VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.6% → 99.7%

Time: 7.3s

Alternatives: 24

Speedup: 1.8×

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: 76.6% 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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4 \cdot 10^{+48}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\frac{\sin B}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}}, F, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (/ x (tan B))))
  (if (<= F -4e+48)
    (- (/ -1.0 (sin B)) t_0)
    (if (<= F 100000000.0)
      (fma
       (/ 1.0 (/ (sin B) (pow (fma x 2.0 (fma F F 2.0)) -0.5)))
       F
       (/ (- x) (tan B)))
      (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -4e+48) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 100000000.0) {
		tmp = fma((1.0 / (sin(B) / pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5))), F, (-x / tan(B)));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -4e+48)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 100000000.0)
		tmp = fma(Float64(1.0 / Float64(sin(B) / (fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5))), F, Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4e+48], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 100000000.0], N[(N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.0000000000000002e48

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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 76.6%

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

   3. Applied rewrites 85.5%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.1%

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

   if -4.0000000000000002e48 < F < 1e8

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.4%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 85.4%

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

      5. lift-pow.f64 N/A

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

      6. sqr-pow N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-+r+ N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

   5. Applied rewrites 85.4%

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

   if 1e8 < F

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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 76.6%

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

   3. Applied rewrites 85.5%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 56.6%

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

Alternative 2: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4 \cdot 10^{+48}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, F, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (/ x (tan B))))
  (if (<= F -4e+48)
    (- (/ -1.0 (sin B)) t_0)
    (if (<= F 100000000.0)
      (fma
       (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B))
       F
       (/ (- x) (tan B)))
      (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -4e+48) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 100000000.0) {
		tmp = fma((pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), F, (-x / tan(B)));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -4e+48)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 100000000.0)
		tmp = fma(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), F, Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4e+48], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 100000000.0], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] * F + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.0000000000000002e48

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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 76.6%

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

   3. Applied rewrites 85.5%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.1%

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

   if -4.0000000000000002e48 < F < 1e8

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.4%

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

   if 1e8 < F

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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 76.6%

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

   3. Applied rewrites 85.5%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 56.6%

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

Alternative 3: 99.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1000000000000:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot \sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (/ x (tan B))))
  (if (<= F -5e+27)
    (- (/ -1.0 (sin B)) t_0)
    (if (<= F 1000000000000.0)
      (- (/ F (* (sqrt (fma 2.0 x (fma F F 2.0))) (sin B))) t_0)
      (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -5e+27) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1000000000000.0) {
		tmp = (F / (sqrt(fma(2.0, x, fma(F, F, 2.0))) * sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -5e+27)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1000000000000.0)
		tmp = Float64(Float64(F / Float64(sqrt(fma(2.0, x, fma(F, F, 2.0))) * sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5e+27], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1000000000000.0], N[(N[(F / N[(N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.9999999999999998e27

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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 76.6%

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

   3. Applied rewrites 85.5%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.1%

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

   if -4.9999999999999998e27 < F < 1e12

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.4%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 85.4%

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

      5. lift-pow.f64 N/A

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

      6. sqr-pow N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-+r+ N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

   5. Applied rewrites 85.4%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

   7. Applied rewrites 85.5%

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

   if 1e12 < F

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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 76.6%

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

   3. Applied rewrites 85.5%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 56.6%

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

Alternative 4: 99.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -5700:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.00036:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, 2\right)} \cdot \sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (/ x (tan B))))
  (if (<= F -5700.0)
    (- (/ -1.0 (sin B)) t_0)
    (if (<= F 0.00036)
      (- (/ F (* (sqrt (fma 2.0 x 2.0)) (sin B))) t_0)
      (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -5700.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.00036) {
		tmp = (F / (sqrt(fma(2.0, x, 2.0)) * sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -5700.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.00036)
		tmp = Float64(Float64(F / Float64(sqrt(fma(2.0, x, 2.0)) * sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5700.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.00036], N[(N[(F / N[(N[Sqrt[N[(2.0 * x + 2.0), $MachinePrecision]], $MachinePrecision] * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5700

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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 76.6%

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

   3. Applied rewrites 85.5%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.1%

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

   if -5700 < F < 3.6000000000000002e-4

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.4%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 85.4%

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

      5. lift-pow.f64 N/A

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

      6. sqr-pow N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-+r+ N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

   5. Applied rewrites 85.4%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

   7. Applied rewrites 85.5%

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

   8. Taylor expanded in F around 0

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

   10. Applied rewrites 56.2%

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

   if 3.6000000000000002e-4 < F

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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 76.6%

