sin(b))
    else if (f <= 2d-9) then
        tmp = -((x * 1.0d0) / tan(b)) + ((f / sin(b)) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
    else
        tmp = -(x * (1.0d0 / tan(b))) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -155000000000.0) {
		tmp = -((1.0 + (Math.cos(B) * x)) / Math.sin(B));
	} else if (F <= 2e-9) {
		tmp = -((x * 1.0) / Math.tan(B)) + ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = -(x * (1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -155000000000.0:
		tmp = -((1.0 + (math.cos(B) * x)) / math.sin(B))
	elif F <= 2e-9:
		tmp = -((x * 1.0) / math.tan(B)) + ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))
	else:
		tmp = -(x * (1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -155000000000.0)
		tmp = Float64(-Float64(Float64(1.0 + Float64(cos(B) * x)) / sin(B)));
	elseif (F <= 2e-9)
		tmp = Float64(Float64(-Float64(Float64(x * 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)))));
	else
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -155000000000.0)
		tmp = -((1.0 + (cos(B) * x)) / sin(B));
	elseif (F <= 2e-9)
		tmp = -((x * 1.0) / tan(B)) + ((F / sin(B)) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
	else
		tmp = -(x * (1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -155000000000.0], (-N[(N[(1.0 + N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 2e-9], N[((-N[(N[(x * 1.0), $MachinePrecision] / N[Tan[B], $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], 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 < -1.55e11

   1. Initial program 59.1%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if -1.55e11 < F < 2.00000000000000012e-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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 99.6

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

   3. Applied rewrites 99.6%

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

   if 2.00000000000000012e-9 < F

   1. Initial program 59.6%

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

   2. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 97.8

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

   4. Applied rewrites 97.8%

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

Alternative 2: 99.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -100000000000:\\ \;\;\;\;-\frac{1 + \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-9}:\\ \;\;\;\;t\_0 + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* x (/ 1.0 (tan B))))))
   (if (<= F -100000000000.0)
     (- (/ (+ 1.0 (* (cos B) x)) (sin B)))
     (if (<= F 2e-9)
       (+
        t_0
        (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B)));
	double tmp;
	if (F <= -100000000000.0) {
		tmp = -((1.0 + (cos(B) * x)) / sin(B));
	} else if (F <= 2e-9) {
		tmp = t_0 + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -(x * (1.0d0 / tan(b)))
    if (f <= (-100000000000.0d0)) then
        tmp = -((1.0d0 + (cos(b) * x)) / sin(b))
    else if (f <= 2d-9) then
        tmp = t_0 + ((f / sin(b)) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / Math.tan(B)));
	double tmp;
	if (F <= -100000000000.0) {
		tmp = -((1.0 + (Math.cos(B) * x)) / Math.sin(B));
	} else if (F <= 2e-9) {
		tmp = t_0 + ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -(x * (1.0 / math.tan(B)))
	tmp = 0
	if F <= -100000000000.0:
		tmp = -((1.0 + (math.cos(B) * x)) / math.sin(B))
	elif F <= 2e-9:
		tmp = t_0 + ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (F <= -100000000000.0)
		tmp = Float64(-Float64(Float64(1.0 + Float64(cos(B) * x)) / sin(B)));
	elseif (F <= 2e-9)
		tmp = Float64(t_0 + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -(x * (1.0 / tan(B)));
	tmp = 0.0;
	if (F <= -100000000000.0)
		tmp = -((1.0 + (cos(B) * x)) / sin(B));
	elseif (F <= 2e-9)
		tmp = t_0 + ((F / sin(B)) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[F, -100000000000.0], (-N[(N[(1.0 + N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 2e-9], N[(t$95$0 + 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], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e11

   1. Initial program 59.2%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if -1e11 < F < 2.00000000000000012e-9

   1. Initial program 99.5%

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

   if 2.00000000000000012e-9 < F

   1. Initial program 59.6%

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

   2. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 97.8

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

   4. Applied rewrites 97.8%

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

Alternative 3: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -100000000000:\\ \;\;\;\;-\frac{1 + \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-9}:\\ \;\;\;\;t\_0 + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* x (/ 1.0 (tan B))))))
   (if (<= F -100000000000.0)
     (- (/ (+ 1.0 (* (cos B) x)) (sin B)))
     (if (<= F 2e-9)
       (+ t_0 (* (/ F (sin B)) (/ 1.0 (sqrt (fma F F 2.0)))))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B)));
	double tmp;
	if (F <= -100000000000.0) {
		tmp = -((1.0 + (cos(B) * x)) / sin(B));
	} else if (F <= 2e-9) {
		tmp = t_0 + ((F / sin(B)) * (1.0 / sqrt(fma(F, F, 2.0))));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (F <= -100000000000.0)
		tmp = Float64(-Float64(Float64(1.0 + Float64(cos(B) * x)) / sin(B)));
	elseif (F <= 2e-9)
		tmp = Float64(t_0 + Float64(Float64(F / sin(B)) * Float64(1.0 / sqrt(fma(F, F, 2.0)))));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[F, -100000000000.0], (-N[(N[(1.0 + N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 2e-9], N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e11

   1. Initial program 59.2%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if -1e11 < F < 2.00000000000000012e-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. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 99.4

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

   4. Applied rewrites 99.4%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lift-fma.f64 99.4

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

   6. Applied rewrites 99.4%

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

   if 2.00000000000000012e-9 < F

   1. Initial program 59.6%

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

   2. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 97.8

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

   4. Applied rewrites 97.8%

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

Alternative 4: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -100000000000:\\ \;\;\;\;-\frac{1 + \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-9}:\\ \;\;\;\;t\_0 + \frac{F}{\sin B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* x (/ 1.0 (tan B))))))
   (if (<= F -100000000000.0)
     (- (/ (+ 1.0 (* (cos B) x)) (sin B)))
     (if (<= F 2e-9)
       (+ t_0 (* (/ F (sin B)) (sqrt (/ 1.0 (fma F F 2.0)))))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B)));
	double tmp;
	if (F <= -100000000000.0) {
		tmp = -((1.0 + (cos(B) * x)) / sin(B));
	} else if (F <= 2e-9) {
		tmp = t_0 + ((F / sin(B)) * sqrt((1.0 / fma(F, F, 2.0))));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (F <= -100000000000.0)
		tmp = Float64(-Float64(Float64(1.0 + Float64(cos(B) * x)) / sin(B)));
	elseif (F <= 2e-9)
		tmp = Float64(t_0 + Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / fma(F, F, 2.0)))));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[F, -100000000000.0], (-N[(N[(1.0 + N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 2e-9], N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e11

   1. Initial program 59.2%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if -1e11 < F < 2.00000000000000012e-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. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 99.4

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

   4. Applied rewrites 99.4%

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

   if 2.00000000000000012e-9 < F

   1. Initial program 59.6%

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

   2. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 97.8

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

   4. Applied rewrites 97.8%

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

Alternative 5: 98.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -195:\\ \;\;\;\;-\frac{1 + \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-9}:\\ \;\;\;\;t\_0 + \frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* x (/ 1.0 (tan B))))))
   (if (<= F -195.0)
     (- (/ (+ 1.0 (* (cos B) x)) (sin B)))
     (if (<= F 2e-9)
       (+ t_0 (* (/ F (sin B)) (sqrt 0.5)))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B)));
	double tmp;
	if (F <= -195.0) {
		tmp = -((1.0 + (cos(B) * x)) / sin(B));
	} else if (F <= 2e-9) {
		tmp = t_0 + ((F / sin(B)) * sqrt(0.5));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -(x * (1.0d0 / tan(b)))
    if (f <= (-195.0d0)) then
        tmp = -((1.0d0 + (cos(b) * x)) / sin(b))
    else if (f <= 2d-9) then
        tmp = t_0 + ((f / sin(b)) * sqrt(0.5d0))
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / Math.tan(B)));
	double tmp;
	if (F <= -195.0) {
		tmp = -((1.0 + (Math.cos(B) * x)) / Math.sin(B));
	} else if (F <= 2e-9) {
		tmp = t_0 + ((F / Math.sin(B)) * Math.sqrt(0.5));
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -(x * (1.0 / math.tan(B)))
	tmp = 0
	if F <= -195.0:
		tmp = -((1.0 + (math.cos(B) * x)) / math.sin(B))
	elif F <= 2e-9:
		tmp = t_0 + ((F / math.sin(B)) * math.sqrt(0.5))
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (F <= -195.0)
		tmp = Float64(-Float64(Float64(1.0 + Float64(cos(B) * x)) / sin(B)));
	elseif (F <= 2e-9)
		tmp = Float64(t_0 + Float64(Float64(F / sin(B)) * sqrt(0.5)));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -(x * (1.0 / tan(B)));
	tmp = 0.0;
	if (F <= -195.0)
		tmp = -((1.0 + (cos(B) * x)) / sin(B));
	elseif (F <= 2e-9)
		tmp = t_0 + ((F / sin(B)) * sqrt(0.5));
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[F, -195.0], (-N[(N[(1.0 + N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 2e-9], N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -195

