VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.3% → 99.6%

Time: 7.9s

Alternatives: 26

Speedup: 1.4×

Specification

Math

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

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: 76.3% accurate, 1.0× speedup

Math

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

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, 0.9× speedup

Math

\[\begin{array}{l} t_0 := \sin B \cdot F\\ \mathbf{if}\;F \leq -5 \cdot 10^{+151}:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, \frac{-1}{t\_0} \cdot \tan B, \left(-x\right) \cdot 1\right)}{\tan B}\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{1}{\sin B}}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{t\_0} \cdot F\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (sin B) F)))
   (if (<= F -5e+151)
     (/ (fma F (* (/ -1.0 t_0) (tan B)) (* (- x) 1.0)) (tan B))
     (if (<= F 3.9e+142)
       (fma
        F
        (/ (/ 1.0 (sin B)) (sqrt (fma 2.0 x (fma F F 2.0))))
        (/ (- x) (tan B)))
       (* (/ (- 1.0 (* (cos B) x)) t_0) F)))))

C

double code(double F, double B, double x) {
	double t_0 = sin(B) * F;
	double tmp;
	if (F <= -5e+151) {
		tmp = fma(F, ((-1.0 / t_0) * tan(B)), (-x * 1.0)) / tan(B);
	} else if (F <= 3.9e+142) {
		tmp = fma(F, ((1.0 / sin(B)) / sqrt(fma(2.0, x, fma(F, F, 2.0)))), (-x / tan(B)));
	} else {
		tmp = ((1.0 - (cos(B) * x)) / t_0) * F;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(sin(B) * F)
	tmp = 0.0
	if (F <= -5e+151)
		tmp = Float64(fma(F, Float64(Float64(-1.0 / t_0) * tan(B)), Float64(Float64(-x) * 1.0)) / tan(B));
	elseif (F <= 3.9e+142)
		tmp = fma(F, Float64(Float64(1.0 / sin(B)) / sqrt(fma(2.0, x, fma(F, F, 2.0)))), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(cos(B) * x)) / t_0) * F);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]}, If[LessEqual[F, -5e+151], N[(N[(F * N[(N[(-1.0 / t$95$0), $MachinePrecision] * N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[((-x) * 1.0), $MachinePrecision]), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.9e+142], N[(F * N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * F), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.0000000000000002e151

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.64 54.6%

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

   6. Applied rewrites 54.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. add-to-fraction N/A

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

      4. div-add N/A

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

      5. mult-flip N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

   8. Applied rewrites 54.6%

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

   if -5.0000000000000002e151 < F < 3.9e142

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 84.2%

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. add-flip N/A

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

      8. sub-flip N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. add-flip N/A

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

      14. sub-negate N/A

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

      15. sub-negate-rev N/A

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

      16. add-flip N/A

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

      17. lift-*.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. lower-fma.f64 84.2%

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

   5. Applied rewrites 84.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-pow.64 N/A

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

      11. pow-flip N/A

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

      12. metadata-eval N/A

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

      13. unpow1/2 N/A

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

      14. lower-sqrt.64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-fma.f64 84.3%

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

   7. Applied rewrites 84.3%

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

   if 3.9e142 < F

   1. Initial program 76.3%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.64 47.0%

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

   4. Applied rewrites 47.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.0%

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

   6. Applied rewrites 49.6%

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

Alternative 2: 99.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \sin B \cdot F\\ \mathbf{if}\;F \leq -4 \cdot 10^{+153}:\\ \;\;\;\;\frac{-1}{t\_0} \cdot F - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{1}{\sin B}}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{t\_0} \cdot F\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (sin B) F)))
   (if (<= F -4e+153)
     (- (* (/ -1.0 t_0) F) (/ x (tan B)))
     (if (<= F 3.9e+142)
       (fma
        F
        (/ (/ 1.0 (sin B)) (sqrt (fma 2.0 x (fma F F 2.0))))
        (/ (- x) (tan B)))
       (* (/ (- 1.0 (* (cos B) x)) t_0) F)))))

C

double code(double F, double B, double x) {
	double t_0 = sin(B) * F;
	double tmp;
	if (F <= -4e+153) {
		tmp = ((-1.0 / t_0) * F) - (x / tan(B));
	} else if (F <= 3.9e+142) {
		tmp = fma(F, ((1.0 / sin(B)) / sqrt(fma(2.0, x, fma(F, F, 2.0)))), (-x / tan(B)));
	} else {
		tmp = ((1.0 - (cos(B) * x)) / t_0) * F;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(sin(B) * F)
	tmp = 0.0
	if (F <= -4e+153)
		tmp = Float64(Float64(Float64(-1.0 / t_0) * F) - Float64(x / tan(B)));
	elseif (F <= 3.9e+142)
		tmp = fma(F, Float64(Float64(1.0 / sin(B)) / sqrt(fma(2.0, x, fma(F, F, 2.0)))), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(cos(B) * x)) / t_0) * F);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]}, If[LessEqual[F, -4e+153], N[(N[(N[(-1.0 / t$95$0), $MachinePrecision] * F), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.9e+142], N[(F * N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * F), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4e153

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.64 54.6%

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

   6. Applied rewrites 54.6%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.64 N/A

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

      11. lift-sin.64 N/A

          \[\leadsto \frac{-1}{\sin B \cdot F} \cdot F - \left(\mathsf{neg}\left(\mathsf{Rewrite=>}\left(lift-/.f64, \left(\frac{-x}{\tan B}\right)\right)\right)\right) \]

   8. Applied rewrites 54.6%

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

   if -4e153 < F < 3.9e142

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 84.2%

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. add-flip N/A

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

      8. sub-flip N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. add-flip N/A

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

      14. sub-negate N/A

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

      15. sub-negate-rev N/A

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

      16. add-flip N/A

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

      17. lift-*.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. lower-fma.f64 84.2%

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

   5. Applied rewrites 84.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-pow.64 N/A

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

      11. pow-flip N/A

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

      12. metadata-eval N/A

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

      13. unpow1/2 N/A

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

      14. lower-sqrt.64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-fma.f64 84.3%

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

   7. Applied rewrites 84.3%

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

   if 3.9e142 < F

   1. Initial program 76.3%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.64 47.0%

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

   4. Applied rewrites 47.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.0%

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

   6. Applied rewrites 49.6%

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

Alternative 3: 99.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \sin B \cdot F\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.5 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{t\_0} \cdot F - t\_1\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{+142}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{t\_0} \cdot F\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (sin B) F)) (t_1 (/ x (tan B))))
   (if (<= F -2.5e+28)
     (- (* (/ -1.0 t_0) F) t_1)
     (if (<= F 3.9e+142)
       (- (/ F (* (sin B) (sqrt (fma 2.0 x (fma F F 2.0))))) t_1)
       (* (/ (- 1.0 (* (cos B) x)) t_0) F)))))

C

double code(double F, double B, double x) {
	double t_0 = sin(B) * F;
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -2.5e+28) {
		tmp = ((-1.0 / t_0) * F) - t_1;
	} else if (F <= 3.9e+142) {
		tmp = (F / (sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - t_1;
	} else {
		tmp = ((1.0 - (cos(B) * x)) / t_0) * F;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(sin(B) * F)
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.5e+28)
		tmp = Float64(Float64(Float64(-1.0 / t_0) * F) - t_1);
	elseif (F <= 3.9e+142)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - t_1);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(cos(B) * x)) / t_0) * F);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.5e+28], N[(N[(N[(-1.0 / t$95$0), $MachinePrecision] * F), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 3.9e+142], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * F), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.49999999999999979e28

