VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.9% → 99.6%

Time: 5.8s

Alternatives: 14

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x \cdot 1}{\tan B}\\ \mathbf{if}\;F \leq -4 \cdot 10^{+54}:\\ \;\;\;\;-\frac{1 + \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 102000000:\\ \;\;\;\;t\_0 + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + {\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ (* x 1.0) (tan B)))))
   (if (<= F -4e+54)
     (- (/ (+ 1.0 (* (cos B) x)) (sin B)))
     (if (<= F 102000000.0)
       (+
        t_0
        (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
       (+ t_0 (pow (sin B) -1.0))))))

C

double code(double F, double B, double x) {
	double t_0 = -((x * 1.0) / tan(B));
	double tmp;
	if (F <= -4e+54) {
		tmp = -((1.0 + (cos(B) * x)) / sin(B));
	} else if (F <= 102000000.0) {
		tmp = t_0 + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = t_0 + 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) :: t_0
    real(8) :: tmp
    t_0 = -((x * 1.0d0) / tan(b))
    if (f <= (-4d+54)) then
        tmp = -((1.0d0 + (cos(b) * x)) / sin(b))
    else if (f <= 102000000.0d0) then
        tmp = t_0 + ((f / sin(b)) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
    else
        tmp = t_0 + (sin(b) ** (-1.0d0))
    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 <= -4e+54) {
		tmp = -((1.0 + (Math.cos(B) * x)) / Math.sin(B));
	} else if (F <= 102000000.0) {
		tmp = t_0 + ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = t_0 + Math.pow(Math.sin(B), -1.0);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -((x * 1.0) / math.tan(B))
	tmp = 0
	if F <= -4e+54:
		tmp = -((1.0 + (math.cos(B) * x)) / math.sin(B))
	elif F <= 102000000.0:
		tmp = t_0 + ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))
	else:
		tmp = t_0 + math.pow(math.sin(B), -1.0)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(Float64(x * 1.0) / tan(B)))
	tmp = 0.0
	if (F <= -4e+54)
		tmp = Float64(-Float64(Float64(1.0 + Float64(cos(B) * x)) / sin(B)));
	elseif (F <= 102000000.0)
		tmp = Float64(t_0 + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))));
	else
		tmp = Float64(t_0 + (sin(B) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -((x * 1.0) / tan(B));
	tmp = 0.0;
	if (F <= -4e+54)
		tmp = -((1.0 + (cos(B) * x)) / sin(B));
	elseif (F <= 102000000.0)
		tmp = t_0 + ((F / sin(B)) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
	else
		tmp = t_0 + (sin(B) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(N[(x * 1.0), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision])}, If[LessEqual[F, -4e+54], (-N[(N[(1.0 + N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 102000000.0], N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.0000000000000003e54

   1. Initial program 53.2%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if -4.0000000000000003e54 < F < 1.02e8

   1. Initial program 99.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 99.4

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

   3. Applied rewrites 99.4%

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

   if 1.02e8 < F

   1. Initial program 56.8%

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

   2. Taylor expanded in F around inf

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 99.8

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

   6. Applied rewrites 99.8%

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

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.5 \cdot 10^{+50}:\\ \;\;\;\;-\frac{1 + \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 102000000:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + {\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.5e+50)
   (- (/ (+ 1.0 (* (cos B) x)) (sin B)))
   (if (<= F 102000000.0)
     (+
      (- (* x (/ 1.0 (tan B))))
      (* (/ F (sin B)) (sqrt (pow (fma F F 2.0) -1.0))))
     (+ (- (/ (* x 1.0) (tan B))) (pow (sin B) -1.0)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.5e+50) {
		tmp = -((1.0 + (cos(B) * x)) / sin(B));
	} else if (F <= 102000000.0) {
		tmp = -(x * (1.0 / tan(B))) + ((F / sin(B)) * sqrt(pow(fma(F, F, 2.0), -1.0)));
	} else {
		tmp = -((x * 1.0) / tan(B)) + pow(sin(B), -1.0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.5e+50)
		tmp = Float64(-Float64(Float64(1.0 + Float64(cos(B) * x)) / sin(B)));
	elseif (F <= 102000000.0)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / sin(B)) * sqrt((fma(F, F, 2.0) ^ -1.0))));
	else
		tmp = Float64(Float64(-Float64(Float64(x * 1.0) / tan(B))) + (sin(B) ^ -1.0));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.5e+50], (-N[(N[(1.0 + N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 102000000.0], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[(N[(x * 1.0), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]) + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.5000000000000003e50

   1. Initial program 53.6%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if -6.5000000000000003e50 < F < 1.02e8

   1. Initial program 99.3%

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

   2. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. inv-pow N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 99.2