   1. Initial program 60.0%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.5

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

   4. Applied rewrites 99.5%

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

   if -195 < F < 2.00000000000000012e-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. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in F around 0

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

   7. Applied rewrites 99.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \sqrt{0.5} \]

   if 2.00000000000000012e-9 < F

   1. Initial program 59.6%

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

   2. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 97.8

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

   4. Applied rewrites 97.8%

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

Alternative 6: 91.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -230:\\ \;\;\;\;-\frac{1 + \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 1.46 \cdot 10^{-12}:\\ \;\;\;\;t\_0 + \frac{F}{B} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* x (/ 1.0 (tan B))))))
   (if (<= F -230.0)
     (- (/ (+ 1.0 (* (cos B) x)) (sin B)))
     (if (<= F 1.46e-12)
       (+ t_0 (* (/ F B) (/ 1.0 (sqrt (fma F F (fma 2.0 x 2.0))))))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B)));
	double tmp;
	if (F <= -230.0) {
		tmp = -((1.0 + (cos(B) * x)) / sin(B));
	} else if (F <= 1.46e-12) {
		tmp = t_0 + ((F / B) * (1.0 / sqrt(fma(F, F, fma(2.0, x, 2.0)))));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (F <= -230.0)
		tmp = Float64(-Float64(Float64(1.0 + Float64(cos(B) * x)) / sin(B)));
	elseif (F <= 1.46e-12)
		tmp = Float64(t_0 + Float64(Float64(F / B) * Float64(1.0 / sqrt(fma(F, F, fma(2.0, x, 2.0))))));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[F, -230.0], (-N[(N[(1.0 + N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 1.46e-12], N[(t$95$0 + N[(N[(F / B), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -230

   1. Initial program 60.0%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.5

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

   4. Applied rewrites 99.5%

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

   if -230 < F < 1.46000000000000004e-12

   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. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 83.2

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

   4. Applied rewrites 83.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

      11. count-2-rev N/A

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

      12. associate-+l+ N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-fma.f64 83.1

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

   6. Applied rewrites 83.1%

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

   if 1.46000000000000004e-12 < F

   1. Initial program 59.8%

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

   2. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 97.3

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

   4. Applied rewrites 97.3%

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

Alternative 7: 91.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -230:\\ \;\;\;\;-\frac{1 + t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 1.46 \cdot 10^{-12}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -230.0)
     (- (/ (+ 1.0 t_0) (sin B)))
     (if (<= F 1.46e-12)
       (+
        (- (* x (/ 1.0 (tan B))))
        (* (/ F B) (/ 1.0 (sqrt (fma F F (fma 2.0 x 2.0))))))
       (/ (- 1.0 t_0) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -230.0) {
		tmp = -((1.0 + t_0) / sin(B));
	} else if (F <= 1.46e-12) {
		tmp = -(x * (1.0 / tan(B))) + ((F / B) * (1.0 / sqrt(fma(F, F, fma(2.0, x, 2.0)))));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -230.0)
		tmp = Float64(-Float64(Float64(1.0 + t_0) / sin(B)));
	elseif (F <= 1.46e-12)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / B) * Float64(1.0 / sqrt(fma(F, F, fma(2.0, x, 2.0))))));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -230.0], (-N[(N[(1.0 + t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 1.46e-12], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / B), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -230

   1. Initial program 60.0%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.5

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

   4. Applied rewrites 99.5%

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

   if -230 < F < 1.46000000000000004e-12

   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. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 83.2

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

   4. Applied rewrites 83.2%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

      11. count-2-rev N/A

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

      12. associate-+l+ N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-fma.f64 83.1

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

   6. Applied rewrites 83.1%

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

   if 1.46000000000000004e-12 < F

   1. Initial program 59.8%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 97.3

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

   4. Applied rewrites 97.3%

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

Alternative 8: 85.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -3.5 \cdot 10^{+281}:\\ \;\;\;\;t\_0 + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -340:\\ \;\;\;\;-\frac{1 + x}{\sin B}\\ \mathbf{elif}\;F \leq 1.46 \cdot 10^{-12}:\\ \;\;\;\;t\_0 + \frac{F}{B} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* x (/ 1.0 (tan B))))))
   (if (<= F -3.5e+281)
     (+ t_0 (/ -1.0 B))
     (if (<= F -340.0)
       (- (/ (+ 1.0 x) (sin B)))
       (if (<= F 1.46e-12)
         (+ t_0 (* (/ F B) (/ 1.0 (sqrt (fma F F (fma 2.0 x 2.0))))))
         (/ (- 1.0 (* (cos B) x)) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B)));
	double tmp;
	if (F <= -3.5e+281) {
		tmp = t_0 + (-1.0 / B);
	} else if (F <= -340.0) {
		tmp = -((1.0 + x) / sin(B));
	} else if (F <= 1.46e-12) {
		tmp = t_0 + ((F / B) * (1.0 / sqrt(fma(F, F, fma(2.0, x, 2.0)))));
	} else {
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (F <= -3.5e+281)
		tmp = Float64(t_0 + Float64(-1.0 / B));
	elseif (F <= -340.0)
		tmp = Float64(-Float64(Float64(1.0 + x) / sin(B)));
	elseif (F <= 1.46e-12)
		tmp = Float64(t_0 + Float64(Float64(F / B) * Float64(1.0 / sqrt(fma(F, F, fma(2.0, x, 2.0))))));
	else
		tmp = Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[F, -3.5e+281], N[(t$95$0 + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -340.0], (-N[(N[(1.0 + x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 1.46e-12], N[(t$95$0 + N[(N[(F / B), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.4999999999999998e281

   1. Initial program 26.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 27.2

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

   4. Applied rewrites 27.2%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 73.8

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

   7. Applied rewrites 73.8%

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

   if -3.4999999999999998e281 < F < -340

   1. Initial program 63.5%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 77.2%

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

   if -340 < F < 1.46000000000000004e-12

   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. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 83.1

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

   4. Applied rewrites 83.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

      11. count-2-rev N/A

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

      12. associate-+l+ N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-fma.f64 83.1