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.64 54.6%

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

   6. Applied rewrites 54.6%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.64 N/A

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

      11. lift-sin.64 N/A

          \[\leadsto \frac{-1}{\sin B \cdot F} \cdot F - \left(\mathsf{neg}\left(\mathsf{Rewrite=>}\left(lift-/.f64, \left(\frac{-x}{\tan B}\right)\right)\right)\right) \]

   8. Applied rewrites 54.6%

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

   if -2.49999999999999979e28 < F < 3.9e142

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lower-unsound-/.f64 N/A

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

      4. lower-unsound-/.f64 84.2%

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. add-flip N/A

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

      8. sub-flip N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. add-flip N/A

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

      14. sub-negate N/A

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

      15. sub-negate-rev N/A

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

      16. add-flip N/A

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

      17. lift-*.f64 N/A

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

      18. lift-fma.f64 N/A

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

      19. lower-fma.f64 84.2%

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

   5. Applied rewrites 84.2%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac2 N/A

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

      5. lift-neg.f64 N/A

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

      6. frac-2neg N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

   7. Applied rewrites 84.3%

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

   if 3.9e142 < F

   1. Initial program 76.3%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.64 47.0%

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

   4. Applied rewrites 47.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.0%

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

   6. Applied rewrites 49.6%

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

Alternative 4: 92.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \sin B \cdot F\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -37:\\ \;\;\;\;\frac{-1}{t\_0} \cdot F - t\_1\\ \mathbf{elif}\;F \leq -2.6 \cdot 10^{-101}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, -\frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 170000:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{t\_0} \cdot F\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (sin B) F)) (t_1 (/ x (tan B))))
   (if (<= F -37.0)
     (- (* (/ -1.0 t_0) F) t_1)
     (if (<= F -2.6e-101)
       (fma F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B)) (- (/ x B)))
       (if (<= F 170000.0)
         (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F B)) t_1)
         (* (/ (- 1.0 (* (cos B) x)) t_0) F))))))

C

double code(double F, double B, double x) {
	double t_0 = sin(B) * F;
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -37.0) {
		tmp = ((-1.0 / t_0) * F) - t_1;
	} else if (F <= -2.6e-101) {
		tmp = fma(F, (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B)), -(x / B));
	} else if (F <= 170000.0) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / B)) - t_1;
	} else {
		tmp = ((1.0 - (cos(B) * x)) / t_0) * F;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(sin(B) * F)
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -37.0)
		tmp = Float64(Float64(Float64(-1.0 / t_0) * F) - t_1);
	elseif (F <= -2.6e-101)
		tmp = fma(F, Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B)), Float64(-Float64(x / B)));
	elseif (F <= 170000.0)
		tmp = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - t_1);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(cos(B) * x)) / t_0) * F);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -37.0], N[(N[(N[(-1.0 / t$95$0), $MachinePrecision] * F), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -2.6e-101], N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + (-N[(x / B), $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 170000.0], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * F), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -37

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.64 54.6%

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

   6. Applied rewrites 54.6%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.64 N/A

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

      7. lift-*.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift-sin.64 N/A

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

      11. lift-sin.64 N/A

          \[\leadsto \frac{-1}{\sin B \cdot F} \cdot F - \left(\mathsf{neg}\left(\mathsf{Rewrite=>}\left(lift-/.f64, \left(\frac{-x}{\tan B}\right)\right)\right)\right) \]

   8. Applied rewrites 54.6%

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

   if -37 < F < -2.6000000000000001e-101

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 49.7%

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

   4. Applied rewrites 49.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 57.6%

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

   if -2.6000000000000001e-101 < F < 1.7e5

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.3%

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

   3. Applied rewrites 76.4%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 61.2%

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

   if 1.7e5 < F

   1. Initial program 76.3%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.64 47.0%

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

   4. Applied rewrites 47.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.0%

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

   6. Applied rewrites 49.6%

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

Alternative 5: 86.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \sin B \cdot F\\ \mathbf{if}\;F \leq -4 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{t\_0}, F, \frac{-x}{B}\right)\\ \mathbf{elif}\;F \leq -2.6 \cdot 10^{-101}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}}\\ \mathbf{elif}\;F \leq 170000:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{t\_0} \cdot F\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (sin B) F)))
   (if (<= F -4e+153)
     (fma (/ -1.0 t_0) F (/ (- x) B))
     (if (<= F -2.6e-101)
       (+
        (- (/ x B))
        (/ 1.0 (/ (sin B) (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F))))
       (if (<= F 170000.0)
         (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F B)) (/ x (tan B)))
         (* (/ (- 1.0 (* (cos B) x)) t_0) F))))))

C

double code(double F, double B, double x) {
	double t_0 = sin(B) * F;
	double tmp;
	if (F <= -4e+153) {
		tmp = fma((-1.0 / t_0), F, (-x / B));
	} else if (F <= -2.6e-101) {
		tmp = -(x / B) + (1.0 / (sin(B) / (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F)));
	} else if (F <= 170000.0) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / B)) - (x / tan(B));
	} else {
		tmp = ((1.0 - (cos(B) * x)) / t_0) * F;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(sin(B) * F)
	tmp = 0.0
	if (F <= -4e+153)
		tmp = fma(Float64(-1.0 / t_0), F, Float64(Float64(-x) / B));
	elseif (F <= -2.6e-101)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(1.0 / Float64(sin(B) / Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F))));
	elseif (F <= 170000.0)
		tmp = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(cos(B) * x)) / t_0) * F);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]}, If[LessEqual[F, -4e+153], N[(N[(-1.0 / t$95$0), $MachinePrecision] * F + N[((-x) / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.6e-101], N[((-N[(x / B), $MachinePrecision]) + N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 170000.0], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] * F), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4e153

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.64 54.6%

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

   6. Applied rewrites 54.6%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 35.8%

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

   9. Applied rewrites 35.8%

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

       1. lift-fma.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 35.8%

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 35.8%

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

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, \mathsf{Rewrite=>}\left(lift-*.f64, \left(-1 \cdot \frac{x}{B}\right)\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, \mathsf{Rewrite=>}\left(mul-1-neg, \left(\mathsf{neg}\left(\frac{x}{B}\right)\right)\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, \mathsf{neg}\left(\mathsf{Rewrite=>}\left(lift-/.f64, \left(\frac{x}{B}\right)\right)\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, \mathsf{Rewrite=>}\left(distribute-neg-frac, \left(\frac{\mathsf{neg}\left(x\right)}{B}\right)\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, \frac{\mathsf{Rewrite<=}\left(lift-neg.f64, \left(-x\right)\right)}{B}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, \mathsf{Rewrite=>}\left(lower-/.f64, \left(\frac{-x}{B}\right)\right)\right) \]

   11. Applied rewrites 35.8%

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

   if -4e153 < F < -2.6000000000000001e-101

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 49.7%

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

   4. Applied rewrites 49.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-pow.64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-fma.f64 N/A

