VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.1% → 99.5%

Time: 6.5s

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

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

   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%

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

   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%

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

   3. Applied rewrites 70.9%

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

   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

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

   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

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

   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

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

   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

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

   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%

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

   3. Applied rewrites 99.5%

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

   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

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

   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

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

   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%

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

   3. Applied rewrites 99.5%

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

   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

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

   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

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

   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%

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

   3. Applied rewrites 99.5%

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

   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

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

   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

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

   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

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

   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

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

   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%

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

   3. Applied rewrites 99.6%

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

   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

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

   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

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

   4. Applied rewrites 85.2%

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

   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

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

   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

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

   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%

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

   3. Applied rewrites 99.5%

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

   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

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

   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

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

   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%

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

   3. Applied rewrites 95.6%

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

   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

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

   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%

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

   3. Applied rewrites 76.7%

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

   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

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

   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

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

   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

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

   7. Applied rewrites 84.7%

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

   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%

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

   3. Applied rewrites 76.8%

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

   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

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

   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

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

   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

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

   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

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

   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

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

   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

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

   10. Applied rewrites 91.7%

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

   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%

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

   3. Applied rewrites 76.8%

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

   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

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

   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

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

   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

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

   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

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

   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

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

   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

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

   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%

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

   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

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

   6. Applied rewrites 53.1%

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

   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

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

   4. Applied rewrites 39.4%

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

   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

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

   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

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

   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

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

   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

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

   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%

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

   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

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

   6. Applied rewrites 53.1%

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

   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

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

   4. Applied rewrites 39.4%

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

   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

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

   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

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

   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

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

   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

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

   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

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

   10. Applied rewrites 91.5%

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

   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%

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

   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

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

   6. Applied rewrites 51.2%

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

   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

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

   4. Applied rewrites 39.4%

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

   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

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

   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

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

   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

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

   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

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

   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

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

   10. Applied rewrites 91.5%

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

   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%

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

   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

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

   6. Applied rewrites 51.2%

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

   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

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

   4. Applied rewrites 39.4%

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

   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

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

   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

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

   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

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

   4. Applied rewrites 58.4%

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

   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

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

   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

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

   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

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

   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

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

   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

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

   4. Applied rewrites 51.6%

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

   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

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

   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

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

   7. Applied rewrites 51.0%

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

   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

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

   4. Applied rewrites 57.2%

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

   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%

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

   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

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

   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

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

   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

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

   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

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

   7. Applied rewrites 75.0%

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

   8. Taylor expanded in B around 0

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

   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

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

   4. Applied rewrites 51.5%

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

   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

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

   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

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

   7. Applied rewrites 51.0%

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

   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

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8.4 \cdot 10^{-44}:\\ \;\;\;\;\frac{-1 + \left(-x\right)}{B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

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(-1.0 + Float64(-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

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

   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

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

   7. Applied rewrites 47.9%

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

   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

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

   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

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

   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

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

   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

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

   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

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

   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

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

   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

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

   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

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

   7. Applied rewrites 15.4%

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

   8. Taylor expanded in x around 0

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

   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

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

   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

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

   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

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

   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

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

   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

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

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

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

7. Applied rewrites 29.7%

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

8. Taylor expanded in x around 0

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

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))))))