VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.1% → 99.5%

Time: 6.9s

Alternatives: 22

Speedup: 2.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{+30}:\\ \;\;\;\;-\frac{1 + \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+40}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot 1}{\sin B \cdot F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3e+30)
   (- (/ (+ 1.0 (* (cos B) x)) (sin B)))
   (if (<= F 6e+40)
     (+
      (- (/ (* 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)))) (/ (* F 1.0) (* (sin B) F))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3e+30) {
		tmp = -((1.0 + (cos(B) * x)) / sin(B));
	} else if (F <= 6e+40) {
		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))) + ((F * 1.0) / (sin(B) * F));
	}
	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 <= (-3d+30)) then
        tmp = -((1.0d0 + (cos(b) * x)) / sin(b))
    else if (f <= 6d+40) 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))) + ((f * 1.0d0) / (sin(b) * f))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3e+30) {
		tmp = -((1.0 + (Math.cos(B) * x)) / Math.sin(B));
	} else if (F <= 6e+40) {
		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))) + ((F * 1.0) / (Math.sin(B) * F));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3e+30:
		tmp = -((1.0 + (math.cos(B) * x)) / math.sin(B))
	elif F <= 6e+40:
		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))) + ((F * 1.0) / (math.sin(B) * F))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3e+30)
		tmp = Float64(-Float64(Float64(1.0 + Float64(cos(B) * x)) / sin(B)));
	elseif (F <= 6e+40)
		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(Float64(F * 1.0) / Float64(sin(B) * F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3e+30)
		tmp = -((1.0 + (cos(B) * x)) / sin(B));
	elseif (F <= 6e+40)
		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))) + ((F * 1.0) / (sin(B) * F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3e+30], (-N[(N[(1.0 + N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 6e+40], 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[(N[(F * 1.0), $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.99999999999999978e30

   1. Initial program 54.5%

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

   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 99.8

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

   4. Applied rewrites 99.8%

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

   if -2.99999999999999978e30 < F < 6.0000000000000004e40

   1. Initial program 99.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 99.3

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

   3. Applied rewrites 99.3%

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

   if 6.0000000000000004e40 < F

   1. Initial program 53.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 70.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-flip N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-*l/ N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. metadata-eval N/A

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

      17. unpow1/2 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lift-fma.f64 N/A

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

   5. Applied rewrites 70.9%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 99.5%

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

Alternative 2: 98.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -5 \cdot 10^{+32}:\\ \;\;\;\;-\frac{1 + t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-22}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -5e+32)
     (- (/ (+ 1.0 t_0) (sin B)))
     (if (<= F 4.2e-22)
       (+
        (- (* x (/ 1.0 (tan B))))
        (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 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 <= -5e+32) {
		tmp = -((1.0 + t_0) / sin(B));
	} else if (F <= 4.2e-22) {
		tmp = -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = (1.0 - t_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 = cos(b) * x
    if (f <= (-5d+32)) then
        tmp = -((1.0d0 + t_0) / sin(b))
    else if (f <= 4.2d-22) then
        tmp = -(x * (1.0d0 / tan(b))) + ((f / sin(b)) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
    else
        tmp = (1.0d0 - t_0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.cos(B) * x;
	double tmp;
	if (F <= -5e+32) {
		tmp = -((1.0 + t_0) / Math.sin(B));
	} else if (F <= 4.2e-22) {
		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 = (1.0 - t_0) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.cos(B) * x
	tmp = 0
	if F <= -5e+32:
		tmp = -((1.0 + t_0) / math.sin(B))
	elif F <= 4.2e-22:
		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 = (1.0 - t_0) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -5e+32)
		tmp = Float64(-Float64(Float64(1.0 + t_0) / sin(B)));
	elseif (F <= 4.2e-22)
		tmp = 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)))));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = cos(B) * x;
	tmp = 0.0;
	if (F <= -5e+32)
		tmp = -((1.0 + t_0) / sin(B));
	elseif (F <= 4.2e-22)
		tmp = -(x * (1.0 / tan(B))) + ((F / sin(B)) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
	else
		tmp = (1.0 - t_0) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -5e+32], (-N[(N[(1.0 + t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 4.2e-22], 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], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.9999999999999997e32

   1. Initial program 54.2%

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

   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 99.8

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

   4. Applied rewrites 99.8%

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

   if -4.9999999999999997e32 < F < 4.20000000000000016e-22

   1. Initial program 99.4%

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

   if 4.20000000000000016e-22 < F

   1. Initial program 60.5%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   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 95.4

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

   4. Applied rewrites 95.4%

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

Alternative 3: 98.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -5.2 \cdot 10^{+17}:\\ \;\;\;\;-\frac{1 + t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-22}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{\frac{F}{\sin B} \cdot 1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -5.2e+17)
     (- (/ (+ 1.0 t_0) (sin B)))
     (if (<= F 4.2e-22)
       (+
        (- (* x (/ 1.0 (tan B))))
        (/ (* (/ F (sin B)) 1.0) (sqrt (fma 2.0 x (fma F F 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 <= -5.2e+17) {
		tmp = -((1.0 + t_0) / sin(B));
	} else if (F <= 4.2e-22) {
		tmp = -(x * (1.0 / tan(B))) + (((F / sin(B)) * 1.0) / sqrt(fma(2.0, x, fma(F, F, 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 <= -5.2e+17)
		tmp = Float64(-Float64(Float64(1.0 + t_0) / sin(B)));
	elseif (F <= 4.2e-22)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(Float64(F / sin(B)) * 1.0) / sqrt(fma(2.0, x, fma(F, F, 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, -5.2e+17], (-N[(N[(1.0 + t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 4.2e-22], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $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 < -5.2e17

   1. Initial program 55.8%

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

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   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 99.8

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

   4. Applied rewrites 99.8%

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

   if -5.2e17 < F < 4.20000000000000016e-22

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-flip N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-*l/ N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. unpow1/2 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lift-fma.f64 N/A

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

   5. Applied rewrites 99.4%

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

   if 4.20000000000000016e-22 < F

   1. Initial program 60.5%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   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 95.4