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

   6. Applied rewrites 57.6%

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

   if -2.6000000000000001e-101 < F < 1.7e5

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.3%

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

   3. Applied rewrites 76.4%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 61.2%

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

   if 1.7e5 < F

   1. Initial program 76.3%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.64 47.0%

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

   4. Applied rewrites 47.0%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.0%

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

   6. Applied rewrites 49.6%

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

Alternative 6: 80.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\sin B \cdot F}, F, \frac{-x}{B}\right)\\ \mathbf{elif}\;F \leq -2.6 \cdot 10^{-101}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}}\\ \mathbf{elif}\;F \leq 4000000000:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{F \cdot \sin B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4e+153)
   (fma (/ -1.0 (* (sin B) F)) F (/ (- x) B))
   (if (<= F -2.6e-101)
     (+
      (- (/ x B))
      (/ 1.0 (/ (sin B) (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F))))
     (if (<= F 4000000000.0)
       (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F B)) (/ x (tan B)))
       (* F (fma -1.0 (/ x (* B F)) (/ 1.0 (* F (sin B)))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4e+153) {
		tmp = fma((-1.0 / (sin(B) * F)), F, (-x / B));
	} else if (F <= -2.6e-101) {
		tmp = -(x / B) + (1.0 / (sin(B) / (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F)));
	} else if (F <= 4000000000.0) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / B)) - (x / tan(B));
	} else {
		tmp = F * fma(-1.0, (x / (B * F)), (1.0 / (F * sin(B))));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4e+153)
		tmp = fma(Float64(-1.0 / Float64(sin(B) * F)), F, Float64(Float64(-x) / B));
	elseif (F <= -2.6e-101)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(1.0 / Float64(sin(B) / Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F))));
	elseif (F <= 4000000000.0)
		tmp = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / tan(B)));
	else
		tmp = Float64(F * fma(-1.0, Float64(x / Float64(B * F)), Float64(1.0 / Float64(F * sin(B)))));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4e+153], N[(N[(-1.0 / N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * F + N[((-x) / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.6e-101], N[((-N[(x / B), $MachinePrecision]) + N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4000000000.0], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(-1.0 * N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4e153

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.64 54.6%

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

   6. Applied rewrites 54.6%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 35.8%

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

   9. Applied rewrites 35.8%

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

       1. lift-fma.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 35.8%

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 35.8%

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

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, \mathsf{Rewrite=>}\left(lift-*.f64, \left(-1 \cdot \frac{x}{B}\right)\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, \mathsf{Rewrite=>}\left(mul-1-neg, \left(\mathsf{neg}\left(\frac{x}{B}\right)\right)\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, \mathsf{neg}\left(\mathsf{Rewrite=>}\left(lift-/.f64, \left(\frac{x}{B}\right)\right)\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, \mathsf{Rewrite=>}\left(distribute-neg-frac, \left(\frac{\mathsf{neg}\left(x\right)}{B}\right)\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, \frac{\mathsf{Rewrite<=}\left(lift-neg.f64, \left(-x\right)\right)}{B}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, \mathsf{Rewrite=>}\left(lower-/.f64, \left(\frac{-x}{B}\right)\right)\right) \]

   11. Applied rewrites 35.8%

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

   if -4e153 < F < -2.6000000000000001e-101

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 49.7%

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

   4. Applied rewrites 49.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-pow.64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-pow.64 N/A

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

      10. lift-+.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lower-+.f64 N/A

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

      14. +-commutative N/A

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

      15. lift-*.f64 N/A

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

      16. lift-fma.f64 N/A

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

   6. Applied rewrites 57.6%

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

   if -2.6000000000000001e-101 < F < 4e9

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.3%

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

   3. Applied rewrites 76.4%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 61.2%

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

   if 4e9 < F

   1. Initial program 76.3%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.64 47.0%

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

   4. Applied rewrites 47.0%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 31.5%

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

   7. Applied rewrites 31.5%

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

Alternative 7: 80.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\sin B \cdot F}, F, \frac{-x}{B}\right)\\ \mathbf{elif}\;F \leq -2.6 \cdot 10^{-101}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, -\frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 4000000000:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{F \cdot \sin B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.5e+19)
   (fma (/ -1.0 (* (sin B) F)) F (/ (- x) B))
   (if (<= F -2.6e-101)
     (fma F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B)) (- (/ x B)))
     (if (<= F 4000000000.0)
       (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F B)) (/ x (tan B)))
       (* F (fma -1.0 (/ x (* B F)) (/ 1.0 (* F (sin B)))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.5e+19) {
		tmp = fma((-1.0 / (sin(B) * F)), F, (-x / B));
	} else if (F <= -2.6e-101) {
		tmp = fma(F, (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B)), -(x / B));
	} else if (F <= 4000000000.0) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / B)) - (x / tan(B));
	} else {
		tmp = F * fma(-1.0, (x / (B * F)), (1.0 / (F * sin(B))));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.5e+19)
		tmp = fma(Float64(-1.0 / Float64(sin(B) * F)), F, Float64(Float64(-x) / B));
	elseif (F <= -2.6e-101)
		tmp = fma(F, Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B)), Float64(-Float64(x / B)));
	elseif (F <= 4000000000.0)
		tmp = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / tan(B)));
	else
		tmp = Float64(F * fma(-1.0, Float64(x / Float64(B * F)), Float64(1.0 / Float64(F * sin(B)))));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.5e+19], N[(N[(-1.0 / N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * F + N[((-x) / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.6e-101], N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + (-N[(x / B), $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 4000000000.0], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(-1.0 * N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.5e19

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.64 54.6%

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

   6. Applied rewrites 54.6%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 35.8%

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

   9. Applied rewrites 35.8%

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

       1. lift-fma.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 35.8%

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lower-*.f64 35.8%

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

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, \mathsf{Rewrite=>}\left(lift-*.f64, \left(-1 \cdot \frac{x}{B}\right)\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, \mathsf{Rewrite=>}\left(mul-1-neg, \left(\mathsf{neg}\left(\frac{x}{B}\right)\right)\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, \mathsf{neg}\left(\mathsf{Rewrite=>}\left(lift-/.f64, \left(\frac{x}{B}\right)\right)\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, \mathsf{Rewrite=>}\left(distribute-neg-frac, \left(\frac{\mathsf{neg}\left(x\right)}{B}\right)\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, \frac{\mathsf{Rewrite<=}\left(lift-neg.f64, \left(-x\right)\right)}{B}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, \mathsf{Rewrite=>}\left(lower-/.f64, \left(\frac{-x}{B}\right)\right)\right) \]

   11. Applied rewrites 35.8%

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

   if -1.5e19 < F < -2.6000000000000001e-101

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 49.7%

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

   4. Applied rewrites 49.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 57.6%

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

   if -2.6000000000000001e-101 < F < 4e9

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.3%

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

   3. Applied rewrites 76.4%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 61.2%

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

   if 4e9 < F

   1. Initial program 76.3%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.64 47.0%

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

   4. Applied rewrites 47.0%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 31.5%

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

   7. Applied rewrites 31.5%

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

Alternative 8: 75.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{+51}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{B \cdot F}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, -\frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{x \cdot \cos B}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -4.7e+51)
   (fma F (/ -1.0 (* B F)) (/ (- x) (tan B)))
   (if (<= x 4.5e-15)
     (fma F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B)) (- (/ x B)))
     (* -1.0 (/ (* x (cos B)) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -4.7e+51) {
		tmp = fma(F, (-1.0 / (B * F)), (-x / tan(B)));
	} else if (x <= 4.5e-15) {
		tmp = fma(F, (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B)), -(x / B));
	} else {
		tmp = -1.0 * ((x * cos(B)) / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -4.7e+51)
		tmp = fma(F, Float64(-1.0 / Float64(B * F)), Float64(Float64(-x) / tan(B)));
	elseif (x <= 4.5e-15)
		tmp = fma(F, Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B)), Float64(-Float64(x / B)));
	else
		tmp = Float64(-1.0 * Float64(Float64(x * cos(B)) / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -4.7e+51], N[(F * N[(-1.0 / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.5e-15], N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + (-N[(x / B), $MachinePrecision])), $MachinePrecision], N[(-1.0 * N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.7000000000000002e51

