VandenBroeck and Keller, Equation (23)

Percentage Accurate: 75.8% → 98.8%

Time: 8.6s

Alternatives: 24

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.8% 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: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -4.3 \cdot 10^{+40}:\\ \;\;\;\;t\_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-18}:\\ \;\;\;\;t\_0 + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -4.3e+40)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 2.6e-18)
       (+
        t_0
        (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (/ -1.0 2.0))))
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -4.3e+40) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 2.6e-18) {
		tmp = t_0 + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), (-1.0 / 2.0)));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x / tan(b)
    if (f <= (-4.3d+40)) then
        tmp = t_0 + ((-1.0d0) / sin(b))
    else if (f <= 2.6d-18) then
        tmp = t_0 + ((f / sin(b)) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** ((-1.0d0) / 2.0d0)))
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -x / Math.tan(B);
	double tmp;
	if (F <= -4.3e+40) {
		tmp = t_0 + (-1.0 / Math.sin(B));
	} else if (F <= 2.6e-18) {
		tmp = t_0 + ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (2.0 * x)), (-1.0 / 2.0)));
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -x / math.tan(B)
	tmp = 0
	if F <= -4.3e+40:
		tmp = t_0 + (-1.0 / math.sin(B))
	elif F <= 2.6e-18:
		tmp = t_0 + ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (2.0 * x)), (-1.0 / 2.0)))
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -4.3e+40)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 2.6e-18)
		tmp = Float64(t_0 + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-1.0 / 2.0))));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -x / tan(B);
	tmp = 0.0;
	if (F <= -4.3e+40)
		tmp = t_0 + (-1.0 / sin(B));
	elseif (F <= 2.6e-18)
		tmp = t_0 + ((F / sin(B)) * ((((F * F) + 2.0) + (2.0 * x)) ^ (-1.0 / 2.0)));
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4.3e+40], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.6e-18], N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], N[(-1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.3000000000000002e40

   1. Initial program 46.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if -4.3000000000000002e40 < F < 2.6e-18

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if 2.6e-18 < F

   1. Initial program 56.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 71.4%

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

   5. Taylor expanded in F around inf

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

   7. Applied rewrites 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.3 \cdot 10^{+40}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -4.3 \cdot 10^{+40}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-18}:\\ \;\;\;\;t\_0 + \frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -4.3e+40)
     (+ (/ (- x) (tan B)) (/ -1.0 (sin B)))
     (if (<= F 2.6e-18)
       (+ t_0 (* (/ 1.0 (sqrt (+ (+ (* 2.0 x) 2.0) (* F F)))) (/ F (sin B))))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -4.3e+40) {
		tmp = (-x / tan(B)) + (-1.0 / sin(B));
	} else if (F <= 2.6e-18) {
		tmp = t_0 + ((1.0 / sqrt((((2.0 * x) + 2.0) + (F * F)))) * (F / sin(B)));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-4.3d+40)) then
        tmp = (-x / tan(b)) + ((-1.0d0) / sin(b))
    else if (f <= 2.6d-18) then
        tmp = t_0 + ((1.0d0 / sqrt((((2.0d0 * x) + 2.0d0) + (f * f)))) * (f / sin(b)))
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -4.3e+40) {
		tmp = (-x / Math.tan(B)) + (-1.0 / Math.sin(B));
	} else if (F <= 2.6e-18) {
		tmp = t_0 + ((1.0 / Math.sqrt((((2.0 * x) + 2.0) + (F * F)))) * (F / Math.sin(B)));
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -4.3e+40:
		tmp = (-x / math.tan(B)) + (-1.0 / math.sin(B))
	elif F <= 2.6e-18:
		tmp = t_0 + ((1.0 / math.sqrt((((2.0 * x) + 2.0) + (F * F)))) * (F / math.sin(B)))
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -4.3e+40)
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(-1.0 / sin(B)));
	elseif (F <= 2.6e-18)
		tmp = Float64(t_0 + Float64(Float64(1.0 / sqrt(Float64(Float64(Float64(2.0 * x) + 2.0) + Float64(F * F)))) * Float64(F / sin(B))));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -4.3e+40)
		tmp = (-x / tan(B)) + (-1.0 / sin(B));
	elseif (F <= 2.6e-18)
		tmp = t_0 + ((1.0 / sqrt((((2.0 * x) + 2.0) + (F * F)))) * (F / sin(B)));
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4.3e+40], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.6e-18], N[(t$95$0 + N[(N[(1.0 / N[Sqrt[N[(N[(N[(2.0 * x), $MachinePrecision] + 2.0), $MachinePrecision] + N[(F * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.3000000000000002e40

   1. Initial program 46.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if -4.3000000000000002e40 < F < 2.6e-18

   1. Initial program 99.5%

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

   3. Taylor expanded in B around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. associate-+r+ N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-/.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 2.6e-18 < F

   1. Initial program 56.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 71.4%

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

   5. Taylor expanded in F around inf

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

   7. Applied rewrites 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.3 \cdot 10^{+40}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-18}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -3.3 \cdot 10^{+44}:\\ \;\;\;\;t\_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-18}:\\ \;\;\;\;t\_0 + \frac{\frac{F}{\sin B}}{\sqrt{\left(F \cdot F + 2\right) - -2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -3.3e+44)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 2.6e-18)
       (+ t_0 (/ (/ F (sin B)) (sqrt (- (+ (* F F) 2.0) (* -2.0 x)))))
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -3.3e+44) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 2.6e-18) {
		tmp = t_0 + ((F / sin(B)) / sqrt((((F * F) + 2.0) - (-2.0 * x))));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x / tan(b)
    if (f <= (-3.3d+44)) then
        tmp = t_0 + ((-1.0d0) / sin(b))
    else if (f <= 2.6d-18) then
        tmp = t_0 + ((f / sin(b)) / sqrt((((f * f) + 2.0d0) - ((-2.0d0) * x))))
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -x / Math.tan(B);
	double tmp;
	if (F <= -3.3e+44) {
		tmp = t_0 + (-1.0 / Math.sin(B));
	} else if (F <= 2.6e-18) {
		tmp = t_0 + ((F / Math.sin(B)) / Math.sqrt((((F * F) + 2.0) - (-2.0 * x))));
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -x / math.tan(B)
	tmp = 0
	if F <= -3.3e+44:
		tmp = t_0 + (-1.0 / math.sin(B))
	elif F <= 2.6e-18:
		tmp = t_0 + ((F / math.sin(B)) / math.sqrt((((F * F) + 2.0) - (-2.0 * x))))
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -3.3e+44)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 2.6e-18)
		tmp = Float64(t_0 + Float64(Float64(F / sin(B)) / sqrt(Float64(Float64(Float64(F * F) + 2.0) - Float64(-2.0 * x)))));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -x / tan(B);
	tmp = 0.0;
	if (F <= -3.3e+44)
		tmp = t_0 + (-1.0 / sin(B));
	elseif (F <= 2.6e-18)
		tmp = t_0 + ((F / sin(B)) / sqrt((((F * F) + 2.0) - (-2.0 * x))));
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.3e+44], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.6e-18], N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] - N[(-2.0 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.30000000000000013e44

   1. Initial program 44.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 44.0

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

   4. Applied rewrites 44.0%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if -3.30000000000000013e44 < F < 2.6e-18

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in B around inf

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

      1. sqrt-div N/A

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

      2. metadata-eval N/A

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

      3. associate-+r+ N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. associate-+r+ N/A