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

   3. Applied rewrites 85.5%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 56.6%

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

Alternative 5: 91.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6800:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 110000000:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (/ x (tan B))))
  (if (<= F -6800.0)
    (- (/ -1.0 (sin B)) t_0)
    (if (<= F 110000000.0)
      (+
       (- (* x (/ 1.0 (tan B))))
       (*
        (/ F B)
        (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
      (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -6800.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 110000000.0) {
		tmp = -(x * (1.0 / tan(B))) + ((F / B) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-6800.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 110000000.0d0) then
        tmp = -(x * (1.0d0 / tan(b))) + ((f / b) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -6800.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 110000000.0) {
		tmp = -(x * (1.0 / Math.tan(B))) + ((F / B) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -6800.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 110000000.0:
		tmp = -(x * (1.0 / math.tan(B))) + ((F / B) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -6800.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 110000000.0)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(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(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -6800.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 110000000.0)
		tmp = -(x * (1.0 / tan(B))) + ((F / B) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -6800.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 110000000.0], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $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[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -6800

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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 76.6%

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

   3. Applied rewrites 85.5%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.1%

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

   if -6800 < F < 1.1e8

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.3%

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

   4. Applied rewrites 62.3%

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

   if 1.1e8 < F

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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 76.6%

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

   3. Applied rewrites 85.5%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 56.6%

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

Alternative 6: 90.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -26:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq -1.42 \cdot 10^{-151}:\\ \;\;\;\;\frac{{t\_0}^{-0.5} \cdot F}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-131}:\\ \;\;\;\;-1 \cdot \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 920000000000:\\ \;\;\;\;\frac{F}{\sqrt{t\_0} \cdot \sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_1\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (fma 2.0 x (fma F F 2.0))) (t_1 (/ x (tan B))))
  (if (<= F -26.0)
    (- (/ -1.0 (sin B)) t_1)
    (if (<= F -1.42e-151)
      (- (/ (* (pow t_0 -0.5) F) (sin B)) (/ x B))
      (if (<= F 6.5e-131)
        (* -1.0 (/ (* x (cos B)) (sin B)))
        (if (<= F 920000000000.0)
          (- (/ F (* (sqrt t_0) (sin B))) (/ x B))
          (- (/ 1.0 (sin B)) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = fma(2.0, x, fma(F, F, 2.0));
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -26.0) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -1.42e-151) {
		tmp = ((pow(t_0, -0.5) * F) / sin(B)) - (x / B);
	} else if (F <= 6.5e-131) {
		tmp = -1.0 * ((x * cos(B)) / sin(B));
	} else if (F <= 920000000000.0) {
		tmp = (F / (sqrt(t_0) * sin(B))) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(2.0, x, fma(F, F, 2.0))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -26.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -1.42e-151)
		tmp = Float64(Float64(Float64((t_0 ^ -0.5) * F) / sin(B)) - Float64(x / B));
	elseif (F <= 6.5e-131)
		tmp = Float64(-1.0 * Float64(Float64(x * cos(B)) / sin(B)));
	elseif (F <= 920000000000.0)
		tmp = Float64(Float64(F / Float64(sqrt(t_0) * sin(B))) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -26.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -1.42e-151], N[(N[(N[(N[Power[t$95$0, -0.5], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.5e-131], N[(-1.0 * N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 920000000000.0], N[(N[(F / N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -26

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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 76.6%

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

   3. Applied rewrites 85.5%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.1%

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

   if -26 < F < -1.42e-151

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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 76.6%

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

   3. Applied rewrites 85.5%

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

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 58.5%

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

   6. Applied rewrites 58.5%

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

   if -1.42e-151 < F < 6.5000000000000002e-131

   1. Initial program 76.6%

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

   2. Taylor expanded in F around 0

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

      1. 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.f64 N/A

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

      5. lower-sin.f64 56.1%

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

   4. Applied rewrites 56.1%

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

   if 6.5000000000000002e-131 < F < 9.2e11

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.4%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 85.4%

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

      5. lift-pow.f64 N/A

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

      6. sqr-pow N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-+r+ N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

   5. Applied rewrites 85.4%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

   7. Applied rewrites 85.5%

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

   8. Taylor expanded in B around 0

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

      1. lower-/.f64 58.5%

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

   10. Applied rewrites 58.5%

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

   if 9.2e11 < F

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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 76.6%