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

   4. Applied rewrites 99.2%

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

   if 1.02e8 < F

   1. Initial program 56.8%

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

   2. Taylor expanded in F around inf

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 99.8

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

   6. Applied rewrites 99.8%

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

Alternative 3: 99.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.95:\\ \;\;\;\;-\frac{1 + \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\mathsf{fma}\left(-x, {\tan B}^{-1}, \sqrt{0.5} \cdot \frac{F}{\sin B}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + {\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.95)
   (- (/ (+ 1.0 (* (cos B) x)) (sin B)))
   (if (<= F 1.4)
     (fma (- x) (pow (tan B) -1.0) (* (sqrt 0.5) (/ F (sin B))))
     (+ (- (/ (* x 1.0) (tan B))) (pow (sin B) -1.0)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.95) {
		tmp = -((1.0 + (cos(B) * x)) / sin(B));
	} else if (F <= 1.4) {
		tmp = fma(-x, pow(tan(B), -1.0), (sqrt(0.5) * (F / sin(B))));
	} else {
		tmp = -((x * 1.0) / tan(B)) + pow(sin(B), -1.0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.95)
		tmp = Float64(-Float64(Float64(1.0 + Float64(cos(B) * x)) / sin(B)));
	elseif (F <= 1.4)
		tmp = fma(Float64(-x), (tan(B) ^ -1.0), Float64(sqrt(0.5) * Float64(F / sin(B))));
	else
		tmp = Float64(Float64(-Float64(Float64(x * 1.0) / tan(B))) + (sin(B) ^ -1.0));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.95], (-N[(N[(1.0 + N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 1.4], N[((-x) * N[Power[N[Tan[B], $MachinePrecision], -1.0], $MachinePrecision] + N[(N[Sqrt[0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[(N[(x * 1.0), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]) + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.94999999999999996

   1. Initial program 60.0%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.2

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

   4. Applied rewrites 99.2%

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

   if -1.94999999999999996 < F < 1.3999999999999999

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. inv-pow N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in F around 0

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

   7. Applied rewrites 98.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \sqrt{0.5} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

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

      2. lift-+.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tan.f64 N/A

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

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

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

   9. Applied rewrites 98.8%

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

   if 1.3999999999999999 < F

   1. Initial program 57.7%

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

   2. Taylor expanded in F around inf

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. lift-sin.f64 99.1

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

   4. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 99.2

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

   6. Applied rewrites 99.2%

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

Alternative 4: 86.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.125:\\ \;\;\;\;-\frac{1 + \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq -5.8 \cdot 10^{-253}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-20}:\\ \;\;\;\;-\cos B \cdot \frac{x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + {\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.125)
   (- (/ (+ 1.0 (* (cos B) x)) (sin B)))
   (if (<= F -5.8e-253)
     (+ (- (/ x B)) (* (/ F (sin B)) (sqrt 0.5)))
     (if (<= F 1.4e-20)
       (- (* (cos B) (/ x (sin B))))
       (+ (- (/ (* x 1.0) (tan B))) (pow (sin B) -1.0))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.125) {
		tmp = -((1.0 + (cos(B) * x)) / sin(B));
	} else if (F <= -5.8e-253) {
		tmp = -(x / B) + ((F / sin(B)) * sqrt(0.5));
	} else if (F <= 1.4e-20) {
		tmp = -(cos(B) * (x / sin(B)));
	} else {
		tmp = -((x * 1.0) / tan(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 <= (-0.125d0)) then
        tmp = -((1.0d0 + (cos(b) * x)) / sin(b))
    else if (f <= (-5.8d-253)) then
        tmp = -(x / b) + ((f / sin(b)) * sqrt(0.5d0))
    else if (f <= 1.4d-20) then
        tmp = -(cos(b) * (x / sin(b)))
    else
        tmp = -((x * 1.0d0) / tan(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 <= -0.125) {
		tmp = -((1.0 + (Math.cos(B) * x)) / Math.sin(B));
	} else if (F <= -5.8e-253) {
		tmp = -(x / B) + ((F / Math.sin(B)) * Math.sqrt(0.5));
	} else if (F <= 1.4e-20) {
		tmp = -(Math.cos(B) * (x / Math.sin(B)));
	} else {
		tmp = -((x * 1.0) / Math.tan(B)) + Math.pow(Math.sin(B), -1.0);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.125:
		tmp = -((1.0 + (math.cos(B) * x)) / math.sin(B))
	elif F <= -5.8e-253:
		tmp = -(x / B) + ((F / math.sin(B)) * math.sqrt(0.5))
	elif F <= 1.4e-20:
		tmp = -(math.cos(B) * (x / math.sin(B)))
	else:
		tmp = -((x * 1.0) / math.tan(B)) + math.pow(math.sin(B), -1.0)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.125)
		tmp = Float64(-Float64(Float64(1.0 + Float64(cos(B) * x)) / sin(B)));
	elseif (F <= -5.8e-253)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F / sin(B)) * sqrt(0.5)));
	elseif (F <= 1.4e-20)
		tmp = Float64(-Float64(cos(B) * Float64(x / sin(B))));
	else
		tmp = Float64(Float64(-Float64(Float64(x * 1.0) / tan(B))) + (sin(B) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.125)
		tmp = -((1.0 + (cos(B) * x)) / sin(B));
	elseif (F <= -5.8e-253)
		tmp = -(x / B) + ((F / sin(B)) * sqrt(0.5));
	elseif (F <= 1.4e-20)
		tmp = -(cos(B) * (x / sin(B)));
	else
		tmp = -((x * 1.0) / tan(B)) + (sin(B) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.125], (-N[(N[(1.0 + N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, -5.8e-253], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4e-20], (-N[(N[Cos[B], $MachinePrecision] * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[((-N[(N[(x * 1.0), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]) + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.125