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

   6. Applied rewrites 83.1%

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

   if 1.46000000000000004e-12 < F

   1. Initial program 59.8%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 97.3

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

   4. Applied rewrites 97.3%

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

Alternative 9: 78.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -3.5 \cdot 10^{+281}:\\ \;\;\;\;t\_0 + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -340:\\ \;\;\;\;-\frac{1 + x}{\sin B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-134}:\\ \;\;\;\;t\_0 + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}\\ \mathbf{elif}\;F \leq 2.15 \cdot 10^{+143}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* x (/ 1.0 (tan B))))))
   (if (<= F -3.5e+281)
     (+ t_0 (/ -1.0 B))
     (if (<= F -340.0)
       (- (/ (+ 1.0 x) (sin B)))
       (if (<= F 1.2e-134)
         (+ t_0 (* (/ F B) (sqrt (/ 1.0 (fma F F 2.0)))))
         (if (<= F 2.15e+143)
           (+
            (- (/ x B))
            (/ (* F (pow (fma 2.0 x (fma F F 2.0)) -0.5)) (sin B)))
           (+ t_0 (/ 1.0 B))))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B)));
	double tmp;
	if (F <= -3.5e+281) {
		tmp = t_0 + (-1.0 / B);
	} else if (F <= -340.0) {
		tmp = -((1.0 + x) / sin(B));
	} else if (F <= 1.2e-134) {
		tmp = t_0 + ((F / B) * sqrt((1.0 / fma(F, F, 2.0))));
	} else if (F <= 2.15e+143) {
		tmp = -(x / B) + ((F * pow(fma(2.0, x, fma(F, F, 2.0)), -0.5)) / sin(B));
	} else {
		tmp = t_0 + (1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (F <= -3.5e+281)
		tmp = Float64(t_0 + Float64(-1.0 / B));
	elseif (F <= -340.0)
		tmp = Float64(-Float64(Float64(1.0 + x) / sin(B)));
	elseif (F <= 1.2e-134)
		tmp = Float64(t_0 + Float64(Float64(F / B) * sqrt(Float64(1.0 / fma(F, F, 2.0)))));
	elseif (F <= 2.15e+143)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F * (fma(2.0, x, fma(F, F, 2.0)) ^ -0.5)) / sin(B)));
	else
		tmp = Float64(t_0 + Float64(1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[F, -3.5e+281], N[(t$95$0 + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -340.0], (-N[(N[(1.0 + x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 1.2e-134], N[(t$95$0 + N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.15e+143], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F * N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -3.4999999999999998e281

   1. Initial program 26.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 27.2

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

   4. Applied rewrites 27.2%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 73.8

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

   7. Applied rewrites 73.8%

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

   if -3.4999999999999998e281 < F < -340

   1. Initial program 63.5%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 77.2%

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

   if -340 < F < 1.20000000000000005e-134

   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. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 84.6%

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

   if 1.20000000000000005e-134 < F < 2.15000000000000001e143

   1. Initial program 93.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 75.7

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

   6. Applied rewrites 75.7%

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

   if 2.15000000000000001e143 < F

   1. Initial program 33.1%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 31.2

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

   4. Applied rewrites 31.2%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 74.4

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

   7. Applied rewrites 74.4%

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

Alternative 10: 74.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -3.5 \cdot 10^{+281}:\\ \;\;\;\;t\_0 + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -340:\\ \;\;\;\;-\frac{1 + x}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+157}:\\ \;\;\;\;t\_0 + \frac{F}{B} \cdot \frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* x (/ 1.0 (tan B))))))
   (if (<= F -3.5e+281)
     (+ t_0 (/ -1.0 B))
     (if (<= F -340.0)
       (- (/ (+ 1.0 x) (sin B)))
       (if (<= F 5e+157)
         (+ t_0 (* (/ F B) (/ 1.0 (sqrt (fma F F (fma 2.0 x 2.0))))))
         (+
          (/ (- (* 0.3333333333333333 (* (* B B) x)) x) B)
          (/ 1.0 (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B)));
	double tmp;
	if (F <= -3.5e+281) {
		tmp = t_0 + (-1.0 / B);
	} else if (F <= -340.0) {
		tmp = -((1.0 + x) / sin(B));
	} else if (F <= 5e+157) {
		tmp = t_0 + ((F / B) * (1.0 / sqrt(fma(F, F, fma(2.0, x, 2.0)))));
	} else {
		tmp = (((0.3333333333333333 * ((B * B) * x)) - x) / B) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (F <= -3.5e+281)
		tmp = Float64(t_0 + Float64(-1.0 / B));
	elseif (F <= -340.0)
		tmp = Float64(-Float64(Float64(1.0 + x) / sin(B)));
	elseif (F <= 5e+157)
		tmp = Float64(t_0 + Float64(Float64(F / B) * Float64(1.0 / sqrt(fma(F, F, fma(2.0, x, 2.0))))));
	else
		tmp = Float64(Float64(Float64(Float64(0.3333333333333333 * Float64(Float64(B * B) * x)) - x) / B) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[F, -3.5e+281], N[(t$95$0 + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -340.0], (-N[(N[(1.0 + x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 5e+157], N[(t$95$0 + N[(N[(F / B), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 * N[(N[(B * B), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.4999999999999998e281

   1. Initial program 26.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 27.2

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

   4. Applied rewrites 27.2%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 73.8

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

   7. Applied rewrites 73.8%

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

   if -3.4999999999999998e281 < F < -340

   1. Initial program 63.5%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 77.2%

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

   if -340 < F < 4.99999999999999976e157

   1. Initial program 96.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 77.1

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

   4. Applied rewrites 77.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

      11. count-2-rev N/A

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

      12. associate-+l+ N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-fma.f64 77.1

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

   6. Applied rewrites 77.1%

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

   if 4.99999999999999976e157 < F

   1. Initial program 30.1%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 30.2

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

   4. Applied rewrites 30.2%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 7.0

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

   7. Applied rewrites 7.0%

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

   8. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 61.5

         \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{\sin B} \]

   10. Applied rewrites 61.5%

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

Alternative 11: 74.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -3.5 \cdot 10^{+281}:\\ \;\;\;\;t\_0 + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -340:\\ \;\;\;\;-\frac{1 + x}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+157}:\\ \;\;\;\;t\_0 + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* x (/ 1.0 (tan B))))))
   (if (<= F -3.5e+281)
     (+ t_0 (/ -1.0 B))
     (if (<= F -340.0)
       (- (/ (+ 1.0 x) (sin B)))
       (if (<= F 5e+157)
         (+ t_0 (* (/ F B) (sqrt (/ 1.0 (fma F F 2.0)))))
         (+
          (/ (- (* 0.3333333333333333 (* (* B B) x)) x) B)
          (/ 1.0 (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B)));
	double tmp;
	if (F <= -3.5e+281) {
		tmp = t_0 + (-1.0 / B);
	} else if (F <= -340.0) {
		tmp = -((1.0 + x) / sin(B));
	} else if (F <= 5e+157) {
		tmp = t_0 + ((F / B) * sqrt((1.0 / fma(F, F, 2.0))));
	} else {
		tmp = (((0.3333333333333333 * ((B * B) * x)) - x) / B) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (F <= -3.5e+281)
		tmp = Float64(t_0 + Float64(-1.0 / B));
	elseif (F <= -340.0)
		tmp = Float64(-Float64(Float64(1.0 + x) / sin(B)));
	elseif (F <= 5e+157)
		tmp = Float64(t_0 + Float64(Float64(F / B) * sqrt(Float64(1.0 / fma(F, F, 2.0)))));
	else
		tmp = Float64(Float64(Float64(Float64(0.3333333333333333 * Float64(Float64(B * B) * x)) - x) / B) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[F, -3.5e+281], N[(t$95$0 + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -340.0], (-N[(N[(1.0 + x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 5e+157], N[(t$95$0 + N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 * N[(N[(B * B), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.4999999999999998e281

   1. Initial program 26.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 27.2

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

   4. Applied rewrites 27.2%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 73.8

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

   7. Applied rewrites 73.8%

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

   if -3.4999999999999998e281 < F < -340

   1. Initial program 63.5%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 77.2%

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

   if -340 < F < 4.99999999999999976e157

   1. Initial program 96.7%

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

   2. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. lower-fma.f64 96.7

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

   4. Applied rewrites 96.7%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 77.1%

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

   if 4.99999999999999976e157 < F

   1. Initial program 30.1%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 30.2

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

   4. Applied rewrites 30.2%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 7.0

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

   7. Applied rewrites 7.0%

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

   8. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 61.5

         \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{\sin B} \]