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

      9. pow2 N/A

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

      10. associate-*r/ N/A

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

      11. lower-/.f64 N/A

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

   7. Applied rewrites 99.7%

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

   if 2.6e-18 < F

   1. Initial program 56.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 71.4%

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

   5. Taylor expanded in F around inf

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

   7. Applied rewrites 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.3 \cdot 10^{+44}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{\frac{F}{\sin B}}{\sqrt{\left(F \cdot F + 2\right) - -2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -80000000000000:\\ \;\;\;\;t\_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-22}:\\ \;\;\;\;t\_0 + {\left(\sqrt{\left(F \cdot F + 2\right) - -2 \cdot x}\right)}^{-1} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -80000000000000.0)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 5e-22)
       (+ t_0 (* (pow (sqrt (- (+ (* F F) 2.0) (* -2.0 x))) -1.0) (/ F B)))
       (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -80000000000000.0) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 5e-22) {
		tmp = t_0 + (pow(sqrt((((F * F) + 2.0) - (-2.0 * x))), -1.0) * (F / B));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x / tan(b)
    if (f <= (-80000000000000.0d0)) then
        tmp = t_0 + ((-1.0d0) / sin(b))
    else if (f <= 5d-22) then
        tmp = t_0 + ((sqrt((((f * f) + 2.0d0) - ((-2.0d0) * x))) ** (-1.0d0)) * (f / b))
    else
        tmp = (x * ((-1.0d0) / tan(b))) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -x / Math.tan(B);
	double tmp;
	if (F <= -80000000000000.0) {
		tmp = t_0 + (-1.0 / Math.sin(B));
	} else if (F <= 5e-22) {
		tmp = t_0 + (Math.pow(Math.sqrt((((F * F) + 2.0) - (-2.0 * x))), -1.0) * (F / B));
	} else {
		tmp = (x * (-1.0 / Math.tan(B))) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -x / math.tan(B)
	tmp = 0
	if F <= -80000000000000.0:
		tmp = t_0 + (-1.0 / math.sin(B))
	elif F <= 5e-22:
		tmp = t_0 + (math.pow(math.sqrt((((F * F) + 2.0) - (-2.0 * x))), -1.0) * (F / B))
	else:
		tmp = (x * (-1.0 / math.tan(B))) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -80000000000000.0)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 5e-22)
		tmp = Float64(t_0 + Float64((sqrt(Float64(Float64(Float64(F * F) + 2.0) - Float64(-2.0 * x))) ^ -1.0) * Float64(F / B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -x / tan(B);
	tmp = 0.0;
	if (F <= -80000000000000.0)
		tmp = t_0 + (-1.0 / sin(B));
	elseif (F <= 5e-22)
		tmp = t_0 + ((sqrt((((F * F) + 2.0) - (-2.0 * x))) ^ -1.0) * (F / B));
	else
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -80000000000000.0], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5e-22], N[(t$95$0 + N[(N[Power[N[Sqrt[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] - N[(-2.0 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], -1.0], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -8e13

   1. Initial program 51.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 51.9

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

   4. Applied rewrites 51.9%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if -8e13 < F < 4.99999999999999954e-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. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 84.4%

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

   if 4.99999999999999954e-22 < F

   1. Initial program 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 71.8%

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

   5. Taylor expanded in F around inf

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

   7. Applied rewrites 98.4%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -80000000000000:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-22}:\\ \;\;\;\;\frac{-x}{\tan B} + {\left(\sqrt{\left(F \cdot F + 2\right) - -2 \cdot x}\right)}^{-1} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -80000000000000:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-22}:\\ \;\;\;\;t\_0 + \frac{F}{B} \cdot \frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -80000000000000.0)
     (+ (/ (- x) (tan B)) (/ -1.0 (sin B)))
     (if (<= F 5e-22)
       (+ t_0 (* (/ F B) (/ 1.0 (sqrt (+ (+ (* 2.0 x) 2.0) (* F F))))))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -80000000000000.0) {
		tmp = (-x / tan(B)) + (-1.0 / sin(B));
	} else if (F <= 5e-22) {
		tmp = t_0 + ((F / B) * (1.0 / sqrt((((2.0 * x) + 2.0) + (F * F)))));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-80000000000000.0d0)) then
        tmp = (-x / tan(b)) + ((-1.0d0) / sin(b))
    else if (f <= 5d-22) then
        tmp = t_0 + ((f / b) * (1.0d0 / sqrt((((2.0d0 * x) + 2.0d0) + (f * f)))))
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -80000000000000.0) {
		tmp = (-x / Math.tan(B)) + (-1.0 / Math.sin(B));
	} else if (F <= 5e-22) {
		tmp = t_0 + ((F / B) * (1.0 / Math.sqrt((((2.0 * x) + 2.0) + (F * F)))));
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -80000000000000.0:
		tmp = (-x / math.tan(B)) + (-1.0 / math.sin(B))
	elif F <= 5e-22:
		tmp = t_0 + ((F / B) * (1.0 / math.sqrt((((2.0 * x) + 2.0) + (F * F)))))
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -80000000000000.0)
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(-1.0 / sin(B)));
	elseif (F <= 5e-22)
		tmp = Float64(t_0 + Float64(Float64(F / B) * Float64(1.0 / sqrt(Float64(Float64(Float64(2.0 * x) + 2.0) + Float64(F * F))))));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -80000000000000.0)
		tmp = (-x / tan(B)) + (-1.0 / sin(B));
	elseif (F <= 5e-22)
		tmp = t_0 + ((F / B) * (1.0 / sqrt((((2.0 * x) + 2.0) + (F * F)))));
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -80000000000000.0], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5e-22], N[(t$95$0 + N[(N[(F / B), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(N[(N[(2.0 * x), $MachinePrecision] + 2.0), $MachinePrecision] + N[(F * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -8e13

   1. Initial program 51.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 51.9

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

   4. Applied rewrites 51.9%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 99.8

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

   7. Applied rewrites 99.8%

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

   if -8e13 < F < 4.99999999999999954e-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. Add Preprocessing

   3. 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. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. associate-+r+ N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 84.3

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

   5. Applied rewrites 84.3%

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

   if 4.99999999999999954e-22 < F

   1. Initial program 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 71.8%

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

   5. Taylor expanded in F around inf

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

   7. Applied rewrites 98.4%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -80000000000000:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-22}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -80000000000000:\\ \;\;\;\;t\_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-22}:\\ \;\;\;\;t\_0 + \frac{F}{B} \cdot \frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -80000000000000.0)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 5e-22)
       (+ t_0 (* (/ F B) (/ 1.0 (sqrt (+ (+ (* 2.0 x) 2.0) (* F F))))))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -80000000000000.0) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 5e-22) {
		tmp = t_0 + ((F / B) * (1.0 / sqrt((((2.0 * x) + 2.0) + (F * F)))));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-80000000000000.0d0)) then
        tmp = t_0 + ((-1.0d0) / sin(b))
    else if (f <= 5d-22) then
        tmp = t_0 + ((f / b) * (1.0d0 / sqrt((((2.0d0 * x) + 2.0d0) + (f * f)))))
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -80000000000000.0) {
		tmp = t_0 + (-1.0 / Math.sin(B));
	} else if (F <= 5e-22) {
		tmp = t_0 + ((F / B) * (1.0 / Math.sqrt((((2.0 * x) + 2.0) + (F * F)))));
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -80000000000000.0:
		tmp = t_0 + (-1.0 / math.sin(B))
	elif F <= 5e-22:
		tmp = t_0 + ((F / B) * (1.0 / math.sqrt((((2.0 * x) + 2.0) + (F * F)))))
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -80000000000000.0)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 5e-22)
		tmp = Float64(t_0 + Float64(Float64(F / B) * Float64(1.0 / sqrt(Float64(Float64(Float64(2.0 * x) + 2.0) + Float64(F * F))))));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -80000000000000.0)
		tmp = t_0 + (-1.0 / sin(B));
	elseif (F <= 5e-22)
		tmp = t_0 + ((F / B) * (1.0 / sqrt((((2.0 * x) + 2.0) + (F * F)))));
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -80000000000000.0], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5e-22], N[(t$95$0 + N[(N[(F / B), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(N[(N[(2.0 * x), $MachinePrecision] + 2.0), $MachinePrecision] + N[(F * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -8e13

   1. Initial program 51.9%

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

   3. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -8e13 < F < 4.99999999999999954e-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. Add Preprocessing

   3. 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. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. associate-+r+ N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 84.3

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

   5. Applied rewrites 84.3%

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

   if 4.99999999999999954e-22 < F

   1. Initial program 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 71.8%

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

   5. Taylor expanded in F around inf

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

   7. Applied rewrites 98.4%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -80000000000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-22}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 91.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -80000000000000:\\ \;\;\;\;t\_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-22}:\\ \;\;\;\;t\_0 + \frac{F}{B} \cdot \frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -80000000000000.0)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 5e-22)
       (+ t_0 (* (/ F B) (/ 1.0 (sqrt (+ (+ (* 2.0 x) 2.0) (* F F))))))
       (/ (- 1.0 (* (cos B) x)) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -80000000000000.0) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 5e-22) {
		tmp = t_0 + ((F / B) * (1.0 / sqrt((((2.0 * x) + 2.0) + (F * F)))));
	} else {
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-80000000000000.0d0)) then
        tmp = t_0 + ((-1.0d0) / sin(b))
    else if (f <= 5d-22) then
        tmp = t_0 + ((f / b) * (1.0d0 / sqrt((((2.0d0 * x) + 2.0d0) + (f * f)))))
    else
        tmp = (1.0d0 - (cos(b) * x)) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -80000000000000.0) {
		tmp = t_0 + (-1.0 / Math.sin(B));
	} else if (F <= 5e-22) {
		tmp = t_0 + ((F / B) * (1.0 / Math.sqrt((((2.0 * x) + 2.0) + (F * F)))));
	} else {
		tmp = (1.0 - (Math.cos(B) * x)) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -80000000000000.0:
		tmp = t_0 + (-1.0 / math.sin(B))
	elif F <= 5e-22:
		tmp = t_0 + ((F / B) * (1.0 / math.sqrt((((2.0 * x) + 2.0) + (F * F)))))
	else:
		tmp = (1.0 - (math.cos(B) * x)) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -80000000000000.0)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 5e-22)
		tmp = Float64(t_0 + Float64(Float64(F / B) * Float64(1.0 / sqrt(Float64(Float64(Float64(2.0 * x) + 2.0) + Float64(F * F))))));
	else
		tmp = Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -80000000000000.0)
		tmp = t_0 + (-1.0 / sin(B));
	elseif (F <= 5e-22)
		tmp = t_0 + ((F / B) * (1.0 / sqrt((((2.0 * x) + 2.0) + (F * F)))));
	else
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -80000000000000.0], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5e-22], N[(t$95$0 + N[(N[(F / B), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(N[(N[(2.0 * x), $MachinePrecision] + 2.0), $MachinePrecision] + N[(F * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -8e13