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

   3. Applied rewrites 85.5%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 56.6%

      \[\leadsto \frac{\color{blue}{1}}{\sin B} - \frac{x}{\tan B} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 7: 83.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -26:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq -1.42 \cdot 10^{-151}:\\ \;\;\;\;\frac{{t\_0}^{-0.5} \cdot F}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-131}:\\ \;\;\;\;-1 \cdot \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 920000000000:\\ \;\;\;\;\frac{F}{\sqrt{t\_0} \cdot \sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(B \cdot B, -0.16666666666666666, 1\right) \cdot B} - t\_1\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (fma 2.0 x (fma F F 2.0))) (t_1 (/ x (tan B))))
  (if (<= F -26.0)
    (- (/ -1.0 (sin B)) t_1)
    (if (<= F -1.42e-151)
      (- (/ (* (pow t_0 -0.5) F) (sin B)) (/ x B))
      (if (<= F 6.5e-131)
        (* -1.0 (/ (* x (cos B)) (sin B)))
        (if (<= F 920000000000.0)
          (- (/ F (* (sqrt t_0) (sin B))) (/ x B))
          (-
           (/ 1.0 (* (fma (* B B) -0.16666666666666666 1.0) B))
           t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = fma(2.0, x, fma(F, F, 2.0));
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -26.0) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -1.42e-151) {
		tmp = ((pow(t_0, -0.5) * F) / sin(B)) - (x / B);
	} else if (F <= 6.5e-131) {
		tmp = -1.0 * ((x * cos(B)) / sin(B));
	} else if (F <= 920000000000.0) {
		tmp = (F / (sqrt(t_0) * sin(B))) - (x / B);
	} else {
		tmp = (1.0 / (fma((B * B), -0.16666666666666666, 1.0) * B)) - t_1;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(2.0, x, fma(F, F, 2.0))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -26.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -1.42e-151)
		tmp = Float64(Float64(Float64((t_0 ^ -0.5) * F) / sin(B)) - Float64(x / B));
	elseif (F <= 6.5e-131)
		tmp = Float64(-1.0 * Float64(Float64(x * cos(B)) / sin(B)));
	elseif (F <= 920000000000.0)
		tmp = Float64(Float64(F / Float64(sqrt(t_0) * sin(B))) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / Float64(fma(Float64(B * B), -0.16666666666666666, 1.0) * B)) - t_1);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -26.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -1.42e-151], N[(N[(N[(N[Power[t$95$0, -0.5], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.5e-131], N[(-1.0 * N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 920000000000.0], N[(N[(F / N[(N[Sqrt[t$95$0], $MachinePrecision] * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(N[(B * B), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -26

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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 76.6%

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

   3. Applied rewrites 85.5%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.1%

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

   if -26 < F < -1.42e-151

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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 76.6%

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

   3. Applied rewrites 85.5%

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

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 58.5%

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

   6. Applied rewrites 58.5%

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

   if -1.42e-151 < F < 6.5000000000000002e-131

   1. Initial program 76.6%

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

   2. Taylor expanded in F around 0

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

      1. 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.f64 N/A

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

      5. lower-sin.f64 56.1%

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

   4. Applied rewrites 56.1%

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

   if 6.5000000000000002e-131 < F < 9.2e11

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.4%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 85.4%

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

      5. lift-pow.f64 N/A

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

      6. sqr-pow N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-+r+ N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

   5. Applied rewrites 85.4%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

   7. Applied rewrites 85.5%

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

   8. Taylor expanded in B around 0

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

      1. lower-/.f64 58.5%

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

   10. Applied rewrites 58.5%

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

   if 9.2e11 < F

   1. Initial program 76.6%

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

   2. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 56.5%

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

   4. Applied rewrites 56.5%

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

   5. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 55.6%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{2}\right)} \]

   7. Applied rewrites 55.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{B \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {B}^{2}\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lift-/.f64 N/A

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

      10. sub-flip-reverse N/A

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

   9. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(B \cdot B, -0.16666666666666666, 1\right) \cdot B} - \frac{x}{\tan B}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 8: 77.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := -1 \cdot \frac{x \cdot \cos B}{\sin B}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-33}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-44}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (* -1.0 (/ (* x (cos B)) (sin B)))))
  (if (<= x -2.5e-33)
    t_0
    (if (<= x 5.2e-44)
      (-
       (/ (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F) (sin B))
       (/ x B))
      t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 * ((x * cos(B)) / sin(B));
	double tmp;
	if (x <= -2.5e-33) {
		tmp = t_0;
	} else if (x <= 5.2e-44) {
		tmp = ((pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F) / sin(B)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 * Float64(Float64(x * cos(B)) / sin(B)))
	tmp = 0.0
	if (x <= -2.5e-33)
		tmp = t_0;
	elseif (x <= 5.2e-44)
		tmp = Float64(Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F) / sin(B)) - Float64(x / B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 * N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.5e-33], t$95$0, If[LessEqual[x, 5.2e-44], N[(N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5000000000000001e-33 or 5.1999999999999996e-44 < x