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

   4. Applied rewrites 95.4%

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

Alternative 4: 98.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -5.2 \cdot 10^{+17}:\\ \;\;\;\;-\frac{1 + t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-22}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot 1}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -5.2e+17)
     (- (/ (+ 1.0 t_0) (sin B)))
     (if (<= F 4.2e-22)
       (+
        (- (* x (/ 1.0 (tan B))))
        (/ (* F 1.0) (* (sin B) (sqrt (fma 2.0 x (fma F F 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 <= -5.2e+17) {
		tmp = -((1.0 + t_0) / sin(B));
	} else if (F <= 4.2e-22) {
		tmp = -(x * (1.0 / tan(B))) + ((F * 1.0) / (sin(B) * sqrt(fma(2.0, x, fma(F, F, 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 <= -5.2e+17)
		tmp = Float64(-Float64(Float64(1.0 + t_0) / sin(B)));
	elseif (F <= 4.2e-22)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F * 1.0) / Float64(sin(B) * sqrt(fma(2.0, x, fma(F, F, 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, -5.2e+17], (-N[(N[(1.0 + t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 4.2e-22], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F * 1.0), $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 * x + N[(F * F + 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 < -5.2e17

   1. Initial program 55.8%

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

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   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 99.8

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

   4. Applied rewrites 99.8%

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

   if -5.2e17 < F < 4.20000000000000016e-22

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-flip N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-*l/ N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. metadata-eval N/A

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

      17. unpow1/2 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lift-fma.f64 N/A

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

   5. Applied rewrites 99.4%

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

   if 4.20000000000000016e-22 < F

   1. Initial program 60.5%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   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 95.4

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

   4. Applied rewrites 95.4%

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

Alternative 5: 98.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -1.5:\\ \;\;\;\;-\frac{1 + t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-22}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot 1}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -1.5)
     (- (/ (+ 1.0 t_0) (sin B)))
     (if (<= F 4.2e-22)
       (+
        (- (* x (/ 1.0 (tan B))))
        (/ (* F 1.0) (* (sin B) (sqrt (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 <= -1.5) {
		tmp = -((1.0 + t_0) / sin(B));
	} else if (F <= 4.2e-22) {
		tmp = -(x * (1.0 / tan(B))) + ((F * 1.0) / (sin(B) * sqrt(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 <= -1.5)
		tmp = Float64(-Float64(Float64(1.0 + t_0) / sin(B)));
	elseif (F <= 4.2e-22)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F * 1.0) / Float64(sin(B) * sqrt(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, -1.5], (-N[(N[(1.0 + t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 4.2e-22], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F * 1.0), $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 * x + 2.0), $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 < -1.5

   1. Initial program 58.0%

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

   2. 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 99.2

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

   4. Applied rewrites 99.2%

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

   if -1.5 < F < 4.20000000000000016e-22

   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

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-flip N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-*l/ N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. metadata-eval N/A

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

      17. unpow1/2 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lift-fma.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 99.1%

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

   if 4.20000000000000016e-22 < F

   1. Initial program 60.5%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   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 95.4

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

   4. Applied rewrites 95.4%

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

Alternative 6: 85.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\\ t_1 := -x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -1.25 \cdot 10^{+154}:\\ \;\;\;\;t\_1 + \frac{-1}{\mathsf{fma}\left(-0.16666666666666666, B \cdot B, 1\right) \cdot B}\\ \mathbf{elif}\;F \leq -2.9 \cdot 10^{-30}:\\ \;\;\;\;\frac{-x}{B} + \frac{F \cdot 1}{\sin B \cdot \sqrt{t\_0}}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-22}:\\ \;\;\;\;t\_1 + \frac{F}{B} \cdot \sqrt{\frac{1}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (fma 2.0 x (fma F F 2.0))) (t_1 (- (* x (/ 1.0 (tan B))))))
   (if (<= F -1.25e+154)
     (+ t_1 (/ -1.0 (* (fma -0.16666666666666666 (* B B) 1.0) B)))
     (if (<= F -2.9e-30)
       (+ (/ (- x) B) (/ (* F 1.0) (* (sin B) (sqrt t_0))))
       (if (<= F 4.2e-22)
         (+ t_1 (* (/ F B) (sqrt (/ 1.0 t_0))))
         (/ (- 1.0 (* (cos B) x)) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = fma(2.0, x, fma(F, F, 2.0));
	double t_1 = -(x * (1.0 / tan(B)));
	double tmp;
	if (F <= -1.25e+154) {
		tmp = t_1 + (-1.0 / (fma(-0.16666666666666666, (B * B), 1.0) * B));
	} else if (F <= -2.9e-30) {
		tmp = (-x / B) + ((F * 1.0) / (sin(B) * sqrt(t_0)));
	} else if (F <= 4.2e-22) {
		tmp = t_1 + ((F / B) * sqrt((1.0 / t_0)));
	} else {
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(2.0, x, fma(F, F, 2.0))
	t_1 = Float64(-Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (F <= -1.25e+154)
		tmp = Float64(t_1 + Float64(-1.0 / Float64(fma(-0.16666666666666666, Float64(B * B), 1.0) * B)));
	elseif (F <= -2.9e-30)
		tmp = Float64(Float64(Float64(-x) / B) + Float64(Float64(F * 1.0) / Float64(sin(B) * sqrt(t_0))));
	elseif (F <= 4.2e-22)
		tmp = Float64(t_1 + Float64(Float64(F / B) * sqrt(Float64(1.0 / t_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[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[F, -1.25e+154], N[(t$95$1 + N[(-1.0 / N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.9e-30], N[(N[((-x) / B), $MachinePrecision] + N[(N[(F * 1.0), $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.2e-22], N[(t$95$1 + N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / t$95$0), $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 < -1.25000000000000001e154

   1. Initial program 30.3%

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

   2. Taylor expanded in 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 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 75.6

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

   7. Applied rewrites 75.6%

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

   if -1.25000000000000001e154 < F < -2.89999999999999989e-30

   1. Initial program 91.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-flip N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-*l/ N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. metadata-eval N/A

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

      17. unpow1/2 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lift-fma.f64 N/A

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

   5. Applied rewrites 99.5%

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

   6. Taylor expanded in B around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-/.f64 78.9

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

   8. Applied rewrites 78.9%

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

   if -2.89999999999999989e-30 < F < 4.20000000000000016e-22

   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. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 85.2

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

   4. Applied rewrites 85.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. inv-pow N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. pow2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-fma.f64 85.2

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

   6. Applied rewrites 85.2%

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

   if 4.20000000000000016e-22 < F

   1. Initial program 60.5%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   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 95.4