   10. Applied rewrites 61.5%

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

Alternative 12: 68.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;x \leq -1.9 \cdot 10^{-82}:\\ \;\;\;\;t\_0 + \frac{1}{B}\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-128}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* x (/ 1.0 (tan B))))))
   (if (<= x -1.9e-82)
     (+ t_0 (/ 1.0 B))
     (if (<= x 2.15e-128)
       (* (sqrt (/ 1.0 (fma F F 2.0))) (/ F (sin B)))
       (+ t_0 (/ -1.0 B))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B)));
	double tmp;
	if (x <= -1.9e-82) {
		tmp = t_0 + (1.0 / B);
	} else if (x <= 2.15e-128) {
		tmp = sqrt((1.0 / fma(F, F, 2.0))) * (F / sin(B));
	} else {
		tmp = t_0 + (-1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (x <= -1.9e-82)
		tmp = Float64(t_0 + Float64(1.0 / B));
	elseif (x <= 2.15e-128)
		tmp = Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) * Float64(F / sin(B)));
	else
		tmp = Float64(t_0 + Float64(-1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[x, -1.9e-82], N[(t$95$0 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.15e-128], N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.9000000000000001e-82

   1. Initial program 71.6%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 65.1

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

   4. Applied rewrites 65.1%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 72.6

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

   7. Applied rewrites 72.6%

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

   if -1.9000000000000001e-82 < x < 2.14999999999999997e-128

   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. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\sin B} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{{F}^{2} + 2}} \cdot \frac{F}{\sin B} \]

      6. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{F \cdot F + 2}} \cdot \frac{F}{\sin B} \]

      7. lower-fma.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-/.f64 55.8

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

   4. Applied rewrites 55.8%

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

   if 2.14999999999999997e-128 < x

   1. Initial program 83.9%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 78.6

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

   4. Applied rewrites 78.6%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 78.9

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

   7. Applied rewrites 78.9%

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

Alternative 13: 68.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B}\\ \mathbf{if}\;x \leq -1.85 \cdot 10^{-76}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-128}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ (- (* x (/ 1.0 (tan B)))) (/ -1.0 B))))
   (if (<= x -1.85e-76)
     t_0
     (if (<= x 2.15e-128)
       (* (sqrt (/ 1.0 (fma F F 2.0))) (/ F (sin B)))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B))) + (-1.0 / B);
	double tmp;
	if (x <= -1.85e-76) {
		tmp = t_0;
	} else if (x <= 2.15e-128) {
		tmp = sqrt((1.0 / fma(F, F, 2.0))) * (F / sin(B));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(-1.0 / B))
	tmp = 0.0
	if (x <= -1.85e-76)
		tmp = t_0;
	elseif (x <= 2.15e-128)
		tmp = Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) * Float64(F / sin(B)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.85e-76], t$95$0, If[LessEqual[x, 2.15e-128], N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.85000000000000006e-76 or 2.14999999999999997e-128 < x

   1. Initial program 80.1%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 74.6

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

   4. Applied rewrites 74.6%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 77.0

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

   7. Applied rewrites 77.0%

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

   if -1.85000000000000006e-76 < x < 2.14999999999999997e-128

   1. Initial program 72.4%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\sin B} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{{F}^{2} + 2}} \cdot \frac{F}{\sin B} \]

      6. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{F \cdot F + 2}} \cdot \frac{F}{\sin B} \]

      7. lower-fma.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-/.f64 55.5

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

   4. Applied rewrites 55.5%

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

Alternative 14: 58.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B}\\ \mathbf{if}\;F \leq -230:\\ \;\;\;\;-\frac{1 + x}{\sin B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{+28}:\\ \;\;\;\;t\_0 + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+86}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- (* 0.3333333333333333 (* (* B B) x)) x) B)))
   (if (<= F -230.0)
     (- (/ (+ 1.0 x) (sin B)))
     (if (<= F 1.9e+28)
       (+ t_0 (* (/ F B) (sqrt (/ 1.0 (+ (fma F F (+ x x)) 2.0)))))
       (if (<= F 2e+86)
         (* (sqrt (/ 1.0 (fma F F 2.0))) (/ F (sin B)))
         (+ t_0 (/ 1.0 B)))))))

C

double code(double F, double B, double x) {
	double t_0 = ((0.3333333333333333 * ((B * B) * x)) - x) / B;
	double tmp;
	if (F <= -230.0) {
		tmp = -((1.0 + x) / sin(B));
	} else if (F <= 1.9e+28) {
		tmp = t_0 + ((F / B) * sqrt((1.0 / (fma(F, F, (x + x)) + 2.0))));
	} else if (F <= 2e+86) {
		tmp = sqrt((1.0 / fma(F, F, 2.0))) * (F / sin(B));
	} else {
		tmp = t_0 + (1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(0.3333333333333333 * Float64(Float64(B * B) * x)) - x) / B)
	tmp = 0.0
	if (F <= -230.0)
		tmp = Float64(-Float64(Float64(1.0 + x) / sin(B)));
	elseif (F <= 1.9e+28)
		tmp = Float64(t_0 + Float64(Float64(F / B) * sqrt(Float64(1.0 / Float64(fma(F, F, Float64(x + x)) + 2.0)))));
	elseif (F <= 2e+86)
		tmp = Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) * Float64(F / sin(B)));
	else
		tmp = Float64(t_0 + Float64(1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(0.3333333333333333 * N[(N[(B * B), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[F, -230.0], (-N[(N[(1.0 + x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 1.9e+28], N[(t$95$0 + N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(F * F + N[(x + x), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2e+86], N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -230

   1. Initial program 60.0%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 77.3%

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

   if -230 < F < 1.8999999999999999e28

   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. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 82.0

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

   4. Applied rewrites 82.0%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 50.2

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

   7. Applied rewrites 50.2%

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

   if 1.8999999999999999e28 < F < 2e86

   1. Initial program 91.2%

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

   2. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\sin B} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{{F}^{2} + 2}} \cdot \frac{F}{\sin B} \]

      6. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{F \cdot F + 2}} \cdot \frac{F}{\sin B} \]

      7. lower-fma.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-/.f64 55.6

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

   4. Applied rewrites 55.6%

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

   if 2e86 < F

   1. Initial program 47.5%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 39.0

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

   4. Applied rewrites 39.0%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 15.5

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

   7. Applied rewrites 15.5%

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

   8. Taylor expanded in F around inf

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

      1. lower-/.f64 50.7

         \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{B} \]

   10. Applied rewrites 50.7%

       \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{\color{blue}{B}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 58.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B}\\ \mathbf{if}\;F \leq -230:\\ \;\;\;\;-\frac{1 + x}{\sin B}\\ \mathbf{elif}\;F \leq 1.62 \cdot 10^{+78}:\\ \;\;\;\;t\_0 + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- (* 0.3333333333333333 (* (* B B) x)) x) B)))
   (if (<= F -230.0)
     (- (/ (+ 1.0 x) (sin B)))
     (if (<= F 1.62e+78)
       (+ t_0 (* (/ F B) (sqrt (/ 1.0 (+ (fma F F (+ x x)) 2.0)))))
       (+ t_0 (/ 1.0 B))))))

C

double code(double F, double B, double x) {
	double t_0 = ((0.3333333333333333 * ((B * B) * x)) - x) / B;
	double tmp;
	if (F <= -230.0) {
		tmp = -((1.0 + x) / sin(B));
	} else if (F <= 1.62e+78) {
		tmp = t_0 + ((F / B) * sqrt((1.0 / (fma(F, F, (x + x)) + 2.0))));
	} else {
		tmp = t_0 + (1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(0.3333333333333333 * Float64(Float64(B * B) * x)) - x) / B)
	tmp = 0.0
	if (F <= -230.0)
		tmp = Float64(-Float64(Float64(1.0 + x) / sin(B)));
	elseif (F <= 1.62e+78)
		tmp = Float64(t_0 + Float64(Float64(F / B) * sqrt(Float64(1.0 / Float64(fma(F, F, Float64(x + x)) + 2.0)))));
	else
		tmp = Float64(t_0 + Float64(1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(0.3333333333333333 * N[(N[(B * B), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[F, -230.0], (-N[(N[(1.0 + x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 1.62e+78], N[(t$95$0 + N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(F * F + N[(x + x), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -230

   1. Initial program 60.0%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 77.3%

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

   if -230 < F < 1.6199999999999999e78

   1. Initial program 98.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 80.1

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

   4. Applied rewrites 80.1%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 49.5

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

   7. Applied rewrites 49.5%

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

   if 1.6199999999999999e78 < F

   1. Initial program 48.8%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 39.8

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

   4. Applied rewrites 39.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 16.3

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

   7. Applied rewrites 16.3%

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

   8. Taylor expanded in F around inf

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

      1. lower-/.f64 50.8

         \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{B} \]