   1. Initial program 76.6%

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

   2. Taylor expanded in F around 0

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

      1. 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.f64 N/A

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

      5. lower-sin.f64 56.1%

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

   4. Applied rewrites 56.1%

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

   if -2.5000000000000001e-33 < x < 5.1999999999999996e-44

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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 76.6%

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

   3. Applied rewrites 85.5%

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

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 58.5%

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

   6. Applied rewrites 58.5%

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

Alternative 9: 77.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{fma}\left(B \cdot B, -0.16666666666666666, 1\right) \cdot B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-33}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 37000000:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0
        (-
         (/ 1.0 (* (fma (* B B) -0.16666666666666666 1.0) B))
         (/ x (tan B)))))
  (if (<= x -2.5e-33)
    t_0
    (if (<= x 37000000.0)
      (-
       (/ (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F) (sin B))
       (/ x B))
      t_0))))

C

double code(double F, double B, double x) {
	double t_0 = (1.0 / (fma((B * B), -0.16666666666666666, 1.0) * B)) - (x / tan(B));
	double tmp;
	if (x <= -2.5e-33) {
		tmp = t_0;
	} else if (x <= 37000000.0) {
		tmp = ((pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F) / sin(B)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(1.0 / Float64(fma(Float64(B * B), -0.16666666666666666, 1.0) * B)) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -2.5e-33)
		tmp = t_0;
	elseif (x <= 37000000.0)
		tmp = Float64(Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F) / sin(B)) - Float64(x / B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(1.0 / N[(N[(N[(B * B), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.5e-33], t$95$0, If[LessEqual[x, 37000000.0], N[(N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5000000000000001e-33 or 3.7e7 < x

   1. Initial program 76.6%

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

   2. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 56.5%

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

   4. Applied rewrites 56.5%

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

   5. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 55.6%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{2}\right)} \]

   7. Applied rewrites 55.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{B \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {B}^{2}\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lift-/.f64 N/A

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

      10. sub-flip-reverse N/A

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

   9. Applied rewrites 55.7%

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

   if -2.5000000000000001e-33 < x < 3.7e7

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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 76.6%

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

   3. Applied rewrites 85.5%

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

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 58.5%

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

   6. Applied rewrites 58.5%

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

Alternative 10: 77.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{fma}\left(B \cdot B, -0.16666666666666666, 1\right) \cdot B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -2.5 \cdot 10^{-33}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 37000000:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot \sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0
        (-
         (/ 1.0 (* (fma (* B B) -0.16666666666666666 1.0) B))
         (/ x (tan B)))))
  (if (<= x -2.5e-33)
    t_0
    (if (<= x 37000000.0)
      (- (/ F (* (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 = (1.0 / (fma((B * B), -0.16666666666666666, 1.0) * B)) - (x / tan(B));
	double tmp;
	if (x <= -2.5e-33) {
		tmp = t_0;
	} else if (x <= 37000000.0) {
		tmp = (F / (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 = Float64(Float64(1.0 / Float64(fma(Float64(B * B), -0.16666666666666666, 1.0) * B)) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -2.5e-33)
		tmp = t_0;
	elseif (x <= 37000000.0)
		tmp = Float64(Float64(F / Float64(sqrt(fma(2.0, x, fma(F, F, 2.0))) * sin(B))) - Float64(x / B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(1.0 / N[(N[(N[(B * B), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.5e-33], t$95$0, If[LessEqual[x, 37000000.0], N[(N[(F / N[(N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.5000000000000001e-33 or 3.7e7 < x

   1. Initial program 76.6%

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

   2. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 56.5%

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

   4. Applied rewrites 56.5%

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

   5. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 55.6%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{2}\right)} \]

   7. Applied rewrites 55.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{B \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {B}^{2}\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lift-/.f64 N/A

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

      10. sub-flip-reverse N/A

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

   9. Applied rewrites 55.7%

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

   if -2.5000000000000001e-33 < x < 3.7e7

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.4%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 85.4%

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

      5. lift-pow.f64 N/A

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

      6. sqr-pow N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-+r+ N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