   1. Initial program 51.9%

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

   3. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -8e13 < F < 4.99999999999999954e-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. Add Preprocessing

   3. 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. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. associate-+r+ N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 84.3

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

   5. Applied rewrites 84.3%

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

   if 4.99999999999999954e-22 < F

   1. Initial program 56.7%

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

   3. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -80000000000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-22}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -80000000000000:\\ \;\;\;\;\frac{-1 - t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-22}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -80000000000000.0)
     (/ (- -1.0 t_0) (sin B))
     (if (<= F 5e-22)
       (+
        (* x (/ -1.0 (tan B)))
        (* (/ F B) (/ 1.0 (sqrt (+ (+ (* 2.0 x) 2.0) (* F F))))))
       (/ (- 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 <= -80000000000000.0) {
		tmp = (-1.0 - t_0) / sin(B);
	} else if (F <= 5e-22) {
		tmp = (x * (-1.0 / tan(B))) + ((F / B) * (1.0 / sqrt((((2.0 * x) + 2.0) + (F * F)))));
	} 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 <= (-80000000000000.0d0)) then
        tmp = ((-1.0d0) - t_0) / sin(b)
    else if (f <= 5d-22) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((f / b) * (1.0d0 / sqrt((((2.0d0 * x) + 2.0d0) + (f * f)))))
    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 <= -80000000000000.0) {
		tmp = (-1.0 - t_0) / Math.sin(B);
	} else if (F <= 5e-22) {
		tmp = (x * (-1.0 / Math.tan(B))) + ((F / B) * (1.0 / Math.sqrt((((2.0 * x) + 2.0) + (F * F)))));
	} 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 <= -80000000000000.0:
		tmp = (-1.0 - t_0) / math.sin(B)
	elif F <= 5e-22:
		tmp = (x * (-1.0 / math.tan(B))) + ((F / B) * (1.0 / math.sqrt((((2.0 * x) + 2.0) + (F * F)))))
	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 <= -80000000000000.0)
		tmp = Float64(Float64(-1.0 - t_0) / sin(B));
	elseif (F <= 5e-22)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / B) * Float64(1.0 / sqrt(Float64(Float64(Float64(2.0 * x) + 2.0) + Float64(F * F))))));
	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 <= -80000000000000.0)
		tmp = (-1.0 - t_0) / sin(B);
	elseif (F <= 5e-22)
		tmp = (x * (-1.0 / tan(B))) + ((F / B) * (1.0 / sqrt((((2.0 * x) + 2.0) + (F * F)))));
	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, -80000000000000.0], N[(N[(-1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5e-22], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(N[(N[(2.0 * x), $MachinePrecision] + 2.0), $MachinePrecision] + N[(F * F), $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 < -8e13

   1. Initial program 51.9%

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

   3. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -8e13 < F < 4.99999999999999954e-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. Add Preprocessing

   3. 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. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. associate-+r+ N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 84.3

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

   5. Applied rewrites 84.3%

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

   if 4.99999999999999954e-22 < F

   1. Initial program 56.7%

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

   3. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -80000000000000:\\ \;\;\;\;\frac{-1 - \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-22}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 85.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -80000000000000:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-22}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -80000000000000.0)
   (/ (- -1.0 x) (sin B))
   (if (<= F 5e-22)
     (+
      (* x (/ -1.0 (tan B)))
      (* (/ F B) (/ 1.0 (sqrt (+ (+ (* 2.0 x) 2.0) (* F F))))))
     (/ (- 1.0 (* (cos B) x)) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -80000000000000.0) {
		tmp = (-1.0 - x) / sin(B);
	} else if (F <= 5e-22) {
		tmp = (x * (-1.0 / tan(B))) + ((F / B) * (1.0 / sqrt((((2.0 * x) + 2.0) + (F * F)))));
	} else {
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-80000000000000.0d0)) then
        tmp = ((-1.0d0) - x) / sin(b)
    else if (f <= 5d-22) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((f / b) * (1.0d0 / sqrt((((2.0d0 * x) + 2.0d0) + (f * f)))))
    else
        tmp = (1.0d0 - (cos(b) * x)) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -80000000000000.0) {
		tmp = (-1.0 - x) / Math.sin(B);
	} else if (F <= 5e-22) {
		tmp = (x * (-1.0 / Math.tan(B))) + ((F / B) * (1.0 / Math.sqrt((((2.0 * x) + 2.0) + (F * F)))));
	} else {
		tmp = (1.0 - (Math.cos(B) * x)) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -80000000000000.0:
		tmp = (-1.0 - x) / math.sin(B)
	elif F <= 5e-22:
		tmp = (x * (-1.0 / math.tan(B))) + ((F / B) * (1.0 / math.sqrt((((2.0 * x) + 2.0) + (F * F)))))
	else:
		tmp = (1.0 - (math.cos(B) * x)) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -80000000000000.0)
		tmp = Float64(Float64(-1.0 - x) / sin(B));
	elseif (F <= 5e-22)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / B) * Float64(1.0 / sqrt(Float64(Float64(Float64(2.0 * x) + 2.0) + Float64(F * F))))));
	else
		tmp = Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -80000000000000.0)
		tmp = (-1.0 - x) / sin(B);
	elseif (F <= 5e-22)
		tmp = (x * (-1.0 / tan(B))) + ((F / B) * (1.0 / sqrt((((2.0 * x) + 2.0) + (F * F)))));
	else
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -80000000000000.0], N[(N[(-1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5e-22], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(N[(N[(2.0 * x), $MachinePrecision] + 2.0), $MachinePrecision] + N[(F * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -8e13

   1. Initial program 51.9%

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

   3. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 84.7%

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

   if -8e13 < F < 4.99999999999999954e-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. Add Preprocessing

   3. 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. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. associate-+r+ N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 84.3

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

   5. Applied rewrites 84.3%

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

   if 4.99999999999999954e-22 < F

   1. Initial program 56.7%

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

   3. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 98.3

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

   5. Applied rewrites 98.3%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -80000000000000:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-22}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 79.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -80000000000000:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + {\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -80000000000000.0)
   (/ (- -1.0 x) (sin B))
   (if (<= F 1.6e-7)
     (+
      (* x (/ -1.0 (tan B)))
      (* (/ F B) (/ 1.0 (sqrt (+ (+ (* 2.0 x) 2.0) (* F F))))))
     (+ (- (/ x B)) (pow (sin B) -1.0)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -80000000000000.0) {
		tmp = (-1.0 - x) / sin(B);
	} else if (F <= 1.6e-7) {
		tmp = (x * (-1.0 / tan(B))) + ((F / B) * (1.0 / sqrt((((2.0 * x) + 2.0) + (F * F)))));
	} else {
		tmp = -(x / B) + pow(sin(B), -1.0);
	}
	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) :: tmp
    if (f <= (-80000000000000.0d0)) then
        tmp = ((-1.0d0) - x) / sin(b)
    else if (f <= 1.6d-7) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((f / b) * (1.0d0 / sqrt((((2.0d0 * x) + 2.0d0) + (f * f)))))
    else
        tmp = -(x / b) + (sin(b) ** (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -80000000000000.0) {
		tmp = (-1.0 - x) / Math.sin(B);
	} else if (F <= 1.6e-7) {
		tmp = (x * (-1.0 / Math.tan(B))) + ((F / B) * (1.0 / Math.sqrt((((2.0 * x) + 2.0) + (F * F)))));
	} else {
		tmp = -(x / B) + Math.pow(Math.sin(B), -1.0);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -80000000000000.0:
		tmp = (-1.0 - x) / math.sin(B)
	elif F <= 1.6e-7:
		tmp = (x * (-1.0 / math.tan(B))) + ((F / B) * (1.0 / math.sqrt((((2.0 * x) + 2.0) + (F * F)))))
	else:
		tmp = -(x / B) + math.pow(math.sin(B), -1.0)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -80000000000000.0)
		tmp = Float64(Float64(-1.0 - x) / sin(B));
	elseif (F <= 1.6e-7)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / B) * Float64(1.0 / sqrt(Float64(Float64(Float64(2.0 * x) + 2.0) + Float64(F * F))))));
	else
		tmp = Float64(Float64(-Float64(x / B)) + (sin(B) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -80000000000000.0)
		tmp = (-1.0 - x) / sin(B);
	elseif (F <= 1.6e-7)
		tmp = (x * (-1.0 / tan(B))) + ((F / B) * (1.0 / sqrt((((2.0 * x) + 2.0) + (F * F)))));
	else
		tmp = -(x / B) + (sin(B) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -80000000000000.0], N[(N[(-1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.6e-7], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(N[(N[(2.0 * x), $MachinePrecision] + 2.0), $MachinePrecision] + N[(F * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[(x / B), $MachinePrecision]) + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -8e13