   10. Applied rewrites 50.8%

       \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{\color{blue}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 57.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -230:\\ \;\;\;\;-\frac{1 + x}{\sin B}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -230.0)
   (- (/ (+ 1.0 x) (sin B)))
   (if (<= F 1.15e+139)
     (/ (- (* (/ 1.0 (sqrt (fma F F (fma 2.0 x 2.0)))) F) x) B)
     (+ (/ (- (* 0.3333333333333333 (* (* B B) x)) x) B) (/ 1.0 B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -230.0) {
		tmp = -((1.0 + x) / sin(B));
	} else if (F <= 1.15e+139) {
		tmp = (((1.0 / sqrt(fma(F, F, fma(2.0, x, 2.0)))) * F) - x) / B;
	} else {
		tmp = (((0.3333333333333333 * ((B * B) * x)) - x) / B) + (1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -230.0)
		tmp = Float64(-Float64(Float64(1.0 + x) / sin(B)));
	elseif (F <= 1.15e+139)
		tmp = Float64(Float64(Float64(Float64(1.0 / sqrt(fma(F, F, fma(2.0, x, 2.0)))) * F) - x) / B);
	else
		tmp = Float64(Float64(Float64(Float64(0.3333333333333333 * Float64(Float64(B * B) * x)) - x) / B) + Float64(1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -230.0], (-N[(N[(1.0 + x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 1.15e+139], N[(N[(N[(N[(1.0 / N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 * N[(N[(B * B), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -230

   1. Initial program 60.0%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 77.3%

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

   if -230 < F < 1.15e139

   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. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 50.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. sqrt-div N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sqrt.f64 N/A

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

      10. pow2 N/A

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

      11. count-2-rev N/A

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

      12. associate-+l+ N/A

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

      13. pow2 N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-fma.f64 50.3

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

   6. Applied rewrites 50.3%

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

   if 1.15e139 < F

   1. Initial program 34.6%

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

   2. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. count-2-rev N/A

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

      11. lower-+.f64 32.0

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

   4. Applied rewrites 32.0%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 9.2

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

   7. Applied rewrites 9.2%

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

   8. Taylor expanded in F around inf

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

      1. lower-/.f64 51.6

         \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{B} \]

   10. Applied rewrites 51.6%

       \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{\color{blue}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 51.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -350000000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -350000000000.0)
   (/ -1.0 (sin B))
   (if (<= F 1.15e+139)
     (/ (- (* (/ 1.0 (sqrt (fma F F (fma 2.0 x 2.0)))) F) x) B)
     (+ (/ (- (* 0.3333333333333333 (* (* B B) x)) x) B) (/ 1.0 B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -350000000000.0) {
		tmp = -1.0 / sin(B);
	} else if (F <= 1.15e+139) {
		tmp = (((1.0 / sqrt(fma(F, F, fma(2.0, x, 2.0)))) * F) - x) / B;
	} else {
		tmp = (((0.3333333333333333 * ((B * B) * x)) - x) / B) + (1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -350000000000.0)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 1.15e+139)
		tmp = Float64(Float64(Float64(Float64(1.0 / sqrt(fma(F, F, fma(2.0, x, 2.0)))) * F) - x) / B);
	else
		tmp = Float64(Float64(Float64(Float64(0.3333333333333333 * Float64(Float64(B * B) * x)) - x) / B) + Float64(1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -350000000000.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.15e+139], N[(N[(N[(N[(1.0 / N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 * N[(N[(B * B), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.5e11

   1. Initial program 59.1%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 52.7

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

   7. Applied rewrites 52.7%

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

   if -3.5e11 < F < 1.15e139

   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. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 50.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\left(F \cdot F + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      6. sqrt-div N/A

         \[\leadsto \frac{\frac{\sqrt{1}}{\sqrt{\left(F \cdot F + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\left(F \cdot F + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\left(F \cdot F + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\left(F \cdot F + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      10. pow2 N/A

          \[\leadsto \frac{\frac{1}{\sqrt{\left({F}^{2} + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      11. count-2-rev N/A

          \[\leadsto \frac{\frac{1}{\sqrt{\left({F}^{2} + 2 \cdot x\right) + 2}} \cdot F - x}{B} \]

      12. associate-+l+ N/A

          \[\leadsto \frac{\frac{1}{\sqrt{{F}^{2} + \left(2 \cdot x + 2\right)}} \cdot F - x}{B} \]

      13. pow2 N/A

          \[\leadsto \frac{\frac{1}{\sqrt{F \cdot F + \left(2 \cdot x + 2\right)}} \cdot F - x}{B} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2 \cdot x + 2\right)}} \cdot F - x}{B} \]

      15. lower-fma.f64 50.2

          \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B} \]

   6. Applied rewrites 50.2%

      \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B} \]

   if 1.15e139 < F

   1. Initial program 34.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left(2 \cdot x + {F}^{2}\right) + 2}} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left(2 \cdot x + {F}^{2}\right) + 2}} \]

      7. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left({F}^{2} + 2 \cdot x\right) + 2}} \]

      8. pow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left(F \cdot F + 2 \cdot x\right) + 2}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2 \cdot x\right) + 2}} \]

      10. count-2-rev N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      11. lower-+.f64 32.0

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   4. Applied rewrites 32.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{\color{blue}{B}} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      6. lower-*.f64 9.2

         \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   7. Applied rewrites 9.2%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B}} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   8. Taylor expanded in F around inf

      \[\leadsto \frac{\frac{1}{3} \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{\color{blue}{B}} \]
   9. Step-by-step derivation

      1. lower-/.f64 51.6

         \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{B} \]

   10. Applied rewrites 51.6%

       \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{\color{blue}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 50.4% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.12 \cdot 10^{+113}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(B \cdot \left(B \cdot x\right)\right) - x}{B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.12e+113)
   (+ (/ (- (* 0.3333333333333333 (* B (* B x))) x) B) (/ -1.0 B))
   (if (<= F 1.15e+139)
     (/ (- (* (/ 1.0 (sqrt (fma F F (fma 2.0 x 2.0)))) F) x) B)
     (+ (/ (- (* 0.3333333333333333 (* (* B B) x)) x) B) (/ 1.0 B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.12e+113) {
		tmp = (((0.3333333333333333 * (B * (B * x))) - x) / B) + (-1.0 / B);
	} else if (F <= 1.15e+139) {
		tmp = (((1.0 / sqrt(fma(F, F, fma(2.0, x, 2.0)))) * F) - x) / B;
	} else {
		tmp = (((0.3333333333333333 * ((B * B) * x)) - x) / B) + (1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.12e+113)
		tmp = Float64(Float64(Float64(Float64(0.3333333333333333 * Float64(B * Float64(B * x))) - x) / B) + Float64(-1.0 / B));
	elseif (F <= 1.15e+139)
		tmp = Float64(Float64(Float64(Float64(1.0 / sqrt(fma(F, F, fma(2.0, x, 2.0)))) * F) - x) / B);
	else
		tmp = Float64(Float64(Float64(Float64(0.3333333333333333 * Float64(Float64(B * B) * x)) - x) / B) + Float64(1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.12e+113], N[(N[(N[(N[(0.3333333333333333 * N[(B * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.15e+139], N[(N[(N[(N[(1.0 / N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 * N[(N[(B * B), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.1200000000000001e113

   1. Initial program 43.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left(2 \cdot x + {F}^{2}\right) + 2}} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left(2 \cdot x + {F}^{2}\right) + 2}} \]

      7. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left({F}^{2} + 2 \cdot x\right) + 2}} \]

      8. pow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left(F \cdot F + 2 \cdot x\right) + 2}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2 \cdot x\right) + 2}} \]

      10. count-2-rev N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      11. lower-+.f64 38.0

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   4. Applied rewrites 38.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{\color{blue}{B}} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      6. lower-*.f64 13.4