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.64 54.6%

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

   6. Applied rewrites 54.6%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 52.1%

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

   9. Applied rewrites 52.1%

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

   if -4.7000000000000002e51 < x < 4.4999999999999998e-15

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 49.7%

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

   4. Applied rewrites 49.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 57.6%

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

   if 4.4999999999999998e-15 < x

   1. Initial program 76.3%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.64 N/A

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

      5. lower-sin.64 54.7%

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

   4. Applied rewrites 54.7%

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

Alternative 9: 74.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(F, \frac{-1}{B \cdot F}, \frac{-x}{\tan B}\right)\\ \mathbf{if}\;x \leq -4.7 \cdot 10^{+51}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, -\frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (fma F (/ -1.0 (* B F)) (/ (- x) (tan B)))))
   (if (<= x -4.7e+51)
     t_0
     (if (<= x 7e-17)
       (fma F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B)) (- (/ x B)))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = fma(F, (-1.0 / (B * F)), (-x / tan(B)));
	double tmp;
	if (x <= -4.7e+51) {
		tmp = t_0;
	} else if (x <= 7e-17) {
		tmp = fma(F, (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B)), -(x / B));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(F, Float64(-1.0 / Float64(B * F)), Float64(Float64(-x) / tan(B)))
	tmp = 0.0
	if (x <= -4.7e+51)
		tmp = t_0;
	elseif (x <= 7e-17)
		tmp = fma(F, Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B)), Float64(-Float64(x / B)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F * N[(-1.0 / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.7e+51], t$95$0, If[LessEqual[x, 7e-17], N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + (-N[(x / B), $MachinePrecision])), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.7000000000000002e51 or 7.0000000000000003e-17 < x

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.64 54.6%

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

   6. Applied rewrites 54.6%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 52.1%

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

   9. Applied rewrites 52.1%

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

   if -4.7000000000000002e51 < x < 7.0000000000000003e-17

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 49.7%

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

   4. Applied rewrites 49.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, -\frac{x}{B}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 74.0% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(F, \frac{-1}{B \cdot F}, \frac{-x}{\tan B}\right)\\ \mathbf{if}\;x \leq -4.7 \cdot 10^{+51}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, \frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F, -\frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (fma F (/ -1.0 (* B F)) (/ (- x) (tan B)))))
   (if (<= x -4.7e+51)
     t_0
     (if (<= x 7e-17)
       (fma
        (/ 1.0 (sin B))
        (* (/ 1.0 (sqrt (fma 2.0 x (fma F F 2.0)))) F)
        (- (/ x B)))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = fma(F, (-1.0 / (B * F)), (-x / tan(B)));
	double tmp;
	if (x <= -4.7e+51) {
		tmp = t_0;
	} else if (x <= 7e-17) {
		tmp = fma((1.0 / sin(B)), ((1.0 / sqrt(fma(2.0, x, fma(F, F, 2.0)))) * F), -(x / B));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(F, Float64(-1.0 / Float64(B * F)), Float64(Float64(-x) / tan(B)))
	tmp = 0.0
	if (x <= -4.7e+51)
		tmp = t_0;
	elseif (x <= 7e-17)
		tmp = fma(Float64(1.0 / sin(B)), Float64(Float64(1.0 / sqrt(fma(2.0, x, fma(F, F, 2.0)))) * F), Float64(-Float64(x / B)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F * N[(-1.0 / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.7e+51], t$95$0, If[LessEqual[x, 7e-17], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] + (-N[(x / B), $MachinePrecision])), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.7000000000000002e51 or 7.0000000000000003e-17 < x

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.64 54.6%

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

   6. Applied rewrites 54.6%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 52.1%

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

   9. Applied rewrites 52.1%

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

   if -4.7000000000000002e51 < x < 7.0000000000000003e-17

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 49.7%

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

   4. Applied rewrites 49.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 57.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip-rev N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. associate-*l* N/A

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

      11. *-commutative N/A

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

   8. Applied rewrites 57.6%

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

Alternative 11: 74.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(F, \frac{-1}{B \cdot F}, \frac{-x}{\tan B}\right)\\ \mathbf{if}\;x \leq -4.7 \cdot 10^{+51}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}, -\frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (fma F (/ -1.0 (* B F)) (/ (- x) (tan B)))))
   (if (<= x -4.7e+51)
     t_0
     (if (<= x 7e-17)
       (fma F (/ (/ 1.0 (sqrt (fma 2.0 x (fma F F 2.0)))) (sin B)) (- (/ x B)))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = fma(F, (-1.0 / (B * F)), (-x / tan(B)));
	double tmp;
	if (x <= -4.7e+51) {
		tmp = t_0;
	} else if (x <= 7e-17) {
		tmp = fma(F, ((1.0 / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / sin(B)), -(x / B));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(F, Float64(-1.0 / Float64(B * F)), Float64(Float64(-x) / tan(B)))
	tmp = 0.0
	if (x <= -4.7e+51)
		tmp = t_0;
	elseif (x <= 7e-17)
		tmp = fma(F, Float64(Float64(1.0 / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / sin(B)), Float64(-Float64(x / B)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F * N[(-1.0 / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.7e+51], t$95$0, If[LessEqual[x, 7e-17], N[(F * N[(N[(1.0 / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + (-N[(x / B), $MachinePrecision])), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.7000000000000002e51 or 7.0000000000000003e-17 < x

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.64 54.6%

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

   6. Applied rewrites 54.6%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 52.1%

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

   9. Applied rewrites 52.1%

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

   if -4.7000000000000002e51 < x < 7.0000000000000003e-17

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 49.7%

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

   4. Applied rewrites 49.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 57.6%

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

      1. lift-pow.64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. pow-flip N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. pow-flip N/A

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

      11. lower-pow.64 N/A

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

      12. *-commutative N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lift-fma.f64 N/A

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

      17. lift-fma.f64 N/A

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

      18. *-commutative N/A

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

   8. Applied rewrites 57.6%

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

Alternative 12: 74.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(F, \frac{-1}{B \cdot F}, \frac{-x}{\tan B}\right)\\ \mathbf{if}\;x \leq -4.7 \cdot 10^{+51}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-17}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (fma F (/ -1.0 (* B F)) (/ (- x) (tan B)))))
   (if (<= x -4.7e+51)
     t_0
     (if (<= x 7e-17)
       (- (/ F (* (sin B) (sqrt (fma 2.0 x (fma F F 2.0))))) (/ x B))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = fma(F, (-1.0 / (B * F)), (-x / tan(B)));
	double tmp;
	if (x <= -4.7e+51) {
		tmp = t_0;
	} else if (x <= 7e-17) {
		tmp = (F / (sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(F, Float64(-1.0 / Float64(B * F)), Float64(Float64(-x) / tan(B)))
	tmp = 0.0
	if (x <= -4.7e+51)
		tmp = t_0;
	elseif (x <= 7e-17)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - Float64(x / B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F * N[(-1.0 / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.7e+51], t$95$0, If[LessEqual[x, 7e-17], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -4.7000000000000002e51 or 7.0000000000000003e-17 < x

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.64 54.6%

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

   6. Applied rewrites 54.6%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 52.1%

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

   9. Applied rewrites 52.1%

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

   if -4.7000000000000002e51 < x < 7.0000000000000003e-17

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 49.7%

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

   4. Applied rewrites 49.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 57.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. sub-flip-reverse N/A