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

   4. Applied rewrites 95.4%

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

Alternative 7: 91.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -7 \cdot 10^{-15}:\\ \;\;\;\;-\frac{1 + t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-22}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot 1}{B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -7e-15)
     (- (/ (+ 1.0 t_0) (sin B)))
     (if (<= F 4.2e-22)
       (+
        (- (* x (/ 1.0 (tan B))))
        (/ (* F 1.0) (* B (sqrt (fma 2.0 x (fma F F 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 <= -7e-15) {
		tmp = -((1.0 + t_0) / sin(B));
	} else if (F <= 4.2e-22) {
		tmp = -(x * (1.0 / tan(B))) + ((F * 1.0) / (B * sqrt(fma(2.0, x, fma(F, F, 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 <= -7e-15)
		tmp = Float64(-Float64(Float64(1.0 + t_0) / sin(B)));
	elseif (F <= 4.2e-22)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F * 1.0) / Float64(B * sqrt(fma(2.0, x, fma(F, F, 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, -7e-15], (-N[(N[(1.0 + t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 4.2e-22], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F * 1.0), $MachinePrecision] / N[(B * N[Sqrt[N[(2.0 * x + N[(F * F + 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 < -7.0000000000000001e-15

   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

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   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 96.7

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

   4. Applied rewrites 96.7%

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

   if -7.0000000000000001e-15 < F < 4.20000000000000016e-22

   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

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-flip N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-*l/ N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. metadata-eval N/A

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

      17. unpow1/2 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lift-fma.f64 N/A

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 84.9%

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

   if 4.20000000000000016e-22 < F

   1. Initial program 60.5%

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

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   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 95.4

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

   4. Applied rewrites 95.4%

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

Alternative 8: 78.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}\\ t_1 := \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot 1}{B \cdot t\_0}\\ \mathbf{if}\;x \leq -1.08 \cdot 10^{-48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{-x}{B} + \frac{F \cdot 1}{\sin B \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (fma 2.0 x (fma F F 2.0))))
        (t_1 (+ (- (* x (/ 1.0 (tan B)))) (/ (* F 1.0) (* B t_0)))))
   (if (<= x -1.08e-48)
     t_1
     (if (<= x 9.8e-27) (+ (/ (- x) B) (/ (* F 1.0) (* (sin B) t_0))) t_1))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt(fma(2.0, x, fma(F, F, 2.0)));
	double t_1 = -(x * (1.0 / tan(B))) + ((F * 1.0) / (B * t_0));
	double tmp;
	if (x <= -1.08e-48) {
		tmp = t_1;
	} else if (x <= 9.8e-27) {
		tmp = (-x / B) + ((F * 1.0) / (sin(B) * t_0));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = sqrt(fma(2.0, x, fma(F, F, 2.0)))
	t_1 = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F * 1.0) / Float64(B * t_0)))
	tmp = 0.0
	if (x <= -1.08e-48)
		tmp = t_1;
	elseif (x <= 9.8e-27)
		tmp = Float64(Float64(Float64(-x) / B) + Float64(Float64(F * 1.0) / Float64(sin(B) * t_0)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F * 1.0), $MachinePrecision] / N[(B * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.08e-48], t$95$1, If[LessEqual[x, 9.8e-27], N[(N[((-x) / B), $MachinePrecision] + N[(N[(F * 1.0), $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.08e-48 or 9.79999999999999952e-27 < x

   1. Initial program 80.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 95.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-flip N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-*l/ N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. metadata-eval N/A

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

      17. unpow1/2 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lift-fma.f64 N/A

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 93.3%

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

   if -1.08e-48 < x < 9.79999999999999952e-27

   1. Initial program 73.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 76.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-flip N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-*l/ N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. metadata-eval N/A

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

      17. unpow1/2 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lift-fma.f64 N/A

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in B around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-/.f64 65.2

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

   8. Applied rewrites 65.2%

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

Alternative 9: 77.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-30}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666, B \cdot B, 1\right) \cdot B}\\ \mathbf{elif}\;x \leq 0.00033:\\ \;\;\;\;\frac{-x}{B} + \frac{F \cdot 1}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -2e-30)
   (+
    (- (/ (* x 1.0) (tan B)))
    (/
     -1.0
     (*
      (fma
       (- (* 0.008333333333333333 (* B B)) 0.16666666666666666)
       (* B B)
       1.0)
      B)))
   (if (<= x 0.00033)
     (+ (/ (- x) B) (/ (* F 1.0) (* (sin B) (sqrt (fma 2.0 x (fma F F 2.0))))))
     (+ (- (* x (/ 1.0 (tan B)))) (/ -1.0 B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -2e-30) {
		tmp = -((x * 1.0) / tan(B)) + (-1.0 / (fma(((0.008333333333333333 * (B * B)) - 0.16666666666666666), (B * B), 1.0) * B));
	} else if (x <= 0.00033) {
		tmp = (-x / B) + ((F * 1.0) / (sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0)))));
	} else {
		tmp = -(x * (1.0 / tan(B))) + (-1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -2e-30)
		tmp = Float64(Float64(-Float64(Float64(x * 1.0) / tan(B))) + Float64(-1.0 / Float64(fma(Float64(Float64(0.008333333333333333 * Float64(B * B)) - 0.16666666666666666), Float64(B * B), 1.0) * B)));
	elseif (x <= 0.00033)
		tmp = Float64(Float64(Float64(-x) / B) + Float64(Float64(F * 1.0) / Float64(sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))));
	else
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(-1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -2e-30], N[((-N[(N[(x * 1.0), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / N[(N[(N[(N[(0.008333333333333333 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00033], N[(N[((-x) / B), $MachinePrecision] + N[(N[(F * 1.0), $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2e-30

   1. Initial program 69.1%

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

   2. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   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 82.7

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

   4. Applied rewrites 82.7%

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

   5. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \color{blue}{\left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right) \cdot B} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right) \cdot B} \]

      3. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\left({B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right) + 1\right) \cdot B} \]

      4. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\left(\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right) \cdot {B}^{2} + 1\right) \cdot B} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      6. lower--.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      10. unpow2 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}, B \cdot B, 1\right) \cdot B} \]

      11. lower-*.f64 84.7

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666, B \cdot B, 1\right) \cdot B} \]

   7. Applied rewrites 84.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666, B \cdot B, 1\right) \cdot \color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\color{blue}{x \cdot \frac{1}{\tan B}}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}, B \cdot B, 1\right) \cdot B} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{\tan B}}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}, B \cdot B, 1\right) \cdot B} \]

      3. lift-tan.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{\tan B}}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}, B \cdot B, 1\right) \cdot B} \]

      4. associate-*r/ N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}, B \cdot B, 1\right) \cdot B} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}, B \cdot B, 1\right) \cdot B} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{x \cdot 1}}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}, B \cdot B, 1\right) \cdot B} \]

      7. lift-tan.f64 84.9

         \[\leadsto \left(-\frac{x \cdot 1}{\color{blue}{\tan B}}\right) + \frac{-1}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666, B \cdot B, 1\right) \cdot B} \]