   1. Initial program 60.0%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.2

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

   4. Applied rewrites 99.2%

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

   if -0.125 < F < -5.7999999999999996e-253

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. inv-pow N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in F around 0

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in B around 0

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

      1. lift-/.f64 69.9

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

   10. Applied rewrites 69.9%

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

   if -5.7999999999999996e-253 < F < 1.4000000000000001e-20

   1. Initial program 99.4%

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

   2. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 72.4

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

   4. Applied rewrites 72.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sin.f64 72.4

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

   6. Applied rewrites 72.4%

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

   if 1.4000000000000001e-20 < F

   1. Initial program 59.5%

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

   2. Taylor expanded in F around inf

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. lift-sin.f64 96.3

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

   4. Applied rewrites 96.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 96.3

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

   6. Applied rewrites 96.3%

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

Alternative 5: 86.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.125:\\ \;\;\;\;-\frac{1 + \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq -5.8 \cdot 10^{-253}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{-21}:\\ \;\;\;\;-\cos B \cdot \frac{x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + {\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.125)
   (- (/ (+ 1.0 (* (cos B) x)) (sin B)))
   (if (<= F -5.8e-253)
     (+ (- (/ x B)) (* (/ F (sin B)) (sqrt 0.5)))
     (if (<= F 1.5e-21)
       (- (* (cos B) (/ x (sin B))))
       (+ (- (* x (/ 1.0 (tan B)))) (pow (sin B) -1.0))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.125) {
		tmp = -((1.0 + (cos(B) * x)) / sin(B));
	} else if (F <= -5.8e-253) {
		tmp = -(x / B) + ((F / sin(B)) * sqrt(0.5));
	} else if (F <= 1.5e-21) {
		tmp = -(cos(B) * (x / sin(B)));
	} else {
		tmp = -(x * (1.0 / tan(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 <= (-0.125d0)) then
        tmp = -((1.0d0 + (cos(b) * x)) / sin(b))
    else if (f <= (-5.8d-253)) then
        tmp = -(x / b) + ((f / sin(b)) * sqrt(0.5d0))
    else if (f <= 1.5d-21) then
        tmp = -(cos(b) * (x / sin(b)))
    else
        tmp = -(x * (1.0d0 / tan(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 <= -0.125) {
		tmp = -((1.0 + (Math.cos(B) * x)) / Math.sin(B));
	} else if (F <= -5.8e-253) {
		tmp = -(x / B) + ((F / Math.sin(B)) * Math.sqrt(0.5));
	} else if (F <= 1.5e-21) {
		tmp = -(Math.cos(B) * (x / Math.sin(B)));
	} else {
		tmp = -(x * (1.0 / Math.tan(B))) + Math.pow(Math.sin(B), -1.0);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.125:
		tmp = -((1.0 + (math.cos(B) * x)) / math.sin(B))
	elif F <= -5.8e-253:
		tmp = -(x / B) + ((F / math.sin(B)) * math.sqrt(0.5))
	elif F <= 1.5e-21:
		tmp = -(math.cos(B) * (x / math.sin(B)))
	else:
		tmp = -(x * (1.0 / math.tan(B))) + math.pow(math.sin(B), -1.0)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.125)
		tmp = Float64(-Float64(Float64(1.0 + Float64(cos(B) * x)) / sin(B)));
	elseif (F <= -5.8e-253)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F / sin(B)) * sqrt(0.5)));
	elseif (F <= 1.5e-21)
		tmp = Float64(-Float64(cos(B) * Float64(x / sin(B))));
	else
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + (sin(B) ^ -1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.125)
		tmp = -((1.0 + (cos(B) * x)) / sin(B));
	elseif (F <= -5.8e-253)
		tmp = -(x / B) + ((F / sin(B)) * sqrt(0.5));
	elseif (F <= 1.5e-21)
		tmp = -(cos(B) * (x / sin(B)));
	else
		tmp = -(x * (1.0 / tan(B))) + (sin(B) ^ -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.125], (-N[(N[(1.0 + N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, -5.8e-253], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.5e-21], (-N[(N[Cos[B], $MachinePrecision] * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.125

   1. Initial program 60.0%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.2

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

   4. Applied rewrites 99.2%

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

   if -0.125 < F < -5.7999999999999996e-253

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. inv-pow N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in F around 0