         \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   7. Applied rewrites 13.4%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B}} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   8. Taylor expanded in F around -inf

      \[\leadsto \frac{\frac{1}{3} \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{-1}{\color{blue}{B}} \]
   9. Step-by-step derivation

      1. lower-/.f64 50.1

         \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{-1}{B} \]

   10. Applied rewrites 50.1%

       \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{-1}{\color{blue}{B}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{-1}{B} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{-1}{B} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \left(B \cdot \left(B \cdot x\right)\right) - x}{B} + \frac{-1}{B} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \left(B \cdot \left(B \cdot x\right)\right) - x}{B} + \frac{-1}{B} \]

       5. lift-*.f64 50.3

          \[\leadsto \frac{0.3333333333333333 \cdot \left(B \cdot \left(B \cdot x\right)\right) - x}{B} + \frac{-1}{B} \]

   12. Applied rewrites 50.3%

       \[\leadsto \frac{0.3333333333333333 \cdot \left(B \cdot \left(B \cdot x\right)\right) - x}{B} + \frac{-1}{B} \]

   if -1.1200000000000001e113 < F < 1.15e139

   1. Initial program 96.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 50.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]
   5. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B} \]

      4. lift-+.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\left(F \cdot F + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      6. sqrt-div N/A

         \[\leadsto \frac{\frac{\sqrt{1}}{\sqrt{\left(F \cdot F + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\left(F \cdot F + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\left(F \cdot F + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      9. lower-sqrt.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{\left(F \cdot F + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      10. pow2 N/A

          \[\leadsto \frac{\frac{1}{\sqrt{\left({F}^{2} + \left(x + x\right)\right) + 2}} \cdot F - x}{B} \]

      11. count-2-rev N/A

          \[\leadsto \frac{\frac{1}{\sqrt{\left({F}^{2} + 2 \cdot x\right) + 2}} \cdot F - x}{B} \]

      12. associate-+l+ N/A

          \[\leadsto \frac{\frac{1}{\sqrt{{F}^{2} + \left(2 \cdot x + 2\right)}} \cdot F - x}{B} \]

      13. pow2 N/A

          \[\leadsto \frac{\frac{1}{\sqrt{F \cdot F + \left(2 \cdot x + 2\right)}} \cdot F - x}{B} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2 \cdot x + 2\right)}} \cdot F - x}{B} \]

      15. lower-fma.f64 50.2

          \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B} \]

   6. Applied rewrites 50.2%

      \[\leadsto \frac{\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B} \]

   if 1.15e139 < F

   1. Initial program 34.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left(2 \cdot x + {F}^{2}\right) + 2}} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left(2 \cdot x + {F}^{2}\right) + 2}} \]

      7. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left({F}^{2} + 2 \cdot x\right) + 2}} \]

      8. pow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left(F \cdot F + 2 \cdot x\right) + 2}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2 \cdot x\right) + 2}} \]

      10. count-2-rev N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      11. lower-+.f64 32.0

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   4. Applied rewrites 32.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{\color{blue}{B}} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      6. lower-*.f64 9.2

         \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   7. Applied rewrites 9.2%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B}} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   8. Taylor expanded in F around inf

      \[\leadsto \frac{\frac{1}{3} \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{\color{blue}{B}} \]
   9. Step-by-step derivation

      1. lower-/.f64 51.6

         \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{B} \]

   10. Applied rewrites 51.6%

       \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{\color{blue}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 50.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.05:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(B \cdot \left(B \cdot x\right)\right) - x}{B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 1850000000000:\\ \;\;\;\;\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, \frac{F}{B}, \frac{-x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.05)
   (+ (/ (- (* 0.3333333333333333 (* B (* B x))) x) B) (/ -1.0 B))
   (if (<= F 1850000000000.0)
     (fma (sqrt (/ 1.0 (fma 2.0 x 2.0))) (/ F B) (/ (- x) B))
     (+ (/ (- (* 0.3333333333333333 (* (* B B) x)) x) B) (/ 1.0 B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.05) {
		tmp = (((0.3333333333333333 * (B * (B * x))) - x) / B) + (-1.0 / B);
	} else if (F <= 1850000000000.0) {
		tmp = fma(sqrt((1.0 / fma(2.0, x, 2.0))), (F / B), (-x / B));
	} else {
		tmp = (((0.3333333333333333 * ((B * B) * x)) - x) / B) + (1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.05)
		tmp = Float64(Float64(Float64(Float64(0.3333333333333333 * Float64(B * Float64(B * x))) - x) / B) + Float64(-1.0 / B));
	elseif (F <= 1850000000000.0)
		tmp = fma(sqrt(Float64(1.0 / fma(2.0, x, 2.0))), Float64(F / B), Float64(Float64(-x) / B));
	else
		tmp = Float64(Float64(Float64(Float64(0.3333333333333333 * Float64(Float64(B * B) * x)) - x) / B) + Float64(1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.05], N[(N[(N[(N[(0.3333333333333333 * N[(B * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1850000000000.0], N[(N[Sqrt[N[(1.0 / N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision] + N[((-x) / B), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 * N[(N[(B * B), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.05000000000000004

   1. Initial program 60.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left(2 \cdot x + {F}^{2}\right) + 2}} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left(2 \cdot x + {F}^{2}\right) + 2}} \]

      7. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left({F}^{2} + 2 \cdot x\right) + 2}} \]

      8. pow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left(F \cdot F + 2 \cdot x\right) + 2}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2 \cdot x\right) + 2}} \]

      10. count-2-rev N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      11. lower-+.f64 46.8

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   4. Applied rewrites 46.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{\color{blue}{B}} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      6. lower-*.f64 23.3

         \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   7. Applied rewrites 23.3%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B}} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   8. Taylor expanded in F around -inf

      \[\leadsto \frac{\frac{1}{3} \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{-1}{\color{blue}{B}} \]
   9. Step-by-step derivation

      1. lower-/.f64 49.7

         \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{-1}{B} \]

   10. Applied rewrites 49.7%

       \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{-1}{\color{blue}{B}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{-1}{B} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{-1}{B} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \left(B \cdot \left(B \cdot x\right)\right) - x}{B} + \frac{-1}{B} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \left(B \cdot \left(B \cdot x\right)\right) - x}{B} + \frac{-1}{B} \]

       5. lift-*.f64 49.9

          \[\leadsto \frac{0.3333333333333333 \cdot \left(B \cdot \left(B \cdot x\right)\right) - x}{B} + \frac{-1}{B} \]

   12. Applied rewrites 49.9%

       \[\leadsto \frac{0.3333333333333333 \cdot \left(B \cdot \left(B \cdot x\right)\right) - x}{B} + \frac{-1}{B} \]

   if -1.05000000000000004 < F < 1.85e12

   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. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 20.4%

      \[\leadsto \frac{1 - x}{B} \]

   8. Taylor expanded in F around 0

      \[\leadsto -1 \cdot \frac{x}{B} + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}}} \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{F}{B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} + -1 \cdot \color{blue}{\frac{x}{B}} \]

      2. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + 2 \cdot x}} \cdot \frac{F}{B} + -1 \cdot \frac{\color{blue}{x}}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{2 + 2 \cdot x}}, \frac{F}{\color{blue}{B}}, -1 \cdot \frac{x}{B}\right) \]

      4. lower-sqrt.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{2 + 2 \cdot x}}, \frac{F}{B}, -1 \cdot \frac{x}{B}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{2 + 2 \cdot x}}, \frac{F}{B}, -1 \cdot \frac{x}{B}\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{2 \cdot x + 2}}, \frac{F}{B}, -1 \cdot \frac{x}{B}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, \frac{F}{B}, -1 \cdot \frac{x}{B}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, \frac{F}{B}, -1 \cdot \frac{x}{B}\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, \frac{F}{B}, \frac{-1 \cdot x}{B}\right) \]

      10. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, \frac{F}{B}, \frac{\mathsf{neg}\left(x\right)}{B}\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, \frac{F}{B}, \frac{\mathsf{neg}\left(x\right)}{B}\right) \]

      12. lower-neg.f64 49.7

          \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, \frac{F}{B}, \frac{-x}{B}\right) \]