   1. Initial program 51.9%

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

   3. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 84.7%

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

   if -8e13 < F < 1.6e-7

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. associate-+r+ N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 84.0

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

   5. Applied rewrites 84.0%

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

   if 1.6e-7 < F

   1. Initial program 53.9%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 31.1

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

   5. Applied rewrites 31.1%

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

   6. Taylor expanded in F around inf

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. lift-sin.f64 76.7

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

   8. Applied rewrites 76.7%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -80000000000000:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + {\sin B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 73.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;F \leq -80000000000000:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-98}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{+222}:\\ \;\;\;\;t\_0 + \frac{F}{\sin B \cdot \sqrt{\left(F \cdot F + 2\right) - -2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{F}{\sin B} \cdot \frac{1 + -0.5 \cdot \frac{2 + 2 \cdot x}{F \cdot F}}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x B))))
   (if (<= F -80000000000000.0)
     (/ (- -1.0 x) (sin B))
     (if (<= F 7.5e-98)
       (+
        (* x (/ -1.0 (tan B)))
        (* (/ F B) (/ 1.0 (sqrt (+ (+ (* 2.0 x) 2.0) (* F F))))))
       (if (<= F 4.5e+222)
         (+ t_0 (/ F (* (sin B) (sqrt (- (+ (* F F) 2.0) (* -2.0 x))))))
         (+
          t_0
          (*
           (/ F (sin B))
           (/ (+ 1.0 (* -0.5 (/ (+ 2.0 (* 2.0 x)) (* F F)))) F))))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (F <= -80000000000000.0) {
		tmp = (-1.0 - x) / sin(B);
	} else if (F <= 7.5e-98) {
		tmp = (x * (-1.0 / tan(B))) + ((F / B) * (1.0 / sqrt((((2.0 * x) + 2.0) + (F * F)))));
	} else if (F <= 4.5e+222) {
		tmp = t_0 + (F / (sin(B) * sqrt((((F * F) + 2.0) - (-2.0 * x)))));
	} else {
		tmp = t_0 + ((F / sin(B)) * ((1.0 + (-0.5 * ((2.0 + (2.0 * x)) / (F * F)))) / 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) :: t_0
    real(8) :: tmp
    t_0 = -(x / b)
    if (f <= (-80000000000000.0d0)) then
        tmp = ((-1.0d0) - x) / sin(b)
    else if (f <= 7.5d-98) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((f / b) * (1.0d0 / sqrt((((2.0d0 * x) + 2.0d0) + (f * f)))))
    else if (f <= 4.5d+222) then
        tmp = t_0 + (f / (sin(b) * sqrt((((f * f) + 2.0d0) - ((-2.0d0) * x)))))
    else
        tmp = t_0 + ((f / sin(b)) * ((1.0d0 + ((-0.5d0) * ((2.0d0 + (2.0d0 * x)) / (f * f)))) / f))
    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 (F <= -80000000000000.0) {
		tmp = (-1.0 - x) / Math.sin(B);
	} else if (F <= 7.5e-98) {
		tmp = (x * (-1.0 / Math.tan(B))) + ((F / B) * (1.0 / Math.sqrt((((2.0 * x) + 2.0) + (F * F)))));
	} else if (F <= 4.5e+222) {
		tmp = t_0 + (F / (Math.sin(B) * Math.sqrt((((F * F) + 2.0) - (-2.0 * x)))));
	} else {
		tmp = t_0 + ((F / Math.sin(B)) * ((1.0 + (-0.5 * ((2.0 + (2.0 * x)) / (F * F)))) / F));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -(x / B)
	tmp = 0
	if F <= -80000000000000.0:
		tmp = (-1.0 - x) / math.sin(B)
	elif F <= 7.5e-98:
		tmp = (x * (-1.0 / math.tan(B))) + ((F / B) * (1.0 / math.sqrt((((2.0 * x) + 2.0) + (F * F)))))
	elif F <= 4.5e+222:
		tmp = t_0 + (F / (math.sin(B) * math.sqrt((((F * F) + 2.0) - (-2.0 * x)))))
	else:
		tmp = t_0 + ((F / math.sin(B)) * ((1.0 + (-0.5 * ((2.0 + (2.0 * x)) / (F * F)))) / F))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	tmp = 0.0
	if (F <= -80000000000000.0)
		tmp = Float64(Float64(-1.0 - x) / sin(B));
	elseif (F <= 7.5e-98)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / B) * Float64(1.0 / sqrt(Float64(Float64(Float64(2.0 * x) + 2.0) + Float64(F * F))))));
	elseif (F <= 4.5e+222)
		tmp = Float64(t_0 + Float64(F / Float64(sin(B) * sqrt(Float64(Float64(Float64(F * F) + 2.0) - Float64(-2.0 * x))))));
	else
		tmp = Float64(t_0 + Float64(Float64(F / sin(B)) * Float64(Float64(1.0 + Float64(-0.5 * Float64(Float64(2.0 + Float64(2.0 * x)) / Float64(F * F)))) / F)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -(x / B);
	tmp = 0.0;
	if (F <= -80000000000000.0)
		tmp = (-1.0 - x) / sin(B);
	elseif (F <= 7.5e-98)
		tmp = (x * (-1.0 / tan(B))) + ((F / B) * (1.0 / sqrt((((2.0 * x) + 2.0) + (F * F)))));
	elseif (F <= 4.5e+222)
		tmp = t_0 + (F / (sin(B) * sqrt((((F * F) + 2.0) - (-2.0 * x)))));
	else
		tmp = t_0 + ((F / sin(B)) * ((1.0 + (-0.5 * ((2.0 + (2.0 * x)) / (F * F)))) / F));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[F, -80000000000000.0], N[(N[(-1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.5e-98], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(N[(N[(2.0 * x), $MachinePrecision] + 2.0), $MachinePrecision] + N[(F * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.5e+222], N[(t$95$0 + N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] - N[(-2.0 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 + N[(-0.5 * N[(N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -8e13

   1. Initial program 51.9%

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

   3. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 84.7%

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

   if -8e13 < F < 7.5000000000000006e-98

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. associate-+r+ N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 85.7

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

   5. Applied rewrites 85.7%

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

   if 7.5000000000000006e-98 < F < 4.49999999999999989e222

   1. Initial program 74.9%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 54.9

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

   5. Applied rewrites 54.9%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 54.9

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

   8. Applied rewrites 54.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. pow2 N/A

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

      16. lower-sqrt.f64 N/A

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

   10. Applied rewrites 72.5%

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

   if 4.49999999999999989e222 < F

   1. Initial program 27.1%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 2.6

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

   5. Applied rewrites 2.6%

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

   6. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 60.7

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \frac{1 + -0.5 \cdot \frac{2 + 2 \cdot x}{F \cdot F}}{F} \]