   5. Applied rewrites 85.4%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

   7. Applied rewrites 85.5%

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

   8. Taylor expanded in B around 0

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

      1. lower-/.f64 58.5%

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

   10. Applied rewrites 58.5%

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

Alternative 11: 68.8% accurate, 1.7× speedup

Math

\[\mathsf{copysign}\left(1, B\right) \cdot \begin{array}{l} \mathbf{if}\;\left|B\right| \leq 8.4 \cdot 10^{+19}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{2 + \mathsf{fma}\left(2, x, {F}^{2}\right)}} - x}{\left|B\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{fma}\left(\left|B\right| \cdot \left|B\right|, -0.16666666666666666, 1\right) \cdot \left|B\right|} - \frac{x}{\tan \left(\left|B\right|\right)}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (*
 (copysign 1.0 B)
 (if (<= (fabs B) 8.4e+19)
   (/ (- (/ F (sqrt (+ 2.0 (fma 2.0 x (pow F 2.0))))) x) (fabs B))
   (-
    (/
     1.0
     (*
      (fma (* (fabs B) (fabs B)) -0.16666666666666666 1.0)
      (fabs B)))
    (/ x (tan (fabs B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (fabs(B) <= 8.4e+19) {
		tmp = ((F / sqrt((2.0 + fma(2.0, x, pow(F, 2.0))))) - x) / fabs(B);
	} else {
		tmp = (1.0 / (fma((fabs(B) * fabs(B)), -0.16666666666666666, 1.0) * fabs(B))) - (x / tan(fabs(B)));
	}
	return copysign(1.0, B) * tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (abs(B) <= 8.4e+19)
		tmp = Float64(Float64(Float64(F / sqrt(Float64(2.0 + fma(2.0, x, (F ^ 2.0))))) - x) / abs(B));
	else
		tmp = Float64(Float64(1.0 / Float64(fma(Float64(abs(B) * abs(B)), -0.16666666666666666, 1.0) * 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], 8.4e+19], N[(N[(N[(F / N[Sqrt[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[Abs[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(N[(N[Abs[B], $MachinePrecision] * N[Abs[B], $MachinePrecision]), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * 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 < 8.4e19

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.4%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 85.4%

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

      5. lift-pow.f64 N/A

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

      6. sqr-pow N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-+r+ N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

   5. Applied rewrites 85.4%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

   7. Applied rewrites 85.5%

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

   8. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-pow.f64 45.0%

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

   10. Applied rewrites 45.0%

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

   if 8.4e19 < B

   1. Initial program 76.6%

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

   2. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 56.5%

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

   4. Applied rewrites 56.5%

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

   5. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 55.6%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{2}\right)} \]

   7. Applied rewrites 55.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{B \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {B}^{2}\right)}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lift-/.f64 N/A

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

      10. sub-flip-reverse N/A

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

   9. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{fma}\left(B \cdot B, -0.16666666666666666, 1\right) \cdot B} - \frac{x}{\tan B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 67.5% accurate, 1.8× speedup

Math

\[\mathsf{copysign}\left(1, B\right) \cdot \begin{array}{l} \mathbf{if}\;\left|B\right| \leq 7.6 \cdot 10^{+18}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{2 + \mathsf{fma}\left(2, x, {F}^{2}\right)}} - x}{\left|B\right|}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\left|B\right| \cdot F}, F, \frac{-x}{\tan \left(\left|B\right|\right)}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (*
 (copysign 1.0 B)
 (if (<= (fabs B) 7.6e+18)
   (/ (- (/ F (sqrt (+ 2.0 (fma 2.0 x (pow F 2.0))))) x) (fabs B))
   (fma (/ 1.0 (* (fabs B) F)) F (/ (- x) (tan (fabs B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (fabs(B) <= 7.6e+18) {
		tmp = ((F / sqrt((2.0 + fma(2.0, x, pow(F, 2.0))))) - x) / fabs(B);
	} else {
		tmp = fma((1.0 / (fabs(B) * F)), F, (-x / tan(fabs(B))));
	}
	return copysign(1.0, B) * tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (abs(B) <= 7.6e+18)
		tmp = Float64(Float64(Float64(F / sqrt(Float64(2.0 + fma(2.0, x, (F ^ 2.0))))) - x) / abs(B));
	else
		tmp = fma(Float64(1.0 / Float64(abs(B) * F)), 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], 7.6e+18], N[(N[(N[(F / N[Sqrt[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[Abs[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[Abs[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * F + N[((-x) / N[Tan[N[Abs[B], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if B < 7.6e18

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.4%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 85.4%

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

      5. lift-pow.f64 N/A

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

      6. sqr-pow N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-+r+ N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