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

      4. lift-/.f64 N/A

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

      5. div-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. lower--.f64 N/A

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

   8. Applied rewrites 57.6%

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

Alternative 13: 67.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{-1}{F}}{\sin B}, -1 \cdot \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq -1.05 \cdot 10^{-190}:\\ \;\;\;\;\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}\\ \mathbf{elif}\;F \leq 4 \cdot 10^{-11}:\\ \;\;\;\;\frac{-1}{F} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \frac{1}{F \cdot \sin B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.3e+14)
   (fma F (/ (/ -1.0 F) (sin B)) (* -1.0 (/ x B)))
   (if (<= F -1.05e-190)
     (/ (- (* F (pow (+ 2.0 (fma 2.0 x (pow F 2.0))) -0.5)) x) B)
     (if (<= F 4e-11)
       (- (* (/ -1.0 F) (/ F B)) (/ x (tan B)))
       (* F (fma -1.0 (/ x (* B F)) (/ 1.0 (* F (sin B)))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.3e+14) {
		tmp = fma(F, ((-1.0 / F) / sin(B)), (-1.0 * (x / B)));
	} else if (F <= -1.05e-190) {
		tmp = ((F * pow((2.0 + fma(2.0, x, pow(F, 2.0))), -0.5)) - x) / B;
	} else if (F <= 4e-11) {
		tmp = ((-1.0 / F) * (F / B)) - (x / tan(B));
	} else {
		tmp = F * fma(-1.0, (x / (B * F)), (1.0 / (F * sin(B))));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.3e+14)
		tmp = fma(F, Float64(Float64(-1.0 / F) / sin(B)), Float64(-1.0 * Float64(x / B)));
	elseif (F <= -1.05e-190)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + fma(2.0, x, (F ^ 2.0))) ^ -0.5)) - x) / B);
	elseif (F <= 4e-11)
		tmp = Float64(Float64(Float64(-1.0 / F) * Float64(F / B)) - Float64(x / tan(B)));
	else
		tmp = Float64(F * fma(-1.0, Float64(x / Float64(B * F)), Float64(1.0 / Float64(F * sin(B)))));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.3e+14], N[(F * N[(N[(-1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.05e-190], N[(N[(N[(F * N[Power[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4e-11], N[(N[(N[(-1.0 / F), $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(-1.0 * N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.3e14

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.64 54.6%

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

   6. Applied rewrites 54.6%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 35.8%

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

   9. Applied rewrites 35.8%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. frac-2neg N/A

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

       5. metadata-eval N/A

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

       6. distribute-neg-frac2 N/A

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

       7. lift-/.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. distribute-neg-frac N/A

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

       11. metadata-eval N/A

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

       12. lift-/.f64 35.8%

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

   11. Applied rewrites 35.8%

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

   if -2.3e14 < F < -1.04999999999999996e-190

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 43.3%

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

   if -1.04999999999999996e-190 < F < 3.99999999999999976e-11

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.3%

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

   3. Applied rewrites 76.4%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 48.0%

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

   6. Applied rewrites 48.0%

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

   7. Taylor expanded in B around 0

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

   9. Applied rewrites 46.4%

      \[\leadsto \frac{-1}{F} \cdot \frac{F}{\color{blue}{B}} - \frac{x}{\tan B} \]

   if 3.99999999999999976e-11 < F

   1. Initial program 76.3%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.64 47.0%

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

   4. Applied rewrites 47.0%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 31.5%

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

   7. Applied rewrites 31.5%

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

Alternative 14: 65.9% accurate, 1.8× speedup

Math

\[\mathsf{copysign}\left(1, B\right) \cdot \begin{array}{l} \mathbf{if}\;\left|B\right| \leq 8 \cdot 10^{-18}:\\ \;\;\;\;\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{\left|B\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{F} \cdot \frac{F}{\left|B\right|} - \frac{x}{\tan \left(\left|B\right|\right)}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (*
  (copysign 1.0 B)
  (if (<= (fabs B) 8e-18)
    (/ (- (* F (pow (+ 2.0 (fma 2.0 x (pow F 2.0))) -0.5)) x) (fabs B))
    (- (* (/ -1.0 F) (/ F (fabs B))) (/ x (tan (fabs B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (fabs(B) <= 8e-18) {
		tmp = ((F * pow((2.0 + fma(2.0, x, pow(F, 2.0))), -0.5)) - x) / fabs(B);
	} else {
		tmp = ((-1.0 / F) * (F / fabs(B))) - (x / tan(fabs(B)));
	}
	return copysign(1.0, B) * tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (abs(B) <= 8e-18)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + fma(2.0, x, (F ^ 2.0))) ^ -0.5)) - x) / abs(B));
	else
		tmp = Float64(Float64(Float64(-1.0 / F) * Float64(F / abs(B))) - Float64(x / tan(abs(B))));
	end
	return Float64(copysign(1.0, B) * tmp)
end

Wolfram

code[F_, B_, x_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[B]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[B], $MachinePrecision], 8e-18], N[(N[(N[(F * N[Power[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[Abs[B], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 / F), $MachinePrecision] * N[(F / N[Abs[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[N[Abs[B], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if B < 8.0000000000000006e-18

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 43.3%

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

   if 8.0000000000000006e-18 < B

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.3%

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

   3. Applied rewrites 76.4%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 48.0%

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

   6. Applied rewrites 48.0%

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

   7. Taylor expanded in B around 0

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

   9. Applied rewrites 46.4%

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

Alternative 15: 64.7% accurate, 1.8× speedup

Math

\[\mathsf{copysign}\left(1, B\right) \cdot \begin{array}{l} \mathbf{if}\;\left|B\right| \leq 4.7 \cdot 10^{-24}:\\ \;\;\;\;\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{\left|B\right|}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{\left|B\right| \cdot F}, \frac{-x}{\tan \left(\left|B\right|\right)}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (*
  (copysign 1.0 B)
  (if (<= (fabs B) 4.7e-24)
    (/ (- (* F (pow (+ 2.0 (fma 2.0 x (pow F 2.0))) -0.5)) x) (fabs B))
    (fma F (/ -1.0 (* (fabs B) F)) (/ (- x) (tan (fabs B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (fabs(B) <= 4.7e-24) {
		tmp = ((F * pow((2.0 + fma(2.0, x, pow(F, 2.0))), -0.5)) - x) / fabs(B);
	} else {
		tmp = fma(F, (-1.0 / (fabs(B) * F)), (-x / tan(fabs(B))));
	}
	return copysign(1.0, B) * tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (abs(B) <= 4.7e-24)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + fma(2.0, x, (F ^ 2.0))) ^ -0.5)) - x) / abs(B));
	else
		tmp = fma(F, Float64(-1.0 / Float64(abs(B) * F)), Float64(Float64(-x) / tan(abs(B))));
	end
	return Float64(copysign(1.0, B) * tmp)
end

Wolfram

code[F_, B_, x_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[B]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[B], $MachinePrecision], 4.7e-24], N[(N[(N[(F * N[Power[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[Abs[B], $MachinePrecision]), $MachinePrecision], N[(F * N[(-1.0 / N[(N[Abs[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[N[Abs[B], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if B < 4.69999999999999992e-24

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 43.3%

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

   if 4.69999999999999992e-24 < B

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.64 54.6%

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

   6. Applied rewrites 54.6%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 52.1%

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

   9. Applied rewrites 52.1%

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

Alternative 16: 58.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{-1}{F}}{\sin B}, -1 \cdot \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{+25}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.3e+14)
   (fma F (/ (/ -1.0 F) (sin B)) (* -1.0 (/ x B)))
   (if (<= F 2.2e+25)
     (+
      (- (/ x B))
      (* (/ F B) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.3e+14) {
		tmp = fma(F, ((-1.0 / F) / sin(B)), (-1.0 * (x / B)));
	} else if (F <= 2.2e+25) {
		tmp = -(x / B) + ((F / B) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.3e+14)
		tmp = fma(F, Float64(Float64(-1.0 / F) / sin(B)), Float64(-1.0 * Float64(x / B)));
	elseif (F <= 2.2e+25)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F / B) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.3e+14], N[(F * N[(N[(-1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.2e+25], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F / B), $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[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.3e14