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in B around 0

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

      1. lift-/.f64 69.9

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

   10. Applied rewrites 69.9%

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

   if -5.7999999999999996e-253 < F < 1.49999999999999996e-21

   1. Initial program 99.4%

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

   2. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 72.4

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

   4. Applied rewrites 72.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sin.f64 72.4

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

   6. Applied rewrites 72.4%

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

   if 1.49999999999999996e-21 < F

   1. Initial program 59.5%

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

   2. Taylor expanded in F around inf

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. lift-sin.f64 96.3

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

   4. Applied rewrites 96.3%

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

Alternative 6: 86.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -0.125:\\ \;\;\;\;-\frac{1 + t\_0}{\sin B}\\ \mathbf{elif}\;F \leq -5.8 \cdot 10^{-253}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-14}:\\ \;\;\;\;-\cos B \cdot \frac{x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -0.125)
     (- (/ (+ 1.0 t_0) (sin B)))
     (if (<= F -5.8e-253)
       (+ (- (/ x B)) (* (/ F (sin B)) (sqrt 0.5)))
       (if (<= F 2e-14)
         (- (* (cos B) (/ x (sin B))))
         (/ (- 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 <= -0.125) {
		tmp = -((1.0 + t_0) / sin(B));
	} else if (F <= -5.8e-253) {
		tmp = -(x / B) + ((F / sin(B)) * sqrt(0.5));
	} else if (F <= 2e-14) {
		tmp = -(cos(B) * (x / sin(B)));
	} 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 <= (-0.125d0)) then
        tmp = -((1.0d0 + t_0) / sin(b))
    else if (f <= (-5.8d-253)) then
        tmp = -(x / b) + ((f / sin(b)) * sqrt(0.5d0))
    else if (f <= 2d-14) then
        tmp = -(cos(b) * (x / sin(b)))
    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 <= -0.125) {
		tmp = -((1.0 + t_0) / Math.sin(B));
	} else if (F <= -5.8e-253) {
		tmp = -(x / B) + ((F / Math.sin(B)) * Math.sqrt(0.5));
	} else if (F <= 2e-14) {
		tmp = -(Math.cos(B) * (x / Math.sin(B)));
	} 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 <= -0.125:
		tmp = -((1.0 + t_0) / math.sin(B))
	elif F <= -5.8e-253:
		tmp = -(x / B) + ((F / math.sin(B)) * math.sqrt(0.5))
	elif F <= 2e-14:
		tmp = -(math.cos(B) * (x / math.sin(B)))
	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 <= -0.125)
		tmp = Float64(-Float64(Float64(1.0 + t_0) / sin(B)));
	elseif (F <= -5.8e-253)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F / sin(B)) * sqrt(0.5)));
	elseif (F <= 2e-14)
		tmp = Float64(-Float64(cos(B) * Float64(x / sin(B))));
	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 <= -0.125)
		tmp = -((1.0 + t_0) / sin(B));
	elseif (F <= -5.8e-253)
		tmp = -(x / B) + ((F / sin(B)) * sqrt(0.5));
	elseif (F <= 2e-14)
		tmp = -(cos(B) * (x / sin(B)));
	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, -0.125], (-N[(N[(1.0 + t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, -5.8e-253], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2e-14], (-N[(N[Cos[B], $MachinePrecision] * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.125

   1. Initial program 60.0%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.2

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

   4. Applied rewrites 99.2%

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

   if -0.125 < F < -5.7999999999999996e-253

   1. Initial program 99.5%

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

   2. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. inv-pow N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 99.3