   9. Applied rewrites 84.9%

      \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{-1}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666, B \cdot B, 1\right) \cdot B} \]

   if -2e-30 < x < 3.3e-4

   1. Initial program 73.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 76.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-flip N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-*l/ N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. metadata-eval N/A

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

      17. unpow1/2 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lift-fma.f64 N/A

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in B around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-/.f64 64.6

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

   8. Applied rewrites 64.6%

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

   if 3.3e-4 < x

   1. Initial program 87.2%

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

   2. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   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 96.9

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

   4. Applied rewrites 96.9%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 97.3%

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

Alternative 10: 77.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-12}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{-1}{B}\\ \mathbf{elif}\;x \leq 0.00033:\\ \;\;\;\;\frac{-x}{B} + \frac{F \cdot 1}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -1.85e-12)
   (+ (- (/ (* x 1.0) (tan B))) (/ -1.0 B))
   (if (<= x 0.00033)
     (+ (/ (- x) B) (/ (* F 1.0) (* (sin B) (sqrt (fma 2.0 x (fma F F 2.0))))))
     (+ (- (* x (/ 1.0 (tan B)))) (/ -1.0 B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -1.85e-12) {
		tmp = -((x * 1.0) / tan(B)) + (-1.0 / B);
	} else if (x <= 0.00033) {
		tmp = (-x / B) + ((F * 1.0) / (sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0)))));
	} else {
		tmp = -(x * (1.0 / tan(B))) + (-1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -1.85e-12)
		tmp = Float64(Float64(-Float64(Float64(x * 1.0) / tan(B))) + Float64(-1.0 / B));
	elseif (x <= 0.00033)
		tmp = Float64(Float64(Float64(-x) / B) + Float64(Float64(F * 1.0) / Float64(sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))));
	else
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(-1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -1.85e-12], N[((-N[(N[(x * 1.0), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.00033], N[(N[((-x) / B), $MachinePrecision] + N[(N[(F * 1.0), $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.84999999999999999e-12

   1. Initial program 68.2%

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

   2. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   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 90.9

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

   4. Applied rewrites 90.9%

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

   5. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \color{blue}{\left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right) \cdot B} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right) \cdot B} \]

      3. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\left({B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right) + 1\right) \cdot B} \]

      4. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\left(\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right) \cdot {B}^{2} + 1\right) \cdot B} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      6. lower--.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      10. unpow2 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}, B \cdot B, 1\right) \cdot B} \]

      11. lower-*.f64 91.7

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666, B \cdot B, 1\right) \cdot B} \]

   7. Applied rewrites 91.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666, B \cdot B, 1\right) \cdot \color{blue}{B}} \]

   8. Taylor expanded in B around 0

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

   10. Applied rewrites 91.7%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-tan.f64 N/A

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

       4. associate-*r/ N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-tan.f64 91.9

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

   12. Applied rewrites 91.9%

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

   if -1.84999999999999999e-12 < x < 3.3e-4

   1. Initial program 73.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 76.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-flip N/A

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

      9. lift-sin.f64 N/A

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

      10. associate-*l/ N/A

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

      11. frac-times N/A

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

      12. lower-/.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lower-*.f64 N/A

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

      15. lift-sin.f64 N/A

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

      16. metadata-eval N/A

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

      17. unpow1/2 N/A

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

      18. lower-sqrt.f64 N/A

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

      19. lift-fma.f64 N/A

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in B around 0

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. lift-neg.f64 N/A

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

      4. lower-/.f64 64.3

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

   8. Applied rewrites 64.3%

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

   if 3.3e-4 < x

   1. Initial program 87.2%

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

   2. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   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 96.9

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

   4. Applied rewrites 96.9%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 97.3%

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

Alternative 11: 69.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-48}:\\ \;\;\;\;t\_0 + \frac{-1}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(B \cdot B\right), B \cdot B, 1\right) \cdot B}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-86}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{F}{B} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* x (/ 1.0 (tan B))))))
   (if (<= x -5e-48)
     (+ t_0 (/ -1.0 (* (fma (* 0.008333333333333333 (* B B)) (* B B) 1.0) B)))
     (if (<= x 5.4e-86)
       (* (/ 1.0 (sqrt (fma F F 2.0))) (/ F (sin B)))
       (if (<= x 1.6e-9)
         (- (* (/ F B) (pow (fma 2.0 x (fma F F 2.0)) -0.5)) (/ x 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 <= -5e-48) {
		tmp = t_0 + (-1.0 / (fma((0.008333333333333333 * (B * B)), (B * B), 1.0) * B));
	} else if (x <= 5.4e-86) {
		tmp = (1.0 / sqrt(fma(F, F, 2.0))) * (F / sin(B));
	} else if (x <= 1.6e-9) {
		tmp = ((F / B) * pow(fma(2.0, x, fma(F, F, 2.0)), -0.5)) - (x / 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 <= -5e-48)
		tmp = Float64(t_0 + Float64(-1.0 / Float64(fma(Float64(0.008333333333333333 * Float64(B * B)), Float64(B * B), 1.0) * B)));
	elseif (x <= 5.4e-86)
		tmp = Float64(Float64(1.0 / sqrt(fma(F, F, 2.0))) * Float64(F / sin(B)));
	elseif (x <= 1.6e-9)
		tmp = Float64(Float64(Float64(F / B) * (fma(2.0, x, fma(F, F, 2.0)) ^ -0.5)) - Float64(x / 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, -5e-48], N[(t$95$0 + N[(-1.0 / N[(N[(N[(0.008333333333333333 * N[(B * B), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.4e-86], N[(N[(1.0 / N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.6e-9], N[(N[(N[(F / B), $MachinePrecision] * N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.9999999999999999e-48

   1. Initial program 69.7%

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

   2. Taylor expanded in 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 76.2

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

   4. Applied rewrites 76.2%

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

   5. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \color{blue}{\left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right) \cdot B} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right) \cdot B} \]

      3. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\left({B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right) + 1\right) \cdot B} \]

      4. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\left(\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right) \cdot {B}^{2} + 1\right) \cdot B} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      6. lower--.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      10. unpow2 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}, B \cdot B, 1\right) \cdot B} \]

      11. lower-*.f64 78.9

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666, B \cdot B, 1\right) \cdot B} \]

   7. Applied rewrites 78.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666, B \cdot B, 1\right) \cdot \color{blue}{B}} \]

   8. Taylor expanded in B around inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot {B}^{2}, B \cdot B, 1\right) \cdot B} \]
   9. Step-by-step derivation