   8. Applied rewrites 60.7%

      \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1 + -0.5 \cdot \frac{2 + 2 \cdot x}{F \cdot F}}{F}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -80000000000000:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-98}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{+222}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{\sin B \cdot \sqrt{\left(F \cdot F + 2\right) - -2 \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \frac{1 + -0.5 \cdot \frac{2 + 2 \cdot x}{F \cdot F}}{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 72.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -80000000000000:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-98} \lor \neg \left(F \leq 1.5 \cdot 10^{+196}\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{\sin B \cdot \sqrt{\left(F \cdot F + 2\right) - -2 \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -80000000000000.0)
   (/ (- -1.0 x) (sin B))
   (if (or (<= F 7.5e-98) (not (<= F 1.5e+196)))
     (+
      (* x (/ -1.0 (tan B)))
      (* (/ F B) (/ 1.0 (sqrt (+ (+ (* 2.0 x) 2.0) (* F F))))))
     (+ (- (/ x B)) (/ F (* (sin B) (sqrt (- (+ (* F F) 2.0) (* -2.0 x)))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -80000000000000.0) {
		tmp = (-1.0 - x) / sin(B);
	} else if ((F <= 7.5e-98) || !(F <= 1.5e+196)) {
		tmp = (x * (-1.0 / tan(B))) + ((F / B) * (1.0 / sqrt((((2.0 * x) + 2.0) + (F * F)))));
	} else {
		tmp = -(x / B) + (F / (sin(B) * sqrt((((F * F) + 2.0) - (-2.0 * x)))));
	}
	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 <= (-80000000000000.0d0)) then
        tmp = ((-1.0d0) - x) / sin(b)
    else if ((f <= 7.5d-98) .or. (.not. (f <= 1.5d+196))) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((f / b) * (1.0d0 / sqrt((((2.0d0 * x) + 2.0d0) + (f * f)))))
    else
        tmp = -(x / b) + (f / (sin(b) * sqrt((((f * f) + 2.0d0) - ((-2.0d0) * x)))))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -80000000000000.0) {
		tmp = (-1.0 - x) / Math.sin(B);
	} else if ((F <= 7.5e-98) || !(F <= 1.5e+196)) {
		tmp = (x * (-1.0 / Math.tan(B))) + ((F / B) * (1.0 / Math.sqrt((((2.0 * x) + 2.0) + (F * F)))));
	} else {
		tmp = -(x / B) + (F / (Math.sin(B) * Math.sqrt((((F * F) + 2.0) - (-2.0 * x)))));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -80000000000000.0:
		tmp = (-1.0 - x) / math.sin(B)
	elif (F <= 7.5e-98) or not (F <= 1.5e+196):
		tmp = (x * (-1.0 / math.tan(B))) + ((F / B) * (1.0 / math.sqrt((((2.0 * x) + 2.0) + (F * F)))))
	else:
		tmp = -(x / B) + (F / (math.sin(B) * math.sqrt((((F * F) + 2.0) - (-2.0 * x)))))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -80000000000000.0)
		tmp = Float64(Float64(-1.0 - x) / sin(B));
	elseif ((F <= 7.5e-98) || !(F <= 1.5e+196))
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / B) * Float64(1.0 / sqrt(Float64(Float64(Float64(2.0 * x) + 2.0) + Float64(F * F))))));
	else
		tmp = Float64(Float64(-Float64(x / B)) + Float64(F / Float64(sin(B) * sqrt(Float64(Float64(Float64(F * F) + 2.0) - Float64(-2.0 * x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -80000000000000.0)
		tmp = (-1.0 - x) / sin(B);
	elseif ((F <= 7.5e-98) || ~((F <= 1.5e+196)))
		tmp = (x * (-1.0 / tan(B))) + ((F / B) * (1.0 / sqrt((((2.0 * x) + 2.0) + (F * F)))));
	else
		tmp = -(x / B) + (F / (sin(B) * sqrt((((F * F) + 2.0) - (-2.0 * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -80000000000000.0], N[(N[(-1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 7.5e-98], N[Not[LessEqual[F, 1.5e+196]], $MachinePrecision]], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(N[(N[(2.0 * x), $MachinePrecision] + 2.0), $MachinePrecision] + N[(F * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[(x / B), $MachinePrecision]) + N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] - N[(-2.0 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -8e13

   1. Initial program 51.9%

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

   3. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 84.7%

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

   if -8e13 < F < 7.5000000000000006e-98 or 1.4999999999999999e196 < F

   1. Initial program 85.1%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. associate-+r+ N/A

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

      8. lower-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-*.f64 74.2

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

   5. Applied rewrites 74.2%

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

   if 7.5000000000000006e-98 < F < 1.4999999999999999e196

   1. Initial program 78.2%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 61.1

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 61.2

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

   8. Applied rewrites 61.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. pow2 N/A

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

      16. lower-sqrt.f64 N/A

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

   10. Applied rewrites 78.8%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -80000000000000:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-98} \lor \neg \left(F \leq 1.5 \cdot 10^{+196}\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{\sin B \cdot \sqrt{\left(F \cdot F + 2\right) - -2 \cdot x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 67.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.9 \cdot 10^{+35}:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{+116}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{2 + F \cdot F}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + -0.5 \cdot \frac{2 + 2 \cdot x}{F \cdot F}\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.9e+35)
   (/ (- -1.0 x) (sin B))
   (if (<= F 3e+116)
     (+ (- (/ x B)) (* (/ F (sin B)) (/ 1.0 (sqrt (+ 2.0 (* F F))))))
     (/ (- (+ 1.0 (* -0.5 (/ (+ 2.0 (* 2.0 x)) (* F F)))) x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.9e+35) {
		tmp = (-1.0 - x) / sin(B);
	} else if (F <= 3e+116) {
		tmp = -(x / B) + ((F / sin(B)) * (1.0 / sqrt((2.0 + (F * F)))));
	} else {
		tmp = ((1.0 + (-0.5 * ((2.0 + (2.0 * x)) / (F * F)))) - x) / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.9d+35)) then
        tmp = ((-1.0d0) - x) / sin(b)
    else if (f <= 3d+116) then
        tmp = -(x / b) + ((f / sin(b)) * (1.0d0 / sqrt((2.0d0 + (f * f)))))
    else
        tmp = ((1.0d0 + ((-0.5d0) * ((2.0d0 + (2.0d0 * x)) / (f * f)))) - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.9e+35) {
		tmp = (-1.0 - x) / Math.sin(B);
	} else if (F <= 3e+116) {
		tmp = -(x / B) + ((F / Math.sin(B)) * (1.0 / Math.sqrt((2.0 + (F * F)))));
	} else {
		tmp = ((1.0 + (-0.5 * ((2.0 + (2.0 * x)) / (F * F)))) - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.9e+35:
		tmp = (-1.0 - x) / math.sin(B)
	elif F <= 3e+116:
		tmp = -(x / B) + ((F / math.sin(B)) * (1.0 / math.sqrt((2.0 + (F * F)))))
	else:
		tmp = ((1.0 + (-0.5 * ((2.0 + (2.0 * x)) / (F * F)))) - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.9e+35)
		tmp = Float64(Float64(-1.0 - x) / sin(B));
	elseif (F <= 3e+116)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F / sin(B)) * Float64(1.0 / sqrt(Float64(2.0 + Float64(F * F))))));
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(-0.5 * Float64(Float64(2.0 + Float64(2.0 * x)) / Float64(F * F)))) - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.9e+35)
		tmp = (-1.0 - x) / sin(B);
	elseif (F <= 3e+116)
		tmp = -(x / B) + ((F / sin(B)) * (1.0 / sqrt((2.0 + (F * F)))));
	else
		tmp = ((1.0 + (-0.5 * ((2.0 + (2.0 * x)) / (F * F)))) - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.9e+35], N[(N[(-1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3e+116], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 + N[(-0.5 * N[(N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.9e35