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{F \cdot \color{blue}{\sin B}}, \frac{-x}{\tan B}\right) \]

      3. lower-sin.64 54.6%

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, \frac{-x}{\tan B}\right) \]

   6. Applied rewrites 54.6%

      \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{-1}{F \cdot \sin B}}, \frac{-x}{\tan B}\right) \]

   7. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, -1 \cdot \color{blue}{\frac{x}{B}}\right) \]

      2. lower-/.f64 35.8%

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, -1 \cdot \frac{x}{\color{blue}{B}}\right) \]

   9. Applied rewrites 35.8%

      \[\leadsto \mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{-1}{\color{blue}{F \cdot \sin B}}, -1 \cdot \frac{x}{B}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{-1}{F \cdot \color{blue}{\sin B}}, -1 \cdot \frac{x}{B}\right) \]

       3. associate-/r* N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{-1}{F}}{\color{blue}{\sin B}}, -1 \cdot \frac{x}{B}\right) \]

       4. frac-2neg N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(F\right)}}{\sin \color{blue}{B}}, -1 \cdot \frac{x}{B}\right) \]

       5. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{\mathsf{neg}\left(F\right)}}{\sin B}, -1 \cdot \frac{x}{B}\right) \]

       6. distribute-neg-frac2 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\mathsf{neg}\left(\frac{1}{F}\right)}{\sin \color{blue}{B}}, -1 \cdot \frac{x}{B}\right) \]

       7. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\mathsf{neg}\left(\frac{1}{F}\right)}{\sin B}, -1 \cdot \frac{x}{B}\right) \]

       8. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\mathsf{neg}\left(\frac{1}{F}\right)}{\color{blue}{\sin B}}, -1 \cdot \frac{x}{B}\right) \]

       9. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{\mathsf{neg}\left(\frac{1}{F}\right)}{\sin B}, -1 \cdot \frac{x}{B}\right) \]

       10. distribute-neg-frac N/A

           \[\leadsto \mathsf{fma}\left(F, \frac{\frac{\mathsf{neg}\left(1\right)}{F}}{\sin \color{blue}{B}}, -1 \cdot \frac{x}{B}\right) \]

       11. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(F, \frac{\frac{-1}{F}}{\sin B}, -1 \cdot \frac{x}{B}\right) \]

       12. lift-/.f64 35.8%

           \[\leadsto \mathsf{fma}\left(F, \frac{\frac{-1}{F}}{\sin \color{blue}{B}}, -1 \cdot \frac{x}{B}\right) \]

   11. Applied rewrites 35.8%

       \[\leadsto \mathsf{fma}\left(F, \frac{\frac{-1}{F}}{\color{blue}{\sin B}}, -1 \cdot \frac{x}{B}\right) \]

   if -2.3e14 < F < 2.2000000000000001e25

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.7%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\color{blue}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 35.5%

      \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\color{blue}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if 2.2000000000000001e25 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.64 17.2%

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 58.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\sin B \cdot F}, F, \frac{-x}{B}\right)\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{+25}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.3e+14)
   (fma (/ -1.0 (* (sin B) F)) F (/ (- x) B))
   (if (<= F 2.2e+25)
     (+
      (- (/ x B))
      (* (/ F B) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.3e+14) {
		tmp = fma((-1.0 / (sin(B) * F)), F, (-x / B));
	} else if (F <= 2.2e+25) {
		tmp = -(x / B) + ((F / B) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.3e+14)
		tmp = fma(Float64(-1.0 / Float64(sin(B) * F)), F, Float64(Float64(-x) / B));
	elseif (F <= 2.2e+25)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F / B) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.3e+14], N[(N[(-1.0 / N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * F + N[((-x) / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.2e+25], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F / B), $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[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.3e14

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. mult-flip N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right)} \]

   3. Applied rewrites 84.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in F around -inf

      \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{-1}{F \cdot \sin B}}, \frac{-x}{\tan B}\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{\color{blue}{F \cdot \sin B}}, \frac{-x}{\tan B}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{F \cdot \color{blue}{\sin B}}, \frac{-x}{\tan B}\right) \]

      3. lower-sin.64 54.6%

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, \frac{-x}{\tan B}\right) \]

   6. Applied rewrites 54.6%

      \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{-1}{F \cdot \sin B}}, \frac{-x}{\tan B}\right) \]

   7. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, -1 \cdot \color{blue}{\frac{x}{B}}\right) \]

      2. lower-/.f64 35.8%

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, -1 \cdot \frac{x}{\color{blue}{B}}\right) \]

   9. Applied rewrites 35.8%

      \[\leadsto \mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]
   10. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto \color{blue}{F \cdot \frac{-1}{F \cdot \sin B} + -1 \cdot \frac{x}{B}} \]

       2. *-commutative N/A

          \[\leadsto \color{blue}{\frac{-1}{F \cdot \sin B} \cdot F} + -1 \cdot \frac{x}{B} \]

       3. lower-fma.f64 35.8%

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{F \cdot \sin B}, F, -1 \cdot \frac{x}{B}\right)} \]

       4. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{F \cdot \color{blue}{\sin B}}, F, -1 \cdot \frac{x}{B}\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, -1 \cdot \frac{x}{B}\right) \]

       6. lower-*.f64 35.8%

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, -1 \cdot \frac{x}{B}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, \mathsf{Rewrite=>}\left(lift-*.f64, \left(-1 \cdot \frac{x}{B}\right)\right)\right) \]

       8. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, \mathsf{Rewrite=>}\left(mul-1-neg, \left(\mathsf{neg}\left(\frac{x}{B}\right)\right)\right)\right) \]

       9. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, \mathsf{neg}\left(\mathsf{Rewrite=>}\left(lift-/.f64, \left(\frac{x}{B}\right)\right)\right)\right) \]

       10. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, \mathsf{Rewrite=>}\left(distribute-neg-frac, \left(\frac{\mathsf{neg}\left(x\right)}{B}\right)\right)\right) \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, \frac{\mathsf{Rewrite<=}\left(lift-neg.f64, \left(-x\right)\right)}{B}\right) \]

       12. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{-1}{\sin B \cdot \color{blue}{F}}, F, \mathsf{Rewrite=>}\left(lower-/.f64, \left(\frac{-x}{B}\right)\right)\right) \]

   11. Applied rewrites 35.8%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{\sin B \cdot F}, F, \frac{-x}{B}\right)} \]

   if -2.3e14 < F < 2.2000000000000001e25

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.7%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\color{blue}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 35.5%

      \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\color{blue}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if 2.2000000000000001e25 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.64 17.2%

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 52.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -2.75 \cdot 10^{+14}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{+25}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.75e+14)
   (/ -1.0 (sin B))
   (if (<= F 2.2e+25)
     (+
      (- (/ x B))
      (* (/ F B) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.75e+14) {
		tmp = -1.0 / sin(B);
	} else if (F <= 2.2e+25) {
		tmp = -(x / B) + ((F / B) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} 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 (f <= (-2.75d+14)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 2.2d+25) then
        tmp = -(x / b) + ((f / b) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
    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 (F <= -2.75e+14) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 2.2e+25) {
		tmp = -(x / B) + ((F / B) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.75e+14:
		tmp = -1.0 / math.sin(B)
	elif F <= 2.2e+25:
		tmp = -(x / B) + ((F / B) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.75e+14)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 2.2e+25)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F / B) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.75e+14)
		tmp = -1.0 / sin(B);
	elseif (F <= 2.2e+25)
		tmp = -(x / B) + ((F / B) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.75e+14], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.2e+25], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F / B), $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[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.75e14