      1. pow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right), B \cdot B, 1\right) \cdot B} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right), B \cdot B, 1\right) \cdot B} \]

      3. lift-*.f64 78.9

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(B \cdot B\right), B \cdot B, 1\right) \cdot B} \]

   10. Applied rewrites 78.9%

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(B \cdot B\right), B \cdot B, 1\right) \cdot B} \]

   if -4.9999999999999999e-48 < x < 5.39999999999999985e-86

   1. Initial program 73.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 77.0%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   5. 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. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-/.f64 53.1

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

   6. Applied rewrites 53.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. unpow-1 N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. pow2 N/A

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

      13. lift-fma.f64 53.0

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

   8. Applied rewrites 53.0%

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

   if 5.39999999999999985e-86 < x < 1.60000000000000006e-9

   1. Initial program 72.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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 39.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. div-sub N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. lower--.f64 N/A

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

   6. Applied rewrites 36.8%

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

   if 1.60000000000000006e-9 < x

   1. Initial program 86.9%

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

   2. Taylor expanded in 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 96.0

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 96.4%

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

Alternative 12: 69.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;x \leq -5 \cdot 10^{-48}:\\ \;\;\;\;t\_0 + \frac{-1}{\mathsf{fma}\left(-0.16666666666666666, B \cdot B, 1\right) \cdot B}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-86}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{F}{B} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* x (/ 1.0 (tan B))))))
   (if (<= x -5e-48)
     (+ t_0 (/ -1.0 (* (fma -0.16666666666666666 (* B B) 1.0) B)))
     (if (<= x 5.4e-86)
       (* (/ 1.0 (sqrt (fma F F 2.0))) (/ F (sin B)))
       (if (<= x 1.6e-9)
         (- (* (/ F B) (pow (fma 2.0 x (fma F F 2.0)) -0.5)) (/ x 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 <= -5e-48) {
		tmp = t_0 + (-1.0 / (fma(-0.16666666666666666, (B * B), 1.0) * B));
	} else if (x <= 5.4e-86) {
		tmp = (1.0 / sqrt(fma(F, F, 2.0))) * (F / sin(B));
	} else if (x <= 1.6e-9) {
		tmp = ((F / B) * pow(fma(2.0, x, fma(F, F, 2.0)), -0.5)) - (x / 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 <= -5e-48)
		tmp = Float64(t_0 + Float64(-1.0 / Float64(fma(-0.16666666666666666, Float64(B * B), 1.0) * B)));
	elseif (x <= 5.4e-86)
		tmp = Float64(Float64(1.0 / sqrt(fma(F, F, 2.0))) * Float64(F / sin(B)));
	elseif (x <= 1.6e-9)
		tmp = Float64(Float64(Float64(F / B) * (fma(2.0, x, fma(F, F, 2.0)) ^ -0.5)) - Float64(x / 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, -5e-48], N[(t$95$0 + N[(-1.0 / N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.4e-86], N[(N[(1.0 / N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.6e-9], N[(N[(N[(F / B), $MachinePrecision] * N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -4.9999999999999999e-48

   1. Initial program 69.7%

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

   2. Taylor expanded in 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 76.2

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

   4. Applied rewrites 76.2%

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

   5. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 78.9

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

   7. Applied rewrites 78.9%

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

   if -4.9999999999999999e-48 < x < 5.39999999999999985e-86

   1. Initial program 73.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 77.0%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   5. 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. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-/.f64 53.1

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

   6. Applied rewrites 53.1%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. unpow-1 N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. pow2 N/A

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

      13. lift-fma.f64 53.0

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

   8. Applied rewrites 53.0%

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

   if 5.39999999999999985e-86 < x < 1.60000000000000006e-9

   1. Initial program 72.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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 39.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. div-sub N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. lower--.f64 N/A

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

   6. Applied rewrites 36.8%

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

   if 1.60000000000000006e-9 < x

   1. Initial program 86.9%

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

   2. Taylor expanded in 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 96.0

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 96.4%

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

Alternative 13: 69.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-12}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{-1}{B}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-86}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{F}{B} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -1.3e-12)
   (+ (- (/ (* x 1.0) (tan B))) (/ -1.0 B))
   (if (<= x 5.4e-86)
     (* (/ 1.0 (sqrt (fma F F 2.0))) (/ F (sin B)))
     (if (<= x 1.6e-9)
       (- (* (/ F B) (pow (fma 2.0 x (fma F F 2.0)) -0.5)) (/ x B))
       (+ (- (* x (/ 1.0 (tan B)))) (/ -1.0 B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -1.3e-12) {
		tmp = -((x * 1.0) / tan(B)) + (-1.0 / B);
	} else if (x <= 5.4e-86) {
		tmp = (1.0 / sqrt(fma(F, F, 2.0))) * (F / sin(B));
	} else if (x <= 1.6e-9) {
		tmp = ((F / B) * pow(fma(2.0, x, fma(F, F, 2.0)), -0.5)) - (x / B);
	} else {
		tmp = -(x * (1.0 / tan(B))) + (-1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -1.3e-12)
		tmp = Float64(Float64(-Float64(Float64(x * 1.0) / tan(B))) + Float64(-1.0 / B));
	elseif (x <= 5.4e-86)
		tmp = Float64(Float64(1.0 / sqrt(fma(F, F, 2.0))) * Float64(F / sin(B)));
	elseif (x <= 1.6e-9)
		tmp = Float64(Float64(Float64(F / B) * (fma(2.0, x, fma(F, F, 2.0)) ^ -0.5)) - Float64(x / B));
	else
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(-1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -1.3e-12], N[((-N[(N[(x * 1.0), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.4e-86], N[(N[(1.0 / N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.6e-9], N[(N[(N[(F / B), $MachinePrecision] * N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.29999999999999991e-12

   1. Initial program 68.2%

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

   2. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   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 90.7

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

   4. Applied rewrites 90.7%

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

   5. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \color{blue}{\left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right) \cdot B} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right) \cdot B} \]

      3. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\left({B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right) + 1\right) \cdot B} \]

      4. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\left(\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right) \cdot {B}^{2} + 1\right) \cdot B} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      6. lower--.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      10. unpow2 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}, B \cdot B, 1\right) \cdot B} \]

      11. lower-*.f64 91.5

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666, B \cdot B, 1\right) \cdot B} \]

   7. Applied rewrites 91.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666, B \cdot B, 1\right) \cdot \color{blue}{B}} \]

   8. Taylor expanded in B around 0

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

   10. Applied rewrites 91.5%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-tan.f64 N/A

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

       4. associate-*r/ N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-tan.f64 91.7