   1. Initial program 48.9%

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

   3. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 83.8%

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

   if -1.9e35 < F < 2.9999999999999999e116

   1. Initial program 96.8%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 67.0

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

   5. Applied rewrites 67.0%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 67.0

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

   8. Applied rewrites 67.0%

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

   9. Taylor expanded in F around inf

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

       1. pow2 N/A

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

       2. lift-*.f64 67.0

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

   11. Applied rewrites 67.0%

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

   if 2.9999999999999999e116 < F

   1. Initial program 31.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 24.9%

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

   6. Taylor expanded in F around inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 35.5

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

   8. Applied rewrites 35.5%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.9 \cdot 10^{+35}:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 3 \cdot 10^{+116}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{2 + F \cdot F}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + -0.5 \cdot \frac{2 + 2 \cdot x}{F \cdot F}\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.2 \cdot 10^{+35}:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{\sin B \cdot \sqrt{\left(F \cdot F + 2\right) - -2 \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.2e+35)
   (/ (- -1.0 x) (sin B))
   (+ (- (/ x B)) (/ F (* (sin B) (sqrt (- (+ (* F F) 2.0) (* -2.0 x))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.2e+35) {
		tmp = (-1.0 - x) / sin(B);
	} else {
		tmp = -(x / B) + (F / (sin(B) * sqrt((((F * F) + 2.0) - (-2.0 * x)))));
	}
	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 <= (-3.2d+35)) then
        tmp = ((-1.0d0) - x) / sin(b)
    else
        tmp = -(x / b) + (f / (sin(b) * sqrt((((f * f) + 2.0d0) - ((-2.0d0) * x)))))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.2e+35) {
		tmp = (-1.0 - x) / Math.sin(B);
	} else {
		tmp = -(x / B) + (F / (Math.sin(B) * Math.sqrt((((F * F) + 2.0) - (-2.0 * x)))));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.2e+35:
		tmp = (-1.0 - x) / math.sin(B)
	else:
		tmp = -(x / B) + (F / (math.sin(B) * math.sqrt((((F * F) + 2.0) - (-2.0 * x)))))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.2e+35)
		tmp = Float64(Float64(-1.0 - x) / sin(B));
	else
		tmp = Float64(Float64(-Float64(x / B)) + Float64(F / Float64(sin(B) * sqrt(Float64(Float64(Float64(F * F) + 2.0) - Float64(-2.0 * x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.2e+35)
		tmp = (-1.0 - x) / sin(B);
	else
		tmp = -(x / B) + (F / (sin(B) * sqrt((((F * F) + 2.0) - (-2.0 * x)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.2e+35], N[(N[(-1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[((-N[(x / B), $MachinePrecision]) + N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] - N[(-2.0 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -3.19999999999999983e35

   1. Initial program 48.9%

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

   3. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 83.8%

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

   if -3.19999999999999983e35 < F

   1. Initial program 83.5%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 55.0

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

   5. Applied rewrites 55.0%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 55.0

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

   8. Applied rewrites 55.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. frac-times N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-*.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. pow2 N/A

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

      16. lower-sqrt.f64 N/A

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

   10. Applied rewrites 60.6%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.2 \cdot 10^{+35}:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{\sin B \cdot \sqrt{\left(F \cdot F + 2\right) - -2 \cdot x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 65.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.26:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{-12}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{\sin B \cdot \sqrt{2 \cdot x + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.26)
   (/ (- -1.0 x) (sin B))
   (if (<= F 6.8e-12)
     (+ (- (/ x B)) (/ F (* (sin B) (sqrt (+ (* 2.0 x) 2.0)))))
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.26) {
		tmp = (-1.0 - x) / sin(B);
	} else if (F <= 6.8e-12) {
		tmp = -(x / B) + (F / (sin(B) * sqrt(((2.0 * x) + 2.0))));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.26d0)) then
        tmp = ((-1.0d0) - x) / sin(b)
    else if (f <= 6.8d-12) then
        tmp = -(x / b) + (f / (sin(b) * sqrt(((2.0d0 * x) + 2.0d0))))
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.26) {
		tmp = (-1.0 - x) / Math.sin(B);
	} else if (F <= 6.8e-12) {
		tmp = -(x / B) + (F / (Math.sin(B) * Math.sqrt(((2.0 * x) + 2.0))));
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.26:
		tmp = (-1.0 - x) / math.sin(B)
	elif F <= 6.8e-12:
		tmp = -(x / B) + (F / (math.sin(B) * math.sqrt(((2.0 * x) + 2.0))))
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.26)
		tmp = Float64(Float64(-1.0 - x) / sin(B));
	elseif (F <= 6.8e-12)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(F / Float64(sin(B) * sqrt(Float64(Float64(2.0 * x) + 2.0)))));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.26)
		tmp = (-1.0 - x) / sin(B);
	elseif (F <= 6.8e-12)
		tmp = -(x / B) + (F / (sin(B) * sqrt(((2.0 * x) + 2.0))));
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.26], N[(N[(-1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.8e-12], N[((-N[(x / B), $MachinePrecision]) + N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * x), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.26000000000000001

   1. Initial program 53.0%

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

   3. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 84.5%

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

   if -1.26000000000000001 < F < 6.8000000000000001e-12

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 66.5

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

   5. Applied rewrites 66.5%

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

   6. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. pow2 N/A

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

      12. lift-*.f64 66.4

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

   8. Applied rewrites 66.4%

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

   9. Taylor expanded in F around 0

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

       1. lift-*.f64 66.4

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

   11. Applied rewrites 66.4%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. lift-sin.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. frac-times N/A

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

       6. lower-/.f64 N/A

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

       7. lower-*.f64 N/A

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

       8. lower-*.f64 N/A

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

       9. lift-sin.f64 66.5

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

       10. lift-+.f64 N/A

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

       11. +-commutative N/A

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

       12. lower-+.f64 66.5

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

   13. Applied rewrites 66.5%

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

   if 6.8000000000000001e-12 < F

   1. Initial program 54.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 38.8%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 44.8%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.26:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{-12}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{\sin B \cdot \sqrt{2 \cdot x + 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 54.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -80000000000000:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{2 + F \cdot F}} \cdot F - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -80000000000000.0)
   (/ (- -1.0 x) (sin B))
   (/ (- (* (/ 1.0 (sqrt (+ 2.0 (* F F)))) F) x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -80000000000000.0) {
		tmp = (-1.0 - x) / sin(B);
	} else {
		tmp = (((1.0 / sqrt((2.0 + (F * F)))) * F) - 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 <= (-80000000000000.0d0)) then
        tmp = ((-1.0d0) - x) / sin(b)
    else
        tmp = (((1.0d0 / sqrt((2.0d0 + (f * f)))) * f) - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -80000000000000.0) {
		tmp = (-1.0 - x) / Math.sin(B);
	} else {
		tmp = (((1.0 / Math.sqrt((2.0 + (F * F)))) * F) - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -80000000000000.0:
		tmp = (-1.0 - x) / math.sin(B)
	else:
		tmp = (((1.0 / math.sqrt((2.0 + (F * F)))) * F) - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -80000000000000.0)
		tmp = Float64(Float64(-1.0 - x) / sin(B));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / sqrt(Float64(2.0 + Float64(F * F)))) * F) - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -80000000000000.0)
		tmp = (-1.0 - x) / sin(B);
	else
		tmp = (((1.0 / sqrt((2.0 + (F * F)))) * F) - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -80000000000000.0], N[(N[(-1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / N[Sqrt[N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -8e13

   1. Initial program 51.9%

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

   3. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 84.7%

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

   if -8e13 < F

   1. Initial program 83.1%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 47.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 47.1%

      \[\leadsto \frac{\frac{1}{\sqrt{2 + F \cdot F}} \cdot F - x}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -80000000000000:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{2 + F \cdot F}} \cdot F - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 48.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -80000000000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{2 + F \cdot F}} \cdot F - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -80000000000000.0)
   (/ -1.0 (sin B))
   (/ (- (* (/ 1.0 (sqrt (+ 2.0 (* F F)))) F) x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -80000000000000.0) {
		tmp = -1.0 / sin(B);
	} else {
		tmp = (((1.0 / sqrt((2.0 + (F * F)))) * F) - 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 <= (-80000000000000.0d0)) then
        tmp = (-1.0d0) / sin(b)
    else
        tmp = (((1.0d0 / sqrt((2.0d0 + (f * f)))) * f) - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -80000000000000.0) {
		tmp = -1.0 / Math.sin(B);
	} else {
		tmp = (((1.0 / Math.sqrt((2.0 + (F * F)))) * F) - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -80000000000000.0:
		tmp = -1.0 / math.sin(B)
	else:
		tmp = (((1.0 / math.sqrt((2.0 + (F * F)))) * F) - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -80000000000000.0)
		tmp = Float64(-1.0 / sin(B));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / sqrt(Float64(2.0 + Float64(F * F)))) * F) - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -80000000000000.0)
		tmp = -1.0 / sin(B);
	else
		tmp = (((1.0 / sqrt((2.0 + (F * F)))) * F) - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -80000000000000.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / N[Sqrt[N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -8e13