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

   4. Applied rewrites 99.3%

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

   5. Taylor expanded in F around 0

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

   7. Applied rewrites 98.9%

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

   8. Taylor expanded in B around 0

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

      1. lift-/.f64 69.9

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

   10. Applied rewrites 69.9%

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

   if -5.7999999999999996e-253 < F < 2e-14

   1. Initial program 99.4%

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

   2. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 71.9

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

   4. Applied rewrites 71.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sin.f64 71.9

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

   6. Applied rewrites 71.9%

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

   if 2e-14 < F

   1. Initial program 59.1%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 97.0

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

   4. Applied rewrites 97.0%

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

Alternative 7: 84.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -2.6 \cdot 10^{-29}:\\ \;\;\;\;-\frac{1 + t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-14}:\\ \;\;\;\;-\cos B \cdot \frac{x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -2.6e-29)
     (- (/ (+ 1.0 t_0) (sin B)))
     (if (<= F 2e-14) (- (* (cos B) (/ x (sin B)))) (/ (- 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 <= -2.6e-29) {
		tmp = -((1.0 + t_0) / sin(B));
	} else if (F <= 2e-14) {
		tmp = -(cos(B) * (x / sin(B)));
	} 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 <= (-2.6d-29)) then
        tmp = -((1.0d0 + t_0) / sin(b))
    else if (f <= 2d-14) then
        tmp = -(cos(b) * (x / sin(b)))
    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 <= -2.6e-29) {
		tmp = -((1.0 + t_0) / Math.sin(B));
	} else if (F <= 2e-14) {
		tmp = -(Math.cos(B) * (x / Math.sin(B)));
	} 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 <= -2.6e-29:
		tmp = -((1.0 + t_0) / math.sin(B))
	elif F <= 2e-14:
		tmp = -(math.cos(B) * (x / math.sin(B)))
	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 <= -2.6e-29)
		tmp = Float64(-Float64(Float64(1.0 + t_0) / sin(B)));
	elseif (F <= 2e-14)
		tmp = Float64(-Float64(cos(B) * Float64(x / sin(B))));
	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 <= -2.6e-29)
		tmp = -((1.0 + t_0) / sin(B));
	elseif (F <= 2e-14)
		tmp = -(cos(B) * (x / sin(B)));
	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, -2.6e-29], (-N[(N[(1.0 + t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 2e-14], (-N[(N[Cos[B], $MachinePrecision] * N[(x / N[Sin[B], $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 < -2.6000000000000002e-29

   1. Initial program 62.7%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 95.1

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

   4. Applied rewrites 95.1%

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

   if -2.6000000000000002e-29 < F < 2e-14

   1. Initial program 99.5%

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

   2. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 68.9

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

   4. Applied rewrites 68.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sin.f64 68.9

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

   6. Applied rewrites 68.9%

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

   if 2e-14 < F

   1. Initial program 59.1%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 97.0

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

   4. Applied rewrites 97.0%

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

Alternative 8: 70.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 2 \cdot 10^{-14}:\\ \;\;\;\;-\cos B \cdot \frac{x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 2e-14)
   (- (* (cos B) (/ x (sin B))))
   (/ (- 1.0 (* (cos B) x)) (sin B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 2e-14) {
		tmp = -(cos(B) * (x / sin(B)));
	} 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 <= 2d-14) then
        tmp = -(cos(b) * (x / sin(b)))
    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 <= 2e-14) {
		tmp = -(Math.cos(B) * (x / Math.sin(B)));
	} else {
		tmp = (1.0 - (Math.cos(B) * x)) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 2e-14:
		tmp = -(math.cos(B) * (x / math.sin(B)))
	else:
		tmp = (1.0 - (math.cos(B) * x)) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 2e-14)
		tmp = Float64(-Float64(cos(B) * Float64(x / sin(B))));
	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 <= 2e-14)
		tmp = -(cos(B) * (x / sin(B)));
	else
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 2e-14], (-N[(N[Cos[B], $MachinePrecision] * N[(x / N[Sin[B], $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 2 regimes

2. if F < 2e-14

   1. Initial program 84.1%

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

   2. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 59.3

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

   4. Applied rewrites 59.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sin.f64 59.3

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

   6. Applied rewrites 59.3%

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

   if 2e-14 < F

   1. Initial program 59.1%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 97.0

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

   4. Applied rewrites 97.0%

      \[\leadsto \color{blue}{\frac{1 - \cos B \cdot x}{\sin B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 62.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{-103}:\\ \;\;\;\;-x \cdot \frac{\cos B}{\sin B}\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-224}:\\ \;\;\;\;\frac{\sqrt{0.5} \cdot F}{\sin B}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-97}:\\ \;\;\;\;{\sin B}^{-1}\\ \mathbf{else}:\\ \;\;\;\;-\cos B \cdot \frac{x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -8.5e-103)
   (- (* x (/ (cos B) (sin B))))
   (if (<= x 7.6e-224)
     (/ (* (sqrt 0.5) F) (sin B))
     (if (<= x 1.55e-97) (pow (sin B) -1.0) (- (* (cos B) (/ x (sin B))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -8.5e-103) {
		tmp = -(x * (cos(B) / sin(B)));
	} else if (x <= 7.6e-224) {
		tmp = (sqrt(0.5) * F) / sin(B);
	} else if (x <= 1.55e-97) {
		tmp = pow(sin(B), -1.0);
	} else {
		tmp = -(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 (x <= (-8.5d-103)) then
        tmp = -(x * (cos(b) / sin(b)))
    else if (x <= 7.6d-224) then
        tmp = (sqrt(0.5d0) * f) / sin(b)
    else if (x <= 1.55d-97) then
        tmp = sin(b) ** (-1.0d0)
    else
        tmp = -(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 (x <= -8.5e-103) {
		tmp = -(x * (Math.cos(B) / Math.sin(B)));
	} else if (x <= 7.6e-224) {
		tmp = (Math.sqrt(0.5) * F) / Math.sin(B);
	} else if (x <= 1.55e-97) {
		tmp = Math.pow(Math.sin(B), -1.0);
	} else {
		tmp = -(Math.cos(B) * (x / Math.sin(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if x <= -8.5e-103:
		tmp = -(x * (math.cos(B) / math.sin(B)))
	elif x <= 7.6e-224:
		tmp = (math.sqrt(0.5) * F) / math.sin(B)
	elif x <= 1.55e-97:
		tmp = math.pow(math.sin(B), -1.0)
	else:
		tmp = -(math.cos(B) * (x / math.sin(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -8.5e-103)
		tmp = Float64(-Float64(x * Float64(cos(B) / sin(B))));
	elseif (x <= 7.6e-224)
		tmp = Float64(Float64(sqrt(0.5) * F) / sin(B));
	elseif (x <= 1.55e-97)
		tmp = sin(B) ^ -1.0;
	else
		tmp = Float64(-Float64(cos(B) * Float64(x / sin(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (x <= -8.5e-103)
		tmp = -(x * (cos(B) / sin(B)));
	elseif (x <= 7.6e-224)
		tmp = (sqrt(0.5) * F) / sin(B);
	elseif (x <= 1.55e-97)
		tmp = sin(B) ^ -1.0;
	else
		tmp = -(cos(B) * (x / sin(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -8.5e-103], (-N[(x * N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), If[LessEqual[x, 7.6e-224], N[(N[(N[Sqrt[0.5], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e-97], N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision], (-N[(N[Cos[B], $MachinePrecision] * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.50000000000000032e-103