   10. Applied rewrites 49.7%

       \[\leadsto \mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, \color{blue}{\frac{F}{B}}, \frac{-x}{B}\right) \]

   if 1.85e12 < F

   1. Initial program 57.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left(2 \cdot x + {F}^{2}\right) + 2}} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left(2 \cdot x + {F}^{2}\right) + 2}} \]

      7. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left({F}^{2} + 2 \cdot x\right) + 2}} \]

      8. pow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left(F \cdot F + 2 \cdot x\right) + 2}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2 \cdot x\right) + 2}} \]

      10. count-2-rev N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      11. lower-+.f64 44.0

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   4. Applied rewrites 44.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{\color{blue}{B}} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      6. lower-*.f64 21.6

         \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   7. Applied rewrites 21.6%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B}} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   8. Taylor expanded in F around inf

      \[\leadsto \frac{\frac{1}{3} \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{\color{blue}{B}} \]
   9. Step-by-step derivation

      1. lower-/.f64 50.6

         \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{B} \]

   10. Applied rewrites 50.6%

       \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{\color{blue}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 50.0% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.05:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(B \cdot \left(B \cdot x\right)\right) - x}{B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 1850000000000:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.05)
   (+ (/ (- (* 0.3333333333333333 (* B (* B x))) x) B) (/ -1.0 B))
   (if (<= F 1850000000000.0)
     (/ (- (* (sqrt (/ 1.0 (fma 2.0 x 2.0))) F) x) B)
     (+ (/ (- (* 0.3333333333333333 (* (* B B) x)) x) B) (/ 1.0 B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.05) {
		tmp = (((0.3333333333333333 * (B * (B * x))) - x) / B) + (-1.0 / B);
	} else if (F <= 1850000000000.0) {
		tmp = ((sqrt((1.0 / fma(2.0, x, 2.0))) * F) - x) / B;
	} else {
		tmp = (((0.3333333333333333 * ((B * B) * x)) - x) / B) + (1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.05)
		tmp = Float64(Float64(Float64(Float64(0.3333333333333333 * Float64(B * Float64(B * x))) - x) / B) + Float64(-1.0 / B));
	elseif (F <= 1850000000000.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, 2.0))) * F) - x) / B);
	else
		tmp = Float64(Float64(Float64(Float64(0.3333333333333333 * Float64(Float64(B * B) * x)) - x) / B) + Float64(1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.05], N[(N[(N[(N[(0.3333333333333333 * N[(B * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1850000000000.0], 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[(0.3333333333333333 * N[(N[(B * B), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.05000000000000004

   1. Initial program 60.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left(2 \cdot x + {F}^{2}\right) + 2}} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left(2 \cdot x + {F}^{2}\right) + 2}} \]

      7. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left({F}^{2} + 2 \cdot x\right) + 2}} \]

      8. pow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left(F \cdot F + 2 \cdot x\right) + 2}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2 \cdot x\right) + 2}} \]

      10. count-2-rev N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      11. lower-+.f64 46.8

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   4. Applied rewrites 46.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{\color{blue}{B}} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      6. lower-*.f64 23.3

         \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   7. Applied rewrites 23.3%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B}} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   8. Taylor expanded in F around -inf

      \[\leadsto \frac{\frac{1}{3} \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{-1}{\color{blue}{B}} \]
   9. Step-by-step derivation

      1. lower-/.f64 49.7

         \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{-1}{B} \]

   10. Applied rewrites 49.7%

       \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{-1}{\color{blue}{B}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{-1}{B} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{-1}{B} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \left(B \cdot \left(B \cdot x\right)\right) - x}{B} + \frac{-1}{B} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \left(B \cdot \left(B \cdot x\right)\right) - x}{B} + \frac{-1}{B} \]

       5. lift-*.f64 49.9

          \[\leadsto \frac{0.3333333333333333 \cdot \left(B \cdot \left(B \cdot x\right)\right) - x}{B} + \frac{-1}{B} \]

   12. Applied rewrites 49.9%

       \[\leadsto \frac{0.3333333333333333 \cdot \left(B \cdot \left(B \cdot x\right)\right) - x}{B} + \frac{-1}{B} \]

   if -1.05000000000000004 < F < 1.85e12

   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. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 20.4%

      \[\leadsto \frac{1 - x}{B} \]

   8. Taylor expanded in F around 0

      \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} - x}{B} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot F - x}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot F - x}{B} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot F - x}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot F - x}{B} \]

      5. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 \cdot x + 2}} \cdot F - x}{B} \]

      6. lower-fma.f64 49.7

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B} \]

   10. Applied rewrites 49.7%

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B} \]

   if 1.85e12 < F

   1. Initial program 57.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left(2 \cdot x + {F}^{2}\right) + 2}} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left(2 \cdot x + {F}^{2}\right) + 2}} \]

      7. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left({F}^{2} + 2 \cdot x\right) + 2}} \]

      8. pow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left(F \cdot F + 2 \cdot x\right) + 2}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2 \cdot x\right) + 2}} \]

      10. count-2-rev N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      11. lower-+.f64 44.0

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   4. Applied rewrites 44.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{\color{blue}{B}} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      6. lower-*.f64 21.6

         \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   7. Applied rewrites 21.6%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B}} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   8. Taylor expanded in F around inf

      \[\leadsto \frac{\frac{1}{3} \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{\color{blue}{B}} \]
   9. Step-by-step derivation

      1. lower-/.f64 50.6

         \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{B} \]

   10. Applied rewrites 50.6%

       \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{1}{\color{blue}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 50.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.05:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(B \cdot \left(B \cdot x\right)\right) - x}{B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.46 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.05)
   (+ (/ (- (* 0.3333333333333333 (* B (* B x))) x) B) (/ -1.0 B))
   (if (<= F 1.46e-12)
     (/ (- (* (sqrt (/ 1.0 (fma 2.0 x 2.0))) F) x) B)
     (- (/ 1.0 B) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.05) {
		tmp = (((0.3333333333333333 * (B * (B * x))) - x) / B) + (-1.0 / B);
	} else if (F <= 1.46e-12) {
		tmp = ((sqrt((1.0 / fma(2.0, x, 2.0))) * F) - x) / B;
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.05)
		tmp = Float64(Float64(Float64(Float64(0.3333333333333333 * Float64(B * Float64(B * x))) - x) / B) + Float64(-1.0 / B));
	elseif (F <= 1.46e-12)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, 2.0))) * F) - x) / B);
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.05], N[(N[(N[(N[(0.3333333333333333 * N[(B * N[(B * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.46e-12], 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 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.05000000000000004

   1. Initial program 60.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left(2 \cdot x + {F}^{2}\right) + 2}} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left(2 \cdot x + {F}^{2}\right) + 2}} \]

      7. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left({F}^{2} + 2 \cdot x\right) + 2}} \]

      8. pow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\left(F \cdot F + 2 \cdot x\right) + 2}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2 \cdot x\right) + 2}} \]

      10. count-2-rev N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      11. lower-+.f64 46.8

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   4. Applied rewrites 46.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{\color{blue}{B}} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

      6. lower-*.f64 23.3

         \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   7. Applied rewrites 23.3%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B}} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \]

   8. Taylor expanded in F around -inf

      \[\leadsto \frac{\frac{1}{3} \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{-1}{\color{blue}{B}} \]
   9. Step-by-step derivation

      1. lower-/.f64 49.7

         \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{-1}{B} \]

   10. Applied rewrites 49.7%

       \[\leadsto \frac{0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{-1}{\color{blue}{B}} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{-1}{B} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \left(\left(B \cdot B\right) \cdot x\right) - x}{B} + \frac{-1}{B} \]

       3. associate-*l* N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \left(B \cdot \left(B \cdot x\right)\right) - x}{B} + \frac{-1}{B} \]

       4. lower-*.f64 N/A

          \[\leadsto \frac{\frac{1}{3} \cdot \left(B \cdot \left(B \cdot x\right)\right) - x}{B} + \frac{-1}{B} \]