   5. Applied rewrites 85.4%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

   7. Applied rewrites 85.5%

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

   8. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-pow.f64 45.0%

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

   10. Applied rewrites 45.0%

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

   if 7.6e18 < B

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.4%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.9%

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

   6. Applied rewrites 52.9%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 50.6%

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

   9. Applied rewrites 50.6%

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

Alternative 13: 52.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.16 \cdot 10^{+171}:\\ \;\;\;\;\frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{2 + \mathsf{fma}\left(2, x, {F}^{2}\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -1.16e+171)
  (/ (* -1.0 (* F (+ (/ 1.0 F) (/ x F)))) B)
  (if (<= F 9.5e+130)
    (/ (- (/ F (sqrt (+ 2.0 (fma 2.0 x (pow F 2.0))))) x) B)
    (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.16e+171) {
		tmp = (-1.0 * (F * ((1.0 / F) + (x / F)))) / B;
	} else if (F <= 9.5e+130) {
		tmp = ((F / sqrt((2.0 + fma(2.0, x, pow(F, 2.0))))) - x) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.16e+171)
		tmp = Float64(Float64(-1.0 * Float64(F * Float64(Float64(1.0 / F) + Float64(x / F)))) / B);
	elseif (F <= 9.5e+130)
		tmp = Float64(Float64(Float64(F / sqrt(Float64(2.0 + fma(2.0, x, (F ^ 2.0))))) - x) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.16e+171], N[(N[(-1.0 * N[(F * N[(N[(1.0 / F), $MachinePrecision] + N[(x / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 9.5e+130], N[(N[(N[(F / N[Sqrt[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.16e171

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 45.0%

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

   5. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 29.0%

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

   7. Applied rewrites 29.0%

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

   if -1.16e171 < F < 9.5000000000000009e130

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.4%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 85.4%

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

      5. lift-pow.f64 N/A

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

      6. sqr-pow N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-+r+ N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

   5. Applied rewrites 85.4%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

   7. Applied rewrites 85.5%

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

   8. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-pow.f64 45.0%

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

   10. Applied rewrites 45.0%

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

   if 9.5000000000000009e130 < F

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.4%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 85.4%

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

      5. lift-pow.f64 N/A

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

      6. sqr-pow N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-+r+ N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 16.6%

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

   8. Applied rewrites 16.6%

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

Alternative 14: 52.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.16 \cdot 10^{+171}:\\ \;\;\;\;\frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+85}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{2 + \mathsf{fma}\left(2, x, {F}^{2}\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -1.16e+171)
  (/ (* -1.0 (* F (+ (/ 1.0 F) (/ x F)))) B)
  (if (<= F 6e+85)
    (/ (- (/ F (sqrt (+ 2.0 (fma 2.0 x (pow F 2.0))))) x) B)
    (/ (* F (fma -1.0 (/ x F) (/ 1.0 F))) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.16e+171) {
		tmp = (-1.0 * (F * ((1.0 / F) + (x / F)))) / B;
	} else if (F <= 6e+85) {
		tmp = ((F / sqrt((2.0 + fma(2.0, x, pow(F, 2.0))))) - x) / B;
	} else {
		tmp = (F * fma(-1.0, (x / F), (1.0 / F))) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.16e+171)
		tmp = Float64(Float64(-1.0 * Float64(F * Float64(Float64(1.0 / F) + Float64(x / F)))) / B);
	elseif (F <= 6e+85)
		tmp = Float64(Float64(Float64(F / sqrt(Float64(2.0 + fma(2.0, x, (F ^ 2.0))))) - x) / B);
	else
		tmp = Float64(Float64(F * fma(-1.0, Float64(x / F), Float64(1.0 / F))) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.16e+171], N[(N[(-1.0 * N[(F * N[(N[(1.0 / F), $MachinePrecision] + N[(x / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 6e+85], N[(N[(N[(F / N[Sqrt[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(F * N[(-1.0 * N[(x / F), $MachinePrecision] + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.16e171

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 45.0%

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

   5. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 29.0%

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

   7. Applied rewrites 29.0%

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

   if -1.16e171 < F < 6.0000000000000001e85

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.4%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 85.4%

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

      5. lift-pow.f64 N/A

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

      6. sqr-pow N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-+r+ N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

   5. Applied rewrites 85.4%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

   7. Applied rewrites 85.5%

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

   8. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-pow.f64 45.0%

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

   10. Applied rewrites 45.0%

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

   if 6.0000000000000001e85 < F

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 45.0%

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

   5. Taylor expanded in F around inf

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]
   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-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 29.1%