   1. Initial program 72.6%

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

   2. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 72.0

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

   4. Applied rewrites 72.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-sin.f64 71.9

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

   6. Applied rewrites 71.9%

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

   if -8.50000000000000032e-103 < x < 7.60000000000000005e-224

   1. Initial program 71.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 75.3%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. unpow-1 N/A

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

      6. sqrt-pow1 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. +-commutative N/A

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

      12. pow2 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-/.f64 57.4

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

   6. Applied rewrites 57.4%

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

   7. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. sqrt-unprod N/A

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

      5. sqrt-unprod N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. metadata-eval N/A

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

      15. lift-sin.f64 37.8

          \[\leadsto \frac{\sqrt{0.5} \cdot F}{\sin B} \]

   9. Applied rewrites 37.8%

      \[\leadsto \frac{\sqrt{0.5} \cdot F}{\color{blue}{\sin B}} \]

   if 7.60000000000000005e-224 < x < 1.55000000000000001e-97

   1. Initial program 71.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 74.2%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. unpow-1 N/A

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

      6. sqrt-pow1 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. +-commutative N/A

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

      12. pow2 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-/.f64 50.5

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

   6. Applied rewrites 50.5%

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

   7. Taylor expanded in F around inf

      \[\leadsto \frac{1}{\color{blue}{\sin B}} \]
   8. Step-by-step derivation

      1. inv-pow N/A

         \[\leadsto {\sin B}^{-1} \]

      2. lower-pow.f64 N/A

         \[\leadsto {\sin B}^{-1} \]

      3. lift-sin.f64 28.3

         \[\leadsto {\sin B}^{-1} \]

   9. Applied rewrites 28.3%

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

   if 1.55000000000000001e-97 < x

   1. Initial program 84.6%

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

   2. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 85.7

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

   4. Applied rewrites 85.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift-cos.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-sin.f64 85.7

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

   6. Applied rewrites 85.7%

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

Alternative 10: 62.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -x \cdot \frac{\cos B}{\sin B}\\ \mathbf{if}\;x \leq -8.5 \cdot 10^{-103}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-224}:\\ \;\;\;\;\frac{\sqrt{0.5} \cdot F}{\sin B}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-97}:\\ \;\;\;\;{\sin B}^{-1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* x (/ (cos B) (sin B))))))
   (if (<= x -8.5e-103)
     t_0
     (if (<= x 7.6e-224)
       (/ (* (sqrt 0.5) F) (sin B))
       (if (<= x 1.55e-97) (pow (sin B) -1.0) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (cos(B) / sin(B)));
	double tmp;
	if (x <= -8.5e-103) {
		tmp = t_0;
	} else if (x <= 7.6e-224) {
		tmp = (sqrt(0.5) * F) / sin(B);
	} else if (x <= 1.55e-97) {
		tmp = pow(sin(B), -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -(x * (cos(b) / sin(b)))
    if (x <= (-8.5d-103)) then
        tmp = t_0
    else if (x <= 7.6d-224) then
        tmp = (sqrt(0.5d0) * f) / sin(b)
    else if (x <= 1.55d-97) then
        tmp = sin(b) ** (-1.0d0)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -(x * (Math.cos(B) / Math.sin(B)));
	double tmp;
	if (x <= -8.5e-103) {
		tmp = t_0;
	} else if (x <= 7.6e-224) {
		tmp = (Math.sqrt(0.5) * F) / Math.sin(B);
	} else if (x <= 1.55e-97) {
		tmp = Math.pow(Math.sin(B), -1.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -(x * (math.cos(B) / math.sin(B)))
	tmp = 0
	if x <= -8.5e-103:
		tmp = t_0
	elif x <= 7.6e-224:
		tmp = (math.sqrt(0.5) * F) / math.sin(B)
	elif x <= 1.55e-97:
		tmp = math.pow(math.sin(B), -1.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x * Float64(cos(B) / sin(B))))
	tmp = 0.0
	if (x <= -8.5e-103)
		tmp = t_0;
	elseif (x <= 7.6e-224)
		tmp = Float64(Float64(sqrt(0.5) * F) / sin(B));
	elseif (x <= 1.55e-97)
		tmp = sin(B) ^ -1.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -(x * (cos(B) / sin(B)));
	tmp = 0.0;
	if (x <= -8.5e-103)
		tmp = t_0;
	elseif (x <= 7.6e-224)
		tmp = (sqrt(0.5) * F) / sin(B);
	elseif (x <= 1.55e-97)
		tmp = sin(B) ^ -1.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x * N[(N[Cos[B], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[x, -8.5e-103], t$95$0, If[LessEqual[x, 7.6e-224], N[(N[(N[Sqrt[0.5], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e-97], N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.50000000000000032e-103 or 1.55000000000000001e-97 < x