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.64 18.0%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 18.0%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -2.75e14 < F < 2.2000000000000001e25

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.7%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\color{blue}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 35.5%

      \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\color{blue}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if 2.2000000000000001e25 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.64 17.2%

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 52.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -2.75 \cdot 10^{+14}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B}, -\frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.75e+14)
   (/ -1.0 (sin B))
   (if (<= F 1.5e+26)
     (fma F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) B) (- (/ x B)))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.75e+14) {
		tmp = -1.0 / sin(B);
	} else if (F <= 1.5e+26) {
		tmp = fma(F, (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / B), -(x / B));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.75e+14)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 1.5e+26)
		tmp = fma(F, Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / B), Float64(-Float64(x / B)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.75e+14], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.5e+26], N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / B), $MachinePrecision] + (-N[(x / B), $MachinePrecision])), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.75e14

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.64 18.0%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 18.0%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -2.75e14 < F < 1.49999999999999999e26

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.7%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-\frac{x}{B}\right)} \]

   6. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, -\frac{x}{B}\right)} \]

   7. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\color{blue}{B}}, -\frac{x}{B}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 43.3%

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\color{blue}{B}}, -\frac{x}{B}\right) \]

   if 1.49999999999999999e26 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.64 17.2%

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 43.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.56 \cdot 10^{-78}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-12}:\\ \;\;\;\;\frac{F \cdot \left(-1 \cdot \frac{x}{F}\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.56e-78)
   (/ -1.0 (sin B))
   (if (<= F 2.3e-12) (/ (* F (* -1.0 (/ x F))) B) (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.56e-78) {
		tmp = -1.0 / sin(B);
	} else if (F <= 2.3e-12) {
		tmp = (F * (-1.0 * (x / F))) / 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 (f <= (-1.56d-78)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 2.3d-12) then
        tmp = (f * ((-1.0d0) * (x / f))) / 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 (F <= -1.56e-78) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 2.3e-12) {
		tmp = (F * (-1.0 * (x / F))) / B;
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.56e-78:
		tmp = -1.0 / math.sin(B)
	elif F <= 2.3e-12:
		tmp = (F * (-1.0 * (x / F))) / B
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.56e-78)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 2.3e-12)
		tmp = Float64(Float64(F * Float64(-1.0 * Float64(x / F))) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.56e-78)
		tmp = -1.0 / sin(B);
	elseif (F <= 2.3e-12)
		tmp = (F * (-1.0 * (x / F))) / B;
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.56e-78], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.3e-12], N[(N[(F * N[(-1.0 * N[(x / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.56000000000000002e-78

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.64 18.0%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 18.0%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -1.56000000000000002e-78 < F < 2.29999999999999989e-12

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.64 47.0%

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.0%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      5. lower-/.f64 28.4%

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   7. Applied rewrites 28.4%

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{\color{blue}{B}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F}\right)}{B} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F}\right)}{B} \]

      2. lower-/.f64 27.7%

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F}\right)}{B} \]

   10. Applied rewrites 27.7%

       \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F}\right)}{B} \]

   if 2.29999999999999989e-12 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.64 17.2%

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 42.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.56 \cdot 10^{-78}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{-79}:\\ \;\;\;\;\frac{F \cdot \left(-1 \cdot \frac{x}{F}\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \left(x - 1\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.56e-78)
   (/ -1.0 (sin B))
   (if (<= F 2.35e-79) (/ (* F (* -1.0 (/ x F))) B) (/ (* -1.0 (- x 1.0)) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.56e-78) {
		tmp = -1.0 / sin(B);
	} else if (F <= 2.35e-79) {
		tmp = (F * (-1.0 * (x / F))) / B;
	} else {
		tmp = (-1.0 * (x - 1.0)) / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.56d-78)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 2.35d-79) then
        tmp = (f * ((-1.0d0) * (x / f))) / b
    else
        tmp = ((-1.0d0) * (x - 1.0d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.56e-78) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 2.35e-79) {
		tmp = (F * (-1.0 * (x / F))) / B;
	} else {
		tmp = (-1.0 * (x - 1.0)) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.56e-78:
		tmp = -1.0 / math.sin(B)
	elif F <= 2.35e-79:
		tmp = (F * (-1.0 * (x / F))) / B
	else:
		tmp = (-1.0 * (x - 1.0)) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.56e-78)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 2.35e-79)
		tmp = Float64(Float64(F * Float64(-1.0 * Float64(x / F))) / B);
	else
		tmp = Float64(Float64(-1.0 * Float64(x - 1.0)) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.56e-78)
		tmp = -1.0 / sin(B);
	elseif (F <= 2.35e-79)
		tmp = (F * (-1.0 * (x / F))) / B;
	else
		tmp = (-1.0 * (x - 1.0)) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.56e-78], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.35e-79], N[(N[(F * N[(-1.0 * N[(x / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(N[(-1.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.56000000000000002e-78

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.64 18.0%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 18.0%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -1.56000000000000002e-78 < F < 2.3500000000000001e-79

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.64 47.0%

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.0%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      5. lower-/.f64 28.4%

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   7. Applied rewrites 28.4%

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{\color{blue}{B}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F}\right)}{B} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F}\right)}{B} \]

      2. lower-/.f64 27.7%

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F}\right)}{B} \]

   10. Applied rewrites 27.7%

       \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F}\right)}{B} \]

   if 2.3500000000000001e-79 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.64 47.0%

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.0%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      5. lower-/.f64 28.4%

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   7. Applied rewrites 28.4%

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{\color{blue}{B}} \]

   8. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]

      2. lower--.f64 29.6%

         \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]

   10. Applied rewrites 29.6%

       \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 40.8% accurate, 5.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{F \cdot B}, -1 \cdot \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{-79}:\\ \;\;\;\;\frac{F \cdot \left(-1 \cdot \frac{x}{F}\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \left(x - 1\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.4e-237)
   (fma F (/ -1.0 (* F B)) (* -1.0 (/ x B)))
   (if (<= F 2.35e-79) (/ (* F (* -1.0 (/ x F))) B) (/ (* -1.0 (- x 1.0)) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.4e-237) {
		tmp = fma(F, (-1.0 / (F * B)), (-1.0 * (x / B)));
	} else if (F <= 2.35e-79) {
		tmp = (F * (-1.0 * (x / F))) / B;
	} else {
		tmp = (-1.0 * (x - 1.0)) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.4e-237)
		tmp = fma(F, Float64(-1.0 / Float64(F * B)), Float64(-1.0 * Float64(x / B)));
	elseif (F <= 2.35e-79)
		tmp = Float64(Float64(F * Float64(-1.0 * Float64(x / F))) / B);
	else
		tmp = Float64(Float64(-1.0 * Float64(x - 1.0)) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.4e-237], N[(F * N[(-1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.35e-79], N[(N[(F * N[(-1.0 * N[(x / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(N[(-1.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.4e-237

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. mult-flip N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right)} \]

   3. Applied rewrites 84.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in F around -inf

      \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{-1}{F \cdot \sin B}}, \frac{-x}{\tan B}\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{\color{blue}{F \cdot \sin B}}, \frac{-x}{\tan B}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{F \cdot \color{blue}{\sin B}}, \frac{-x}{\tan B}\right) \]

      3. lower-sin.64 54.6%

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, \frac{-x}{\tan B}\right) \]

   6. Applied rewrites 54.6%

      \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{-1}{F \cdot \sin B}}, \frac{-x}{\tan B}\right) \]

   7. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, -1 \cdot \color{blue}{\frac{x}{B}}\right) \]

      2. lower-/.f64 35.8%

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, -1 \cdot \frac{x}{\color{blue}{B}}\right) \]