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

   7. Applied rewrites 29.1%

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

Alternative 15: 51.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.92 \cdot 10^{+146}:\\ \;\;\;\;-1 \cdot \left(F \cdot \left(\frac{1}{B \cdot F} + \frac{x}{B \cdot F}\right)\right)\\ \mathbf{elif}\;F \leq 10^{+124}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -1.92e+146)
  (* -1.0 (* F (+ (/ 1.0 (* B F)) (/ x (* B F)))))
  (if (<= F 1e+124)
    (/ (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F) x) B)
    (/ (* F (fma -1.0 (/ x F) (/ 1.0 F))) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.92e+146) {
		tmp = -1.0 * (F * ((1.0 / (B * F)) + (x / (B * F))));
	} else if (F <= 1e+124) {
		tmp = ((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F) - x) / B;
	} else {
		tmp = (F * fma(-1.0, (x / F), (1.0 / F))) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.92e+146)
		tmp = Float64(-1.0 * Float64(F * Float64(Float64(1.0 / Float64(B * F)) + Float64(x / Float64(B * F)))));
	elseif (F <= 1e+124)
		tmp = Float64(Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F) - x) / B);
	else
		tmp = Float64(Float64(F * fma(-1.0, Float64(x / F), Float64(1.0 / F))) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.92e+146], N[(-1.0 * N[(F * N[(N[(1.0 / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e+124], N[(N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(F * N[(-1.0 * N[(x / F), $MachinePrecision] + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.9199999999999999e146

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 45.0%

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

   5. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 27.7%

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

   7. Applied rewrites 27.7%

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

   if -1.9199999999999999e146 < F < 9.9999999999999995e123

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 45.0%

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

   6. Applied rewrites 45.0%

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

   if 9.9999999999999995e123 < F

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 45.0%

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

   5. Taylor expanded in F around inf

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]
   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-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 29.1%

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

   7. Applied rewrites 29.1%

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

Alternative 16: 51.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -5700:\\ \;\;\;\;\frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B}\\ \mathbf{elif}\;F \leq 0.00036:\\ \;\;\;\;\frac{F \cdot {\left(2 + 2 \cdot x\right)}^{-0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -5700.0)
  (/ (* -1.0 (* F (+ (/ 1.0 F) (/ x F)))) B)
  (if (<= F 0.00036)
    (/ (- (* F (pow (+ 2.0 (* 2.0 x)) -0.5)) x) B)
    (/ (* F (fma -1.0 (/ x F) (/ 1.0 F))) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5700.0) {
		tmp = (-1.0 * (F * ((1.0 / F) + (x / F)))) / B;
	} else if (F <= 0.00036) {
		tmp = ((F * pow((2.0 + (2.0 * x)), -0.5)) - x) / B;
	} else {
		tmp = (F * fma(-1.0, (x / F), (1.0 / F))) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5700.0)
		tmp = Float64(Float64(-1.0 * Float64(F * Float64(Float64(1.0 / F) + Float64(x / F)))) / B);
	elseif (F <= 0.00036)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + Float64(2.0 * x)) ^ -0.5)) - x) / B);
	else
		tmp = Float64(Float64(F * fma(-1.0, Float64(x / F), Float64(1.0 / F))) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5700.0], N[(N[(-1.0 * N[(F * N[(N[(1.0 / F), $MachinePrecision] + N[(x / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 0.00036], N[(N[(N[(F * N[Power[N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(F * N[(-1.0 * N[(x / F), $MachinePrecision] + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5700

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 45.0%

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

   5. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 29.0%

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

   7. Applied rewrites 29.0%

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

   if -5700 < F < 3.6000000000000002e-4

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 45.0%

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

   5. Taylor expanded in F around 0

      \[\leadsto \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{-0.5} - x}{B} \]
   6. Step-by-step derivation

      1. lower-*.f64 30.7%

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

   7. Applied rewrites 30.7%

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

   if 3.6000000000000002e-4 < F

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 45.0%

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

   5. Taylor expanded in F around inf

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]
   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-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 29.1%