   1. Initial program 51.9%

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

   3. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 67.5

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

   8. Applied rewrites 67.5%

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

   if -8e13 < F

   1. Initial program 83.1%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 47.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 47.1%

      \[\leadsto \frac{\frac{1}{\sqrt{2 + F \cdot F}} \cdot F - x}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 47.3% accurate, 6.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{+33}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{B \cdot \left(-0.16666666666666666 \cdot \left(B \cdot B\right) + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{2 + F \cdot F}} \cdot F - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.5e+33)
   (+ (- (/ x B)) (/ -1.0 (* B (+ (* -0.16666666666666666 (* B B)) 1.0))))
   (/ (- (* (/ 1.0 (sqrt (+ 2.0 (* F F)))) F) x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.5e+33) {
		tmp = -(x / B) + (-1.0 / (B * ((-0.16666666666666666 * (B * B)) + 1.0)));
	} else {
		tmp = (((1.0 / sqrt((2.0 + (F * F)))) * F) - x) / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.5d+33)) then
        tmp = -(x / b) + ((-1.0d0) / (b * (((-0.16666666666666666d0) * (b * b)) + 1.0d0)))
    else
        tmp = (((1.0d0 / sqrt((2.0d0 + (f * f)))) * f) - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.5e+33) {
		tmp = -(x / B) + (-1.0 / (B * ((-0.16666666666666666 * (B * B)) + 1.0)));
	} else {
		tmp = (((1.0 / Math.sqrt((2.0 + (F * F)))) * F) - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.5e+33:
		tmp = -(x / B) + (-1.0 / (B * ((-0.16666666666666666 * (B * B)) + 1.0)))
	else:
		tmp = (((1.0 / math.sqrt((2.0 + (F * F)))) * F) - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.5e+33)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(-1.0 / Float64(B * Float64(Float64(-0.16666666666666666 * Float64(B * B)) + 1.0))));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / sqrt(Float64(2.0 + Float64(F * F)))) * F) - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.5e+33)
		tmp = -(x / B) + (-1.0 / (B * ((-0.16666666666666666 * (B * B)) + 1.0)));
	else
		tmp = (((1.0 / sqrt((2.0 + (F * F)))) * F) - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.5e+33], N[((-N[(x / B), $MachinePrecision]) + N[(-1.0 / N[(B * N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / N[Sqrt[N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -2.49999999999999986e33

   1. Initial program 50.1%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 33.5

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

   5. Applied rewrites 33.5%

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

   6. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 83.4

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

   8. Applied rewrites 83.4%

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

   9. Taylor expanded in B around 0

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

       1. lower-*.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-+.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. unpow2 N/A

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

       6. lower-*.f64 57.1

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

   11. Applied rewrites 57.1%

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

   if -2.49999999999999986e33 < F

   1. Initial program 83.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 46.4%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 46.4%

      \[\leadsto \frac{\frac{1}{\sqrt{2 + F \cdot F}} \cdot F - x}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 50.7% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.86:\\ \;\;\;\;\frac{\left(0.5 \cdot \frac{2 + 2 \cdot x}{F \cdot F} - 1\right) - x}{B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{2}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.86)
   (/ (- (- (* 0.5 (/ (+ 2.0 (* 2.0 x)) (* F F))) 1.0) x) B)
   (if (<= F 2.6e-18) (/ (- (* (/ 1.0 (sqrt 2.0)) F) x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.86) {
		tmp = (((0.5 * ((2.0 + (2.0 * x)) / (F * F))) - 1.0) - x) / B;
	} else if (F <= 2.6e-18) {
		tmp = (((1.0 / sqrt(2.0)) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.86d0)) then
        tmp = (((0.5d0 * ((2.0d0 + (2.0d0 * x)) / (f * f))) - 1.0d0) - x) / b
    else if (f <= 2.6d-18) then
        tmp = (((1.0d0 / sqrt(2.0d0)) * f) - x) / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.86) {
		tmp = (((0.5 * ((2.0 + (2.0 * x)) / (F * F))) - 1.0) - x) / B;
	} else if (F <= 2.6e-18) {
		tmp = (((1.0 / Math.sqrt(2.0)) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.86:
		tmp = (((0.5 * ((2.0 + (2.0 * x)) / (F * F))) - 1.0) - x) / B
	elif F <= 2.6e-18:
		tmp = (((1.0 / math.sqrt(2.0)) * F) - x) / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.86)
		tmp = Float64(Float64(Float64(Float64(0.5 * Float64(Float64(2.0 + Float64(2.0 * x)) / Float64(F * F))) - 1.0) - x) / B);
	elseif (F <= 2.6e-18)
		tmp = Float64(Float64(Float64(Float64(1.0 / sqrt(2.0)) * F) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.86)
		tmp = (((0.5 * ((2.0 + (2.0 * x)) / (F * F))) - 1.0) - x) / B;
	elseif (F <= 2.6e-18)
		tmp = (((1.0 / sqrt(2.0)) * F) - x) / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.86], N[(N[(N[(N[(0.5 * N[(N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.6e-18], N[(N[(N[(N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.8600000000000001

   1. Initial program 53.0%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 37.5%

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

   6. Taylor expanded in F around -inf

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

      1. metadata-eval N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 56.0

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

   8. Applied rewrites 56.0%

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

   if -1.8600000000000001 < F < 2.6e-18

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 51.0%

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

   6. Taylor expanded in F around 0

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

      1. lower-+.f64 N/A

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

      2. lift-*.f64 51.0

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

   8. Applied rewrites 51.0%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 51.0%

       \[\leadsto \frac{\frac{1}{\sqrt{2}} \cdot F - x}{B} \]

   if 2.6e-18 < F

   1. Initial program 56.0%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 39.2%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.0%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.86:\\ \;\;\;\;\frac{\left(0.5 \cdot \frac{2 + 2 \cdot x}{F \cdot F} - 1\right) - x}{B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{2}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 50.6% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.7:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{B \cdot \left(-0.16666666666666666 \cdot \left(B \cdot B\right) + 1\right)}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{2}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.7)
   (+ (- (/ x B)) (/ -1.0 (* B (+ (* -0.16666666666666666 (* B B)) 1.0))))
   (if (<= F 2.6e-18) (/ (- (* (/ 1.0 (sqrt 2.0)) F) x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.7) {
		tmp = -(x / B) + (-1.0 / (B * ((-0.16666666666666666 * (B * B)) + 1.0)));
	} else if (F <= 2.6e-18) {
		tmp = (((1.0 / sqrt(2.0)) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.7d0)) then
        tmp = -(x / b) + ((-1.0d0) / (b * (((-0.16666666666666666d0) * (b * b)) + 1.0d0)))
    else if (f <= 2.6d-18) then
        tmp = (((1.0d0 / sqrt(2.0d0)) * f) - x) / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.7) {
		tmp = -(x / B) + (-1.0 / (B * ((-0.16666666666666666 * (B * B)) + 1.0)));
	} else if (F <= 2.6e-18) {
		tmp = (((1.0 / Math.sqrt(2.0)) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.7:
		tmp = -(x / B) + (-1.0 / (B * ((-0.16666666666666666 * (B * B)) + 1.0)))
	elif F <= 2.6e-18:
		tmp = (((1.0 / math.sqrt(2.0)) * F) - x) / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.7)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(-1.0 / Float64(B * Float64(Float64(-0.16666666666666666 * Float64(B * B)) + 1.0))));
	elseif (F <= 2.6e-18)
		tmp = Float64(Float64(Float64(Float64(1.0 / sqrt(2.0)) * F) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.7)
		tmp = -(x / B) + (-1.0 / (B * ((-0.16666666666666666 * (B * B)) + 1.0)));
	elseif (F <= 2.6e-18)
		tmp = (((1.0 / sqrt(2.0)) * F) - x) / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.7], N[((-N[(x / B), $MachinePrecision]) + N[(-1.0 / N[(B * N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.6e-18], N[(N[(N[(N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.69999999999999996

   1. Initial program 53.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   4. Step-by-step derivation

      1. lower-/.f64 37.4

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Applied rewrites 37.4%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{-1}{\color{blue}{\sin B}} \]

      2. lift-sin.f64 83.8

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{-1}{\sin B} \]

   8. Applied rewrites 83.8%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(-\frac{x}{B}\right) + \frac{-1}{B \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {B}^{2}\right)}} \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \left(-\frac{x}{B}\right) + \frac{-1}{B \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {B}^{2}}\right)} \]

       2. +-commutative N/A

          \[\leadsto \left(-\frac{x}{B}\right) + \frac{-1}{B \cdot \left(\frac{-1}{6} \cdot {B}^{2} + 1\right)} \]

       3. lower-+.f64 N/A

          \[\leadsto \left(-\frac{x}{B}\right) + \frac{-1}{B \cdot \left(\frac{-1}{6} \cdot {B}^{2} + 1\right)} \]

       4. lower-*.f64 N/A

          \[\leadsto \left(-\frac{x}{B}\right) + \frac{-1}{B \cdot \left(\frac{-1}{6} \cdot {B}^{2} + 1\right)} \]

       5. unpow2 N/A

          \[\leadsto \left(-\frac{x}{B}\right) + \frac{-1}{B \cdot \left(\frac{-1}{6} \cdot \left(B \cdot B\right) + 1\right)} \]