   9. Applied rewrites 35.8%

      \[\leadsto \mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]

   10. Taylor expanded in B around 0

       \[\leadsto \mathsf{fma}\left(F, \frac{-1}{F \cdot B}, -1 \cdot \frac{x}{B}\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 28.6%

       \[\leadsto \mathsf{fma}\left(F, \frac{-1}{F \cdot B}, -1 \cdot \frac{x}{B}\right) \]

   if -2.4e-237 < F < 2.3500000000000001e-79

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.64 47.0%

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.0%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      5. lower-/.f64 28.4%

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   7. Applied rewrites 28.4%

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{\color{blue}{B}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F}\right)}{B} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F}\right)}{B} \]

      2. lower-/.f64 27.7%

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F}\right)}{B} \]

   10. Applied rewrites 27.7%

       \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F}\right)}{B} \]

   if 2.3500000000000001e-79 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.64 47.0%

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.0%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      5. lower-/.f64 28.4%

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   7. Applied rewrites 28.4%

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{\color{blue}{B}} \]

   8. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]

      2. lower--.f64 29.6%

         \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]

   10. Applied rewrites 29.6%

       \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 35.3% accurate, 5.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{-78}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{-79}:\\ \;\;\;\;\frac{F \cdot \left(-1 \cdot \frac{x}{F}\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \left(x - 1\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.7e-78)
   (/ -1.0 B)
   (if (<= F 2.35e-79) (/ (* F (* -1.0 (/ x F))) B) (/ (* -1.0 (- x 1.0)) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.7e-78) {
		tmp = -1.0 / B;
	} else if (F <= 2.35e-79) {
		tmp = (F * (-1.0 * (x / F))) / B;
	} else {
		tmp = (-1.0 * (x - 1.0)) / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.7d-78)) then
        tmp = (-1.0d0) / b
    else if (f <= 2.35d-79) then
        tmp = (f * ((-1.0d0) * (x / f))) / b
    else
        tmp = ((-1.0d0) * (x - 1.0d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.7e-78) {
		tmp = -1.0 / B;
	} else if (F <= 2.35e-79) {
		tmp = (F * (-1.0 * (x / F))) / B;
	} else {
		tmp = (-1.0 * (x - 1.0)) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.7e-78:
		tmp = -1.0 / B
	elif F <= 2.35e-79:
		tmp = (F * (-1.0 * (x / F))) / B
	else:
		tmp = (-1.0 * (x - 1.0)) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.7e-78)
		tmp = Float64(-1.0 / B);
	elseif (F <= 2.35e-79)
		tmp = Float64(Float64(F * Float64(-1.0 * Float64(x / F))) / B);
	else
		tmp = Float64(Float64(-1.0 * Float64(x - 1.0)) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.7e-78)
		tmp = -1.0 / B;
	elseif (F <= 2.35e-79)
		tmp = (F * (-1.0 * (x / F))) / B;
	else
		tmp = (-1.0 * (x - 1.0)) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.7e-78], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 2.35e-79], N[(N[(F * N[(-1.0 * N[(x / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(N[(-1.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.70000000000000006e-78

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.64 18.0%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 18.0%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-1}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 10.8%

      \[\leadsto \frac{-1}{B} \]

   if -1.70000000000000006e-78 < F < 2.3500000000000001e-79

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.64 47.0%

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.0%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      5. lower-/.f64 28.4%

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   7. Applied rewrites 28.4%

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{\color{blue}{B}} \]

   8. Taylor expanded in x around inf

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F}\right)}{B} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F}\right)}{B} \]

      2. lower-/.f64 27.7%

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F}\right)}{B} \]

   10. Applied rewrites 27.7%

       \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F}\right)}{B} \]

   if 2.3500000000000001e-79 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.64 47.0%

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.0%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      5. lower-/.f64 28.4%

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   7. Applied rewrites 28.4%

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{\color{blue}{B}} \]

   8. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]

      2. lower--.f64 29.6%

         \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]

   10. Applied rewrites 29.6%

       \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 30.3% accurate, 8.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.85 \cdot 10^{-75}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot \left(x - 1\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.85e-75) (/ -1.0 B) (/ (* -1.0 (- x 1.0)) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.85e-75) {
		tmp = -1.0 / B;
	} else {
		tmp = (-1.0 * (x - 1.0)) / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.85d-75)) then
        tmp = (-1.0d0) / b
    else
        tmp = ((-1.0d0) * (x - 1.0d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.85e-75) {
		tmp = -1.0 / B;
	} else {
		tmp = (-1.0 * (x - 1.0)) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.85e-75:
		tmp = -1.0 / B
	else:
		tmp = (-1.0 * (x - 1.0)) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.85e-75)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(Float64(-1.0 * Float64(x - 1.0)) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.85e-75)
		tmp = -1.0 / B;
	else
		tmp = (-1.0 * (x - 1.0)) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.85e-75], N[(-1.0 / B), $MachinePrecision], N[(N[(-1.0 * N[(x - 1.0), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.85000000000000012e-75

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.64 18.0%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 18.0%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-1}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 10.8%

      \[\leadsto \frac{-1}{B} \]

   if -1.85000000000000012e-75 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.64 47.0%

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.0%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      5. lower-/.f64 28.4%

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   7. Applied rewrites 28.4%

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{\color{blue}{B}} \]

   8. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]

      2. lower--.f64 29.6%

         \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]

   10. Applied rewrites 29.6%

       \[\leadsto \frac{-1 \cdot \left(x - 1\right)}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 18.0% accurate, 14.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq 0.00082:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (if (<= F 0.00082) (/ -1.0 B) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 0.00082) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / 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 <= 0.00082d0) then
        tmp = (-1.0d0) / b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 0.00082) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 0.00082:
		tmp = -1.0 / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 0.00082)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 0.00082)
		tmp = -1.0 / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 0.00082], N[(-1.0 / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 8.1999999999999998e-4

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.64 18.0%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 18.0%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-1}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 10.8%

      \[\leadsto \frac{-1}{B} \]

   if 8.1999999999999998e-4 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.64 47.0%

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.0%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      5. lower-/.f64 28.4%

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   7. Applied rewrites 28.4%

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{\color{blue}{B}} \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{1}{B} \]
   9. Step-by-step derivation

   10. Applied rewrites 10.1%

       \[\leadsto \frac{1}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 10.1% accurate, 26.5× speedup

Math

\[\frac{1}{B} \]

FPCore

(FPCore (F B x) :precision binary64 (/ 1.0 B))

C

double code(double F, double B, double x) {
	return 1.0 / B;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = 1.0d0 / b
end function

Java

public static double code(double F, double B, double x) {
	return 1.0 / B;
}

Python

def code(F, B, x):
	return 1.0 / B

Julia

function code(F, B, x)
	return Float64(1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = 1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(1.0 / B), $MachinePrecision]

Derivation

1. Initial program 76.3%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in F around inf

   \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

   3. lower-/.f64 N/A

      \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

   4. lower-*.f64 N/A

      \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   5. lower-cos.64 N/A

      \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   6. lower-*.f64 N/A

      \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

   7. lower-sin.64 N/A

      \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   8. lower-/.f64 N/A

      \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   9. lower-*.f64 N/A

      \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   10. lower-sin.64 47.0%

       \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

4. Applied rewrites 47.0%

   \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

5. Taylor expanded in B around 0

   \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{\color{blue}{B}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   4. lower-/.f64 N/A

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   5. lower-/.f64 28.4%

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

7. Applied rewrites 28.4%

   \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{\color{blue}{B}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{1}{B} \]
9. Step-by-step derivation

10. Applied rewrites 10.1%

    \[\leadsto \frac{1}{B} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025183 
(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))))))