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

   12. Applied rewrites 91.7%

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

   if -1.29999999999999991e-12 < x < 5.39999999999999985e-86

   1. Initial program 73.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 77.2%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   5. 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. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-/.f64 51.2

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

   6. Applied rewrites 51.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. unpow-1 N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. lower-sqrt.f64 N/A

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

      11. +-commutative N/A

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

      12. pow2 N/A

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

      13. lift-fma.f64 51.1

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

   8. Applied rewrites 51.1%

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

   if 5.39999999999999985e-86 < x < 1.60000000000000006e-9

   1. Initial program 72.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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 39.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. div-sub N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. lower--.f64 N/A

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

   6. Applied rewrites 36.8%

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

   if 1.60000000000000006e-9 < x

   1. Initial program 86.9%

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

   2. Taylor expanded in 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 96.0

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 96.4%

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

Alternative 14: 69.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-12}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{-1}{B}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-86}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{F}{B} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -1.3e-12)
   (+ (- (/ (* x 1.0) (tan B))) (/ -1.0 B))
   (if (<= x 5.4e-86)
     (* (sqrt (/ 1.0 (fma F F 2.0))) (/ F (sin B)))
     (if (<= x 1.6e-9)
       (- (* (/ F B) (pow (fma 2.0 x (fma F F 2.0)) -0.5)) (/ x B))
       (+ (- (* x (/ 1.0 (tan B)))) (/ -1.0 B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -1.3e-12) {
		tmp = -((x * 1.0) / tan(B)) + (-1.0 / B);
	} else if (x <= 5.4e-86) {
		tmp = sqrt((1.0 / fma(F, F, 2.0))) * (F / sin(B));
	} else if (x <= 1.6e-9) {
		tmp = ((F / B) * pow(fma(2.0, x, fma(F, F, 2.0)), -0.5)) - (x / B);
	} else {
		tmp = -(x * (1.0 / tan(B))) + (-1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -1.3e-12)
		tmp = Float64(Float64(-Float64(Float64(x * 1.0) / tan(B))) + Float64(-1.0 / B));
	elseif (x <= 5.4e-86)
		tmp = Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) * Float64(F / sin(B)));
	elseif (x <= 1.6e-9)
		tmp = Float64(Float64(Float64(F / B) * (fma(2.0, x, fma(F, F, 2.0)) ^ -0.5)) - Float64(x / B));
	else
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(-1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -1.3e-12], N[((-N[(N[(x * 1.0), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.4e-86], N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.6e-9], N[(N[(N[(F / B), $MachinePrecision] * N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.29999999999999991e-12

   1. Initial program 68.2%

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

   2. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   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 90.7

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

   4. Applied rewrites 90.7%

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

   5. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \color{blue}{\left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right) \cdot B} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right) \cdot B} \]

      3. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\left({B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right) + 1\right) \cdot B} \]

      4. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\left(\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right) \cdot {B}^{2} + 1\right) \cdot B} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      6. lower--.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      10. unpow2 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}, B \cdot B, 1\right) \cdot B} \]

      11. lower-*.f64 91.5

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666, B \cdot B, 1\right) \cdot B} \]

   7. Applied rewrites 91.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666, B \cdot B, 1\right) \cdot \color{blue}{B}} \]

   8. Taylor expanded in B around 0

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

   10. Applied rewrites 91.5%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-tan.f64 N/A

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

       4. associate-*r/ N/A

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

       5. lower-/.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lift-tan.f64 91.7

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

   12. Applied rewrites 91.7%

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

   if -1.29999999999999991e-12 < x < 5.39999999999999985e-86

   1. Initial program 73.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 77.2%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   5. 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. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-/.f64 51.2

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

   6. Applied rewrites 51.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. unpow-1 N/A

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

      4. pow2 N/A

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

      5. +-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lift-fma.f64 51.2

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

   8. Applied rewrites 51.2%

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

   if 5.39999999999999985e-86 < x < 1.60000000000000006e-9

   1. Initial program 72.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   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 39.4%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-sqrt.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. div-sub N/A

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

      10. *-commutative N/A

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

      11. associate-*l/ N/A

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

      12. lower--.f64 N/A

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

   6. Applied rewrites 36.8%

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

   if 1.60000000000000006e-9 < x

   1. Initial program 86.9%

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

   2. Taylor expanded in 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 96.0

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 96.4%

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

Alternative 15: 57.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.17:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.17)
   (/ (- (* (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0)))) F) x) B)
   (+ (- (* x (/ 1.0 (tan B)))) (/ -1.0 B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.17) {
		tmp = ((sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B;
	} else {
		tmp = -(x * (1.0 / tan(B))) + (-1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.17)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B);
	else
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(-1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.17], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.170000000000000012

   1. Initial program 74.0%

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

   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 58.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. inv-pow N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. pow2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-fma.f64 58.4

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

   6. Applied rewrites 58.4%

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

   if 0.170000000000000012 < B

   1. Initial program 86.1%

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

   2. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   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 57.7

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

   4. Applied rewrites 57.7%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 53.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 57.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7 \cdot 10^{-15}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{+43}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - x \cdot x}{1 + x}}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7e-15)
   (+ (- (/ x B)) (/ -1.0 (sin B)))
   (if (<= F 7.2e+43)
     (/ (- (* (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0)))) F) x) B)
     (/ (/ (- 1.0 (* x x)) (+ 1.0 x)) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7e-15) {
		tmp = -(x / B) + (-1.0 / sin(B));
	} else if (F <= 7.2e+43) {
		tmp = ((sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B;
	} else {
		tmp = ((1.0 - (x * x)) / (1.0 + x)) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7e-15)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(-1.0 / sin(B)));
	elseif (F <= 7.2e+43)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(x * x)) / Float64(1.0 + x)) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7e-15], N[((-N[(x / B), $MachinePrecision]) + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.2e+43], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.0000000000000001e-15

   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 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 74.3

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

   7. Applied rewrites 74.3%

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

   if -7.0000000000000001e-15 < F < 7.2000000000000002e43

   1. Initial program 99.2%

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

   2. 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 51.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-fma.f64 N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. inv-pow N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. pow2 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-fma.f64 51.6

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

   6. Applied rewrites 51.6%

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

   if 7.2000000000000002e43 < F

   1. Initial program 53.1%

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

   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 36.4%

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

   5. Taylor expanded in F around inf

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

      1. lower--.f64 51.0

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

   7. Applied rewrites 51.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1 - {x}^{2}}{1 + x}}{B} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\frac{1 - {x}^{2}}{1 + x}}{B} \]