   1. Initial program 80.4%

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

   2. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 80.9

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

   4. Applied rewrites 80.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-sin.f64 80.8

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

   6. Applied rewrites 80.8%

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

   if -8.50000000000000032e-103 < x < 7.60000000000000005e-224

   1. Initial program 71.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 75.3%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. unpow-1 N/A

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

      6. sqrt-pow1 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. +-commutative N/A

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

      12. pow2 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-/.f64 57.4

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

   6. Applied rewrites 57.4%

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

   7. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

      4. sqrt-unprod N/A

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

      5. sqrt-unprod N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. lower-sqrt.f64 N/A

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

      14. metadata-eval N/A

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

      15. lift-sin.f64 37.8

          \[\leadsto \frac{\sqrt{0.5} \cdot F}{\sin B} \]

   9. Applied rewrites 37.8%

      \[\leadsto \frac{\sqrt{0.5} \cdot F}{\color{blue}{\sin B}} \]

   if 7.60000000000000005e-224 < x < 1.55000000000000001e-97

   1. Initial program 71.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 74.2%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. unpow-1 N/A

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

      6. sqrt-pow1 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. +-commutative N/A

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

      12. pow2 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-/.f64 50.5

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

   6. Applied rewrites 50.5%

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

   7. Taylor expanded in F around inf

      \[\leadsto \frac{1}{\color{blue}{\sin B}} \]
   8. Step-by-step derivation

      1. inv-pow N/A

         \[\leadsto {\sin B}^{-1} \]

      2. lower-pow.f64 N/A

         \[\leadsto {\sin B}^{-1} \]

      3. lift-sin.f64 28.3

         \[\leadsto {\sin B}^{-1} \]

   9. Applied rewrites 28.3%

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

Alternative 11: 46.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -52000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 0.0027:\\ \;\;\;\;-\frac{x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;{\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -52000.0)
   (/ -1.0 (sin B))
   (if (<= F 0.0027) (- (/ x (sin B))) (pow (sin B) -1.0))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -52000.0) {
		tmp = -1.0 / sin(B);
	} else if (F <= 0.0027) {
		tmp = -(x / sin(B));
	} else {
		tmp = 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 <= (-52000.0d0)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 0.0027d0) then
        tmp = -(x / sin(b))
    else
        tmp = 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 <= -52000.0) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 0.0027) {
		tmp = -(x / Math.sin(B));
	} else {
		tmp = Math.pow(Math.sin(B), -1.0);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -52000.0:
		tmp = -1.0 / math.sin(B)
	elif F <= 0.0027:
		tmp = -(x / math.sin(B))
	else:
		tmp = math.pow(math.sin(B), -1.0)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -52000.0)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 0.0027)
		tmp = Float64(-Float64(x / sin(B)));
	else
		tmp = sin(B) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -52000.0)
		tmp = -1.0 / sin(B);
	elseif (F <= 0.0027)
		tmp = -(x / sin(B));
	else
		tmp = sin(B) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -52000.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0027], (-N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -52000

   1. Initial program 59.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 73.5%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. unpow-1 N/A

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

      6. sqrt-pow1 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. +-commutative N/A

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

      12. pow2 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-/.f64 26.2

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

   6. Applied rewrites 26.2%

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

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 54.3

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

   9. Applied rewrites 54.3%

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

   if -52000 < F < 0.0027000000000000001

   1. Initial program 99.4%

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

   2. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 66.6

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

   4. Applied rewrites 66.6%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 37.3%

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

   if 0.0027000000000000001 < F

   1. Initial program 58.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 73.4%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. unpow-1 N/A

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

      6. sqrt-pow1 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. +-commutative N/A

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

      12. pow2 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-/.f64 25.8

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

   6. Applied rewrites 25.8%

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

   7. Taylor expanded in F around inf

      \[\leadsto \frac{1}{\color{blue}{\sin B}} \]
   8. Step-by-step derivation

      1. inv-pow N/A

         \[\leadsto {\sin B}^{-1} \]