       5. lift-*.f64 49.9

          \[\leadsto \frac{0.3333333333333333 \cdot \left(B \cdot \left(B \cdot x\right)\right) - x}{B} + \frac{-1}{B} \]

   12. Applied rewrites 49.9%

       \[\leadsto \frac{0.3333333333333333 \cdot \left(B \cdot \left(B \cdot x\right)\right) - x}{B} + \frac{-1}{B} \]

   if -1.05000000000000004 < F < 1.46000000000000004e-12

   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. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 50.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 20.0%

      \[\leadsto \frac{1 - x}{B} \]

   8. Taylor expanded in F around 0

      \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} - x}{B} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot F - x}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot F - x}{B} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot F - x}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot F - x}{B} \]

      5. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 \cdot x + 2}} \cdot F - x}{B} \]

      6. lower-fma.f64 50.5

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B} \]

   10. Applied rewrites 50.5%

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B} \]

   if 1.46000000000000004e-12 < F

   1. Initial program 59.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 37.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.4%

      \[\leadsto \frac{1 - x}{B} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1 - x}{\color{blue}{B}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{1 - x}{B} \]

      3. div-sub N/A

         \[\leadsto \frac{1}{B} - \color{blue}{\frac{x}{B}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{1}{B} - \color{blue}{\frac{x}{B}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{1}{B} - \frac{\color{blue}{x}}{B} \]

      6. lower-/.f64 49.4

         \[\leadsto \frac{1}{B} - \frac{x}{\color{blue}{B}} \]

   9. Applied rewrites 49.4%

      \[\leadsto \frac{1}{B} - \color{blue}{\frac{x}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 49.9% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -150000000000:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.46 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -150000000000.0)
   (/ (- -1.0 x) B)
   (if (<= F 1.46e-12)
     (/ (- (* (sqrt (/ 1.0 (fma 2.0 x 2.0))) F) x) B)
     (- (/ 1.0 B) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -150000000000.0) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.46e-12) {
		tmp = ((sqrt((1.0 / fma(2.0, x, 2.0))) * F) - x) / B;
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -150000000000.0)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.46e-12)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, 2.0))) * F) - x) / B);
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -150000000000.0], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.46e-12], 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 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5e11

   1. Initial program 59.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 36.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.2%

      \[\leadsto \frac{-1 - x}{B} \]

   if -1.5e11 < F < 1.46000000000000004e-12

   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. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 50.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 20.0%

      \[\leadsto \frac{1 - x}{B} \]

   8. Taylor expanded in F around 0

      \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + 2 \cdot x}} - x}{B} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot F - x}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot F - x}{B} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot F - x}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot F - x}{B} \]

      5. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 \cdot x + 2}} \cdot F - x}{B} \]

      6. lower-fma.f64 50.0

         \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B} \]

   10. Applied rewrites 50.0%

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B} \]

   if 1.46000000000000004e-12 < F

   1. Initial program 59.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 37.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.4%

      \[\leadsto \frac{1 - x}{B} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1 - x}{\color{blue}{B}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{1 - x}{B} \]

      3. div-sub N/A

         \[\leadsto \frac{1}{B} - \color{blue}{\frac{x}{B}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{1}{B} - \color{blue}{\frac{x}{B}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{1}{B} - \frac{\color{blue}{x}}{B} \]

      6. lower-/.f64 49.4

         \[\leadsto \frac{1}{B} - \frac{x}{\color{blue}{B}} \]

   9. Applied rewrites 49.4%

      \[\leadsto \frac{1}{B} - \color{blue}{\frac{x}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 43.3% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-15}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.5e-29)
   (/ (- -1.0 x) B)
   (if (<= F 3.4e-15) (/ (- x) B) (- (/ 1.0 B) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.5e-29) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.4e-15) {
		tmp = -x / B;
	} else {
		tmp = (1.0 / B) - (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 <= (-2.5d-29)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 3.4d-15) then
        tmp = -x / b
    else
        tmp = (1.0d0 / b) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.5e-29) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.4e-15) {
		tmp = -x / B;
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.5e-29:
		tmp = (-1.0 - x) / B
	elif F <= 3.4e-15:
		tmp = -x / B
	else:
		tmp = (1.0 / B) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.5e-29)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 3.4e-15)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.5e-29)
		tmp = (-1.0 - x) / B;
	elseif (F <= 3.4e-15)
		tmp = -x / B;
	else
		tmp = (1.0 / B) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.5e-29], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3.4e-15], N[((-x) / B), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.49999999999999993e-29

   1. Initial program 63.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 37.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 47.9%

      \[\leadsto \frac{-1 - x}{B} \]

   if -2.49999999999999993e-29 < F < 3.4e-15

   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. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

      2. lower-neg.f64 35.8

         \[\leadsto \frac{-x}{B} \]

   7. Applied rewrites 35.8%

      \[\leadsto \frac{-x}{B} \]

   if 3.4e-15 < F

   1. Initial program 60.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 37.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.2%

      \[\leadsto \frac{1 - x}{B} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{1 - x}{\color{blue}{B}} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{1 - x}{B} \]

      3. div-sub N/A

         \[\leadsto \frac{1}{B} - \color{blue}{\frac{x}{B}} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{1}{B} - \color{blue}{\frac{x}{B}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{1}{B} - \frac{\color{blue}{x}}{B} \]

      6. lower-/.f64 49.2

         \[\leadsto \frac{1}{B} - \frac{x}{\color{blue}{B}} \]

   9. Applied rewrites 49.2%

      \[\leadsto \frac{1}{B} - \color{blue}{\frac{x}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 43.3% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-15}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.5e-29)
   (/ (- -1.0 x) B)
   (if (<= F 3.4e-15) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.5e-29) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.4e-15) {
		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 <= (-2.5d-29)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 3.4d-15) 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 <= -2.5e-29) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.4e-15) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.5e-29:
		tmp = (-1.0 - x) / B
	elif F <= 3.4e-15:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.5e-29)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 3.4e-15)
		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 <= -2.5e-29)
		tmp = (-1.0 - x) / B;
	elseif (F <= 3.4e-15)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.5e-29], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3.4e-15], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.49999999999999993e-29

   1. Initial program 63.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 37.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 47.9%

      \[\leadsto \frac{-1 - x}{B} \]

   if -2.49999999999999993e-29 < F < 3.4e-15

   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. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

      2. lower-neg.f64 35.8

         \[\leadsto \frac{-x}{B} \]

   7. Applied rewrites 35.8%

      \[\leadsto \frac{-x}{B} \]

   if 3.4e-15 < F

   1. Initial program 60.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 37.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 49.2%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 25: 36.1% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.5e-29) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.5e-29) {
		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 <= (-2.5d-29)) 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 <= -2.5e-29) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.5e-29:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.5e-29)
		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 <= -2.5e-29)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.5e-29], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -2.49999999999999993e-29

   1. Initial program 63.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 37.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 47.9%

      \[\leadsto \frac{-1 - x}{B} \]

   if -2.49999999999999993e-29 < F

   1. Initial program 83.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 45.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

      2. lower-neg.f64 30.9

         \[\leadsto \frac{-x}{B} \]

   7. Applied rewrites 30.9%

      \[\leadsto \frac{-x}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 28.9% accurate, 21.7× speedup

Math

\[\begin{array}{l} \\ \frac{-x}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ (- x) B))

C

double code(double F, double B, double x) {
	return -x / B;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -x / b
end function

Java

public static double code(double F, double B, double x) {
	return -x / B;
}

Python

def code(F, B, x):
	return -x / B

Julia

function code(F, B, x)
	return Float64(Float64(-x) / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -x / B;
end

Wolfram

code[F_, B_, x_] := N[((-x) / B), $MachinePrecision]

Derivation

1. Initial program 77.0%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

4. Applied rewrites 43.0%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, x + x\right) + 2}} \cdot F - x}{B}} \]

5. Taylor expanded in F around 0

   \[\leadsto \frac{-1 \cdot x}{B} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

   2. lower-neg.f64 28.9

      \[\leadsto \frac{-x}{B} \]

7. Applied rewrites 28.9%

   \[\leadsto \frac{-x}{B} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025115 
(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))))))