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 48.6

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

   9. Applied rewrites 48.6%

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

Alternative 17: 47.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 2.55 \cdot 10^{+22}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 2.55e+22)
   (/ (- (* (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0)))) F) x) B)
   (/ -1.0 (sin B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 2.55e+22) {
		tmp = ((sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B;
	} else {
		tmp = -1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 2.55e+22)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B);
	else
		tmp = Float64(-1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 2.55e+22], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.5500000000000001e22

   1. Initial program 74.3%

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

   2. 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 57.2%

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

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{{\left(\left(2 \cdot x + F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B} \]

      5. pow2 N/A

         \[\leadsto \frac{\sqrt{{\left(\left(2 \cdot x + {F}^{2}\right) + 2\right)}^{-1}} \cdot F - x}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\sqrt{{\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{-1}} \cdot F - x}{B} \]

      7. inv-pow N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\left(2 \cdot x + {F}^{2}\right) + 2}} \cdot F - x}{B} \]

      9. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 \cdot x + \left({F}^{2} + 2\right)}} \cdot F - x}{B} \]

      10. pow2 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{2 \cdot x + \left(F \cdot F + 2\right)}} \cdot F - x}{B} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{2 \cdot x + \left(F \cdot F + 2\right)}} \cdot F - x}{B} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{2 \cdot x + \mathsf{fma}\left(F, F, 2\right)}} \cdot F - x}{B} \]

      13. lift-fma.f64 57.2

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B} \]

   6. Applied rewrites 57.2%

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B} \]

   if 2.5500000000000001e22 < B

   1. Initial program 86.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. lift-sin.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\color{blue}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      5. lift-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\left(-\frac{1}{2}\right)} \]

      6. lift-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\color{blue}{\left(F \cdot F + 2\right)} + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      7. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(\color{blue}{F \cdot F} + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      8. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + \color{blue}{2 \cdot x}\right)}^{\left(-\frac{1}{2}\right)} \]

      9. lift-neg.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      10. lift-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \]

      11. associate-*l/ N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}} \]

      12. lower-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}} \]

   3. Applied rewrites 86.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   5. 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. inv-pow N/A

         \[\leadsto \sqrt{{\left(2 + {F}^{2}\right)}^{-1}} \cdot \frac{F}{\sin B} \]

      5. lower-pow.f64 N/A

         \[\leadsto \sqrt{{\left(2 + {F}^{2}\right)}^{-1}} \cdot \frac{F}{\sin B} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{{\left({F}^{2} + 2\right)}^{-1}} \cdot \frac{F}{\sin B} \]

      7. pow2 N/A

         \[\leadsto \sqrt{{\left(F \cdot F + 2\right)}^{-1}} \cdot \frac{F}{\sin B} \]

      8. lift-fma.f64 N/A

         \[\leadsto \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B} \]

      9. lift-sin.f64 N/A

         \[\leadsto \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B} \]

      10. lift-/.f64 30.1

          \[\leadsto \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\color{blue}{\sin B}} \]

   6. Applied rewrites 30.1%

      \[\leadsto \color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \frac{-1}{\sin B} \]

      2. lift-/.f64 17.4

         \[\leadsto \frac{-1}{\sin B} \]

   9. Applied rewrites 17.4%

      \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 50.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{+43}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - x \cdot x}{1 + x}}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.35e+154)
   (+ (- (/ x B)) (/ (- (* -0.16666666666666666 (* B B)) 1.0) B))
   (if (<= F 7.2e+43)
     (/ (- (* (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0)))) F) x) B)
     (/ (/ (- 1.0 (* x x)) (+ 1.0 x)) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.35e+154) {
		tmp = -(x / B) + (((-0.16666666666666666 * (B * B)) - 1.0) / B);
	} else if (F <= 7.2e+43) {
		tmp = ((sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B;
	} else {
		tmp = ((1.0 - (x * x)) / (1.0 + x)) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.35e+154)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(Float64(-0.16666666666666666 * Float64(B * B)) - 1.0) / B));
	elseif (F <= 7.2e+43)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(x * x)) / Float64(1.0 + x)) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.35e+154], N[((-N[(x / B), $MachinePrecision]) + N[(N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.2e+43], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.35000000000000003e154

   1. Initial program 30.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in 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 99.6

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B} \]

   4. Applied rewrites 99.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]
   6. Step-by-step derivation

      1. lower-/.f64 75.0

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{-1}{\sin B} \]

   7. Applied rewrites 75.0%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   8. Taylor expanded in B around 0

      \[\leadsto \left(-\frac{x}{B}\right) + \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{\color{blue}{B}} \]
   9. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]

      2. lower--.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]

      4. pow2 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{\frac{-1}{6} \cdot \left(B \cdot B\right) - 1}{B} \]

      5. lift-*.f64 52.0

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   10. Applied rewrites 52.0%

       \[\leadsto \left(-\frac{x}{B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{\color{blue}{B}} \]

   if -1.35000000000000003e154 < F < 7.2000000000000002e43

   1. Initial program 97.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   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 51.5%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]
   5. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{{\left(\left(2 \cdot x + F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B} \]

      5. pow2 N/A

         \[\leadsto \frac{\sqrt{{\left(\left(2 \cdot x + {F}^{2}\right) + 2\right)}^{-1}} \cdot F - x}{B} \]

      6. +-commutative N/A

         \[\leadsto \frac{\sqrt{{\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{-1}} \cdot F - x}{B} \]

      7. inv-pow N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\left(2 \cdot x + {F}^{2}\right) + 2}} \cdot F - x}{B} \]

      9. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2 \cdot x + \left({F}^{2} + 2\right)}} \cdot F - x}{B} \]

      10. pow2 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{2 \cdot x + \left(F \cdot F + 2\right)}} \cdot F - x}{B} \]

      11. lower-/.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{2 \cdot x + \left(F \cdot F + 2\right)}} \cdot F - x}{B} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{2 \cdot x + \mathsf{fma}\left(F, F, 2\right)}} \cdot F - x}{B} \]

      13. lift-fma.f64 51.5

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B} \]

   6. Applied rewrites 51.5%

      \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B} \]

   if 7.2000000000000002e43 < F

   1. Initial program 53.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   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 36.4%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   6. Step-by-step derivation

      1. lower--.f64 51.0

         \[\leadsto \frac{1 - x}{B} \]

   7. Applied rewrites 51.0%

      \[\leadsto \frac{1 - x}{B} \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{1 - x}{B} \]

      2. flip-- N/A

         \[\leadsto \frac{\frac{1 \cdot 1 - x \cdot x}{1 + x}}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1 \cdot 1 - x \cdot x}{1 + x}}{B} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\frac{1 - x \cdot x}{1 + x}}{B} \]