       6. lower-*.f64 55.7

          \[\leadsto \left(-\frac{x}{B}\right) + \frac{-1}{B \cdot \left(-0.16666666666666666 \cdot \left(B \cdot B\right) + 1\right)} \]

   11. Applied rewrites 55.7%

       \[\leadsto \left(-\frac{x}{B}\right) + \frac{-1}{B \cdot \color{blue}{\left(-0.16666666666666666 \cdot \left(B \cdot B\right) + 1\right)}} \]

   if -1.69999999999999996 < F < 2.6e-18

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{\frac{1}{\sqrt{2 + 2 \cdot x}} \cdot F - x}{B} \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{2 + 2 \cdot x}} \cdot F - x}{B} \]

      2. lift-*.f64 51.0

         \[\leadsto \frac{\frac{1}{\sqrt{2 + 2 \cdot x}} \cdot F - x}{B} \]

   8. Applied rewrites 51.0%

      \[\leadsto \frac{\frac{1}{\sqrt{2 + 2 \cdot x}} \cdot F - x}{B} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{\sqrt{2}} \cdot F - x}{B} \]
   10. Step-by-step derivation

   11. Applied rewrites 51.0%

       \[\leadsto \frac{\frac{1}{\sqrt{2}} \cdot F - x}{B} \]

   if 2.6e-18 < F

   1. Initial program 56.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 39.2%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.0%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.7:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{B \cdot \left(-0.16666666666666666 \cdot \left(B \cdot B\right) + 1\right)}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{2}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 49.6% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.55 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{-1 \cdot -1 - x \cdot x}{-1 + x}}{B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{2}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.55e-13)
   (/ (/ (- (* -1.0 -1.0) (* x x)) (+ -1.0 x)) B)
   (if (<= F 2.6e-18) (/ (- (* (/ 1.0 (sqrt 2.0)) F) x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.55e-13) {
		tmp = (((-1.0 * -1.0) - (x * x)) / (-1.0 + x)) / B;
	} else if (F <= 2.6e-18) {
		tmp = (((1.0 / sqrt(2.0)) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.55d-13)) then
        tmp = ((((-1.0d0) * (-1.0d0)) - (x * x)) / ((-1.0d0) + x)) / b
    else if (f <= 2.6d-18) then
        tmp = (((1.0d0 / sqrt(2.0d0)) * f) - x) / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.55e-13) {
		tmp = (((-1.0 * -1.0) - (x * x)) / (-1.0 + x)) / B;
	} else if (F <= 2.6e-18) {
		tmp = (((1.0 / Math.sqrt(2.0)) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.55e-13:
		tmp = (((-1.0 * -1.0) - (x * x)) / (-1.0 + x)) / B
	elif F <= 2.6e-18:
		tmp = (((1.0 / math.sqrt(2.0)) * F) - x) / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.55e-13)
		tmp = Float64(Float64(Float64(Float64(-1.0 * -1.0) - Float64(x * x)) / Float64(-1.0 + x)) / B);
	elseif (F <= 2.6e-18)
		tmp = Float64(Float64(Float64(Float64(1.0 / sqrt(2.0)) * F) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.55e-13)
		tmp = (((-1.0 * -1.0) - (x * x)) / (-1.0 + x)) / B;
	elseif (F <= 2.6e-18)
		tmp = (((1.0 / sqrt(2.0)) * F) - x) / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.55e-13], N[(N[(N[(N[(-1.0 * -1.0), $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.6e-18], N[(N[(N[(N[(1.0 / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.55e-13

   1. Initial program 54.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 37.5%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.0%

      \[\leadsto \frac{-1 - x}{B} \]
   9. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{-1 - x}{B} \]

      2. flip-- N/A

         \[\leadsto \frac{\frac{-1 \cdot -1 - x \cdot x}{-1 + x}}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1 \cdot -1 - x \cdot x}{-1 + x}}{B} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{\frac{-1 \cdot -1 - x \cdot x}{-1 + x}}{B} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1 \cdot -1 - x \cdot x}{-1 + x}}{B} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1 \cdot -1 - x \cdot x}{-1 + x}}{B} \]

      7. lower-+.f64 55.1

         \[\leadsto \frac{\frac{-1 \cdot -1 - x \cdot x}{-1 + x}}{B} \]

   10. Applied rewrites 55.1%

       \[\leadsto \frac{\frac{-1 \cdot -1 - x \cdot x}{-1 + x}}{B} \]

   if -1.55e-13 < F < 2.6e-18

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 51.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{\frac{1}{\sqrt{2 + 2 \cdot x}} \cdot F - x}{B} \]
   7. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{\frac{1}{\sqrt{2 + 2 \cdot x}} \cdot F - x}{B} \]

      2. lift-*.f64 51.4

         \[\leadsto \frac{\frac{1}{\sqrt{2 + 2 \cdot x}} \cdot F - x}{B} \]

   8. Applied rewrites 51.4%

      \[\leadsto \frac{\frac{1}{\sqrt{2 + 2 \cdot x}} \cdot F - x}{B} \]

   9. Taylor expanded in x around 0

      \[\leadsto \frac{\frac{1}{\sqrt{2}} \cdot F - x}{B} \]
   10. Step-by-step derivation

   11. Applied rewrites 51.4%

       \[\leadsto \frac{\frac{1}{\sqrt{2}} \cdot F - x}{B} \]

   if 2.6e-18 < F

   1. Initial program 56.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 39.2%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 45.0%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.55 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{-1 \cdot -1 - x \cdot x}{-1 + x}}{B}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{2}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 42.6% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.9 \cdot 10^{-168}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 3.3 \cdot 10^{-102}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.9e-168)
   (/ (- -1.0 x) B)
   (if (<= F 3.3e-102) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.9e-168) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.3e-102) {
		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 <= (-3.9d-168)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 3.3d-102) 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 <= -3.9e-168) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 3.3e-102) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.9e-168:
		tmp = (-1.0 - x) / B
	elif F <= 3.3e-102:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.9e-168)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 3.3e-102)
		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 <= -3.9e-168)
		tmp = (-1.0 - x) / B;
	elseif (F <= 3.3e-102)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.9e-168], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3.3e-102], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.90000000000000012e-168

   1. Initial program 63.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 37.8%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.9%

      \[\leadsto \frac{-1 - x}{B} \]

   if -3.90000000000000012e-168 < F < 3.3e-102

   1. Initial program 99.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 54.0%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 25.7%

      \[\leadsto \frac{-1 - x}{B} \]

   9. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

       2. lower-neg.f64 46.1

          \[\leadsto \frac{-x}{B} \]

   11. Applied rewrites 46.1%

       \[\leadsto \frac{-x}{B} \]

   if 3.3e-102 < F

   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. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 42.4%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 40.2%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 35.3% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.9 \cdot 10^{-168}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.9e-168) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.9e-168) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3.9d-168)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.9e-168) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.9e-168:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.9e-168)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.9e-168)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.9e-168], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -3.90000000000000012e-168

   1. Initial program 63.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 37.8%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.9%

      \[\leadsto \frac{-1 - x}{B} \]

   if -3.90000000000000012e-168 < F

   1. Initial program 80.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   5. Applied rewrites 47.8%

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 24.8%

      \[\leadsto \frac{-1 - x}{B} \]

   9. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

       2. lower-neg.f64 34.5

          \[\leadsto \frac{-x}{B} \]

   11. Applied rewrites 34.5%

       \[\leadsto \frac{-x}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 29.0% accurate, 26.3× speedup

Math

\[\begin{array}{l} \\ \frac{-x}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ (- x) B))

C

double code(double F, double B, double x) {
	return -x / B;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -x / b
end function

Java

public static double code(double F, double B, double x) {
	return -x / B;
}

Python

def code(F, B, x):
	return -x / B

Julia

function code(F, B, x)
	return Float64(Float64(-x) / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -x / B;
end

Wolfram

code[F_, B_, x_] := N[((-x) / B), $MachinePrecision]

Derivation

1. Initial program 72.8%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

5. Applied rewrites 43.5%

   \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\left(2 \cdot x + 2\right) + F \cdot F}} \cdot F - x}{B}} \]

6. Taylor expanded in F around -inf

   \[\leadsto \frac{-1 - x}{B} \]
7. Step-by-step derivation

8. Applied rewrites 34.8%

   \[\leadsto \frac{-1 - x}{B} \]

9. Taylor expanded in F around 0

   \[\leadsto \frac{-1 \cdot x}{B} \]
10. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

    2. lower-neg.f64 27.5

       \[\leadsto \frac{-x}{B} \]

11. Applied rewrites 27.5%

    \[\leadsto \frac{-x}{B} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025058 
(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))))))