      2. lower-pow.f64 N/A

         \[\leadsto {\sin B}^{-1} \]

      3. lift-sin.f64 52.8

         \[\leadsto {\sin B}^{-1} \]

   9. Applied rewrites 52.8%

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

Alternative 12: 45.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -52000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 0.0027:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;{\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -52000.0)
   (/ -1.0 (sin B))
   (if (<= F 0.0027) (- (/ x B)) (pow (sin B) -1.0))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -52000.0) {
		tmp = -1.0 / sin(B);
	} else if (F <= 0.0027) {
		tmp = -(x / B);
	} else {
		tmp = 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 <= (-52000.0d0)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 0.0027d0) then
        tmp = -(x / b)
    else
        tmp = 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 <= -52000.0) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 0.0027) {
		tmp = -(x / B);
	} else {
		tmp = Math.pow(Math.sin(B), -1.0);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -52000.0:
		tmp = -1.0 / math.sin(B)
	elif F <= 0.0027:
		tmp = -(x / B)
	else:
		tmp = math.pow(math.sin(B), -1.0)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -52000.0)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 0.0027)
		tmp = Float64(-Float64(x / B));
	else
		tmp = sin(B) ^ -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -52000.0)
		tmp = -1.0 / sin(B);
	elseif (F <= 0.0027)
		tmp = -(x / B);
	else
		tmp = sin(B) ^ -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -52000.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0027], (-N[(x / B), $MachinePrecision]), N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -52000

   1. Initial program 59.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 73.5%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. unpow-1 N/A

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

      6. sqrt-pow1 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. +-commutative N/A

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

      12. pow2 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-/.f64 26.2

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

   6. Applied rewrites 26.2%

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

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 54.3

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

   9. Applied rewrites 54.3%

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

   if -52000 < F < 0.0027000000000000001

   1. Initial program 99.4%

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

   2. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 66.6

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

   4. Applied rewrites 66.6%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 34.6

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

   7. Applied rewrites 34.6%

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

   if 0.0027000000000000001 < F

   1. Initial program 58.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 73.4%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. unpow-1 N/A

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

      6. sqrt-pow1 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. +-commutative N/A

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

      12. pow2 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-/.f64 25.8

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

   6. Applied rewrites 25.8%

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

   7. Taylor expanded in F around inf

      \[\leadsto \frac{1}{\color{blue}{\sin B}} \]
   8. Step-by-step derivation

      1. inv-pow N/A

         \[\leadsto {\sin B}^{-1} \]

      2. lower-pow.f64 N/A

         \[\leadsto {\sin B}^{-1} \]

      3. lift-sin.f64 52.8

         \[\leadsto {\sin B}^{-1} \]

   9. Applied rewrites 52.8%

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

Alternative 13: 32.2% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 1.36 \cdot 10^{-8}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 1.36e-8) (- (/ x B)) (/ -1.0 (sin B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 1.36e-8) {
		tmp = -(x / B);
	} else {
		tmp = -1.0 / sin(B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 1.36d-8) then
        tmp = -(x / b)
    else
        tmp = (-1.0d0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 1.36e-8) {
		tmp = -(x / B);
	} else {
		tmp = -1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 1.36e-8:
		tmp = -(x / B)
	else:
		tmp = -1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 1.36e-8)
		tmp = Float64(-Float64(x / B));
	else
		tmp = Float64(-1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 1.36e-8)
		tmp = -(x / B);
	else
		tmp = -1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 1.36e-8], (-N[(x / B), $MachinePrecision]), N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.3599999999999999e-8

   1. Initial program 73.4%

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

   2. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 55.9

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

   4. Applied rewrites 55.9%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 37.4

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

   7. Applied rewrites 37.4%

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

   if 1.3599999999999999e-8 < B

   1. Initial program 86.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 87.0%

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

   4. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. pow2 N/A

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

      5. unpow-1 N/A

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

      6. sqrt-pow1 N/A

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

      7. pow2 N/A

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

      8. +-commutative N/A

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

      9. metadata-eval N/A

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

      10. lower-pow.f64 N/A

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

      11. +-commutative N/A

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

      12. pow2 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-sin.f64 N/A

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

      15. lift-/.f64 33.1

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

   6. Applied rewrites 33.1%

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

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 17.2

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

   9. Applied rewrites 17.2%

      \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 28.9% accurate, 6.8× 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 76.9%

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

2. Taylor expanded in F around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lift-sin.f64 55.8

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

4. Applied rewrites 55.8%

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

5. Taylor expanded in B around 0

   \[\leadsto -\frac{x}{B} \]
6. Step-by-step derivation

   1. lower-/.f64 28.9

      \[\leadsto -\frac{x}{B} \]

7. Applied rewrites 28.9%

   \[\leadsto -\frac{x}{B} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025101 
(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))))))