      5. unpow2 N/A

         \[\leadsto \frac{\frac{1 - {x}^{2}}{1 + x}}{B} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\frac{1 - {x}^{2}}{1 + x}}{B} \]

      7. unpow2 N/A

         \[\leadsto \frac{\frac{1 - x \cdot x}{1 + x}}{B} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1 - x \cdot x}{1 + x}}{B} \]

      9. lower-+.f64 48.6

         \[\leadsto \frac{\frac{1 - x \cdot x}{1 + x}}{B} \]

   9. Applied rewrites 48.6%

      \[\leadsto \frac{\frac{1 - x \cdot x}{1 + x}}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 43.7% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8.4 \cdot 10^{-44}:\\ \;\;\;\;\frac{-\left(1 + x\right)}{B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8.4e-44)
   (/ (- (+ 1.0 x)) B)
   (if (<= F 5.8e-52) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.4e-44) {
		tmp = -(1.0 + x) / B;
	} else if (F <= 5.8e-52) {
		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 <= (-8.4d-44)) then
        tmp = -(1.0d0 + x) / b
    else if (f <= 5.8d-52) 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 <= -8.4e-44) {
		tmp = -(1.0 + x) / B;
	} else if (F <= 5.8e-52) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -8.4e-44:
		tmp = -(1.0 + x) / B
	elif F <= 5.8e-52:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8.4e-44)
		tmp = Float64(Float64(-Float64(1.0 + x)) / B);
	elseif (F <= 5.8e-52)
		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 <= -8.4e-44)
		tmp = -(1.0 + x) / B;
	elseif (F <= 5.8e-52)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8.4e-44], N[((-N[(1.0 + x), $MachinePrecision]) / B), $MachinePrecision], If[LessEqual[F, 5.8e-52], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.40000000000000005e-44

   1. Initial program 62.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   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 40.8%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 \cdot \left(1 + x\right)}{B} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(\left(1 + x\right)\right)}{B} \]

      2. lower-neg.f64 N/A

         \[\leadsto \frac{-\left(1 + x\right)}{B} \]

      3. lower-+.f64 47.9

         \[\leadsto \frac{-\left(1 + x\right)}{B} \]

   7. Applied rewrites 47.9%

      \[\leadsto \frac{-\left(1 + x\right)}{B} \]

   if -8.40000000000000005e-44 < F < 5.8000000000000003e-52

   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 51.4%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   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 37.5

         \[\leadsto \frac{-x}{B} \]

   7. Applied rewrites 37.5%

      \[\leadsto \frac{-x}{B} \]

   if 5.8000000000000003e-52 < F

   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 40.0%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   6. Step-by-step derivation

      1. lower--.f64 47.3

         \[\leadsto \frac{1 - x}{B} \]

   7. Applied rewrites 47.3%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 32.1% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{B}\\ \mathbf{if}\;x \leq -8 \cdot 10^{-128}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-119}:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) B)))
   (if (<= x -8e-128) t_0 (if (<= x 8.2e-119) (/ 1.0 B) t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -x / B;
	double tmp;
	if (x <= -8e-128) {
		tmp = t_0;
	} else if (x <= 8.2e-119) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	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 / b
    if (x <= (-8d-128)) then
        tmp = t_0
    else if (x <= 8.2d-119) then
        tmp = 1.0d0 / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -x / B;
	double tmp;
	if (x <= -8e-128) {
		tmp = t_0;
	} else if (x <= 8.2e-119) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -x / B
	tmp = 0
	if x <= -8e-128:
		tmp = t_0
	elif x <= 8.2e-119:
		tmp = 1.0 / B
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / B)
	tmp = 0.0
	if (x <= -8e-128)
		tmp = t_0;
	elseif (x <= 8.2e-119)
		tmp = Float64(1.0 / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -x / B;
	tmp = 0.0;
	if (x <= -8e-128)
		tmp = t_0;
	elseif (x <= 8.2e-119)
		tmp = 1.0 / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / B), $MachinePrecision]}, If[LessEqual[x, -8e-128], t$95$0, If[LessEqual[x, 8.2e-119], N[(1.0 / B), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.00000000000000043e-128 or 8.20000000000000041e-119 < x

   1. Initial program 79.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   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 47.3%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   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 41.2

         \[\leadsto \frac{-x}{B} \]

   7. Applied rewrites 41.2%

      \[\leadsto \frac{-x}{B} \]

   if -8.00000000000000043e-128 < x < 8.20000000000000041e-119

   1. Initial program 73.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   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 39.8%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   6. Step-by-step derivation

      1. lower--.f64 15.4

         \[\leadsto \frac{1 - x}{B} \]

   7. Applied rewrites 15.4%

      \[\leadsto \frac{1 - x}{B} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{B} \]
   9. Step-by-step derivation

   10. Applied rewrites 15.4%

       \[\leadsto \frac{1}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 37.1% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 5.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 5.8e-52) (/ (- x) B) (/ (- 1.0 x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 5.8e-52) {
		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 <= 5.8d-52) 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 <= 5.8e-52) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 5.8e-52:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 5.8e-52)
		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 <= 5.8e-52)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 5.8e-52], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 5.8000000000000003e-52

   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 46.7%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   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 32.6

         \[\leadsto \frac{-x}{B} \]

   7. Applied rewrites 32.6%

      \[\leadsto \frac{-x}{B} \]

   if 5.8000000000000003e-52 < F

   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 40.0%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   6. Step-by-step derivation

      1. lower--.f64 47.3

         \[\leadsto \frac{1 - x}{B} \]

   7. Applied rewrites 47.3%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 9.7% accurate, 30.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ 1.0 B))

C

double code(double F, double B, double x) {
	return 1.0 / B;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = 1.0d0 / b
end function

Java

public static double code(double F, double B, double x) {
	return 1.0 / B;
}

Python

def code(F, B, x):
	return 1.0 / B

Julia

function code(F, B, x)
	return Float64(1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = 1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(1.0 / B), $MachinePrecision]

Derivation

1. Initial program 77.1%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

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 44.7%

   \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

5. Taylor expanded in F around inf

   \[\leadsto \frac{1 - x}{B} \]
6. Step-by-step derivation

   1. lower--.f64 29.7

      \[\leadsto \frac{1 - x}{B} \]

7. Applied rewrites 29.7%

   \[\leadsto \frac{1 - x}{B} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{1}{B} \]
9. Step-by-step derivation

10. Applied rewrites 9.7%

    \[\leadsto \frac{1}{B} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025106 
(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))))))