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

   7. Applied rewrites 29.1%

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

Alternative 17: 50.7% accurate, 2.4× speedup

Math

\[\mathsf{copysign}\left(1, B\right) \cdot \begin{array}{l} \mathbf{if}\;\left|B\right| \leq 4.5 \cdot 10^{+20}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{2 + \mathsf{fma}\left(2, x, {F}^{2}\right)}} - x}{\left|B\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin \left(\left|B\right|\right)}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (*
 (copysign 1.0 B)
 (if (<= (fabs B) 4.5e+20)
   (/ (- (/ F (sqrt (+ 2.0 (fma 2.0 x (pow F 2.0))))) x) (fabs B))
   (/ -1.0 (sin (fabs B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (fabs(B) <= 4.5e+20) {
		tmp = ((F / sqrt((2.0 + fma(2.0, x, pow(F, 2.0))))) - x) / fabs(B);
	} else {
		tmp = -1.0 / sin(fabs(B));
	}
	return copysign(1.0, B) * tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (abs(B) <= 4.5e+20)
		tmp = Float64(Float64(Float64(F / sqrt(Float64(2.0 + fma(2.0, x, (F ^ 2.0))))) - x) / abs(B));
	else
		tmp = Float64(-1.0 / sin(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], 4.5e+20], N[(N[(N[(F / N[Sqrt[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[Abs[B], $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[Sin[N[Abs[B], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if B < 4.5e20

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.4%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 85.4%

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

      5. lift-pow.f64 N/A

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

      6. sqr-pow N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-+r+ N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

   5. Applied rewrites 85.4%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

   7. Applied rewrites 85.5%

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

   8. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-pow.f64 45.0%

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

   10. Applied rewrites 45.0%

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

   if 4.5e20 < B

   1. Initial program 76.6%

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

      1. lift-+.f64 N/A

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

      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. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.4%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 85.4%

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

      5. lift-pow.f64 N/A

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

      6. sqr-pow N/A

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

      7. lift-fma.f64 N/A

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

      8. lift-fma.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-+r+ N/A

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

      11. lift-*.f64 N/A

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

      12. pow2 N/A

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

      13. lift-pow.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-+.f64 N/A

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 16.7%

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

   8. Applied rewrites 16.7%

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

Alternative 18: 44.5% accurate, 4.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1 \cdot \left(F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)\right)}{B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1 \cdot x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -1e-58)
  (/ (* -1.0 (* F (+ (/ 1.0 F) (/ x F)))) B)
  (if (<= F 7.8e-47)
    (/ (* -1.0 x) B)
    (/ (* F (fma -1.0 (/ x F) (/ 1.0 F))) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e-58) {
		tmp = (-1.0 * (F * ((1.0 / F) + (x / F)))) / B;
	} else if (F <= 7.8e-47) {
		tmp = (-1.0 * x) / B;
	} else {
		tmp = (F * fma(-1.0, (x / F), (1.0 / F))) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1e-58)
		tmp = Float64(Float64(-1.0 * Float64(F * Float64(Float64(1.0 / F) + Float64(x / F)))) / B);
	elseif (F <= 7.8e-47)
		tmp = Float64(Float64(-1.0 * x) / B);
	else
		tmp = Float64(Float64(F * fma(-1.0, Float64(x / F), Float64(1.0 / F))) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1e-58], N[(N[(-1.0 * N[(F * N[(N[(1.0 / F), $MachinePrecision] + N[(x / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 7.8e-47], N[(N[(-1.0 * x), $MachinePrecision] / B), $MachinePrecision], N[(N[(F * N[(-1.0 * N[(x / F), $MachinePrecision] + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e-58

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 45.0%

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

   5. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 29.0%

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

   7. Applied rewrites 29.0%

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

   if -1e-58 < F < 7.7999999999999996e-47

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 45.0%

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

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. lower-*.f64 30.0%

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

   7. Applied rewrites 30.0%

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

   if 7.7999999999999996e-47 < F

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 45.0%

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

   5. Taylor expanded in F around inf

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]
   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-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 29.1%

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

   7. Applied rewrites 29.1%

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

Alternative 19: 38.1% accurate, 4.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{F} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1 \cdot x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -1e-58)
  (- (* (/ -1.0 F) (/ F B)) (/ x B))
  (if (<= F 7.8e-47)
    (/ (* -1.0 x) B)
    (/ (* F (fma -1.0 (/ x F) (/ 1.0 F))) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e-58) {
		tmp = ((-1.0 / F) * (F / B)) - (x / B);
	} else if (F <= 7.8e-47) {
		tmp = (-1.0 * x) / B;
	} else {
		tmp = (F * fma(-1.0, (x / F), (1.0 / F))) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1e-58)
		tmp = Float64(Float64(Float64(-1.0 / F) * Float64(F / B)) - Float64(x / B));
	elseif (F <= 7.8e-47)
		tmp = Float64(Float64(-1.0 * x) / B);
	else
		tmp = Float64(Float64(F * fma(-1.0, Float64(x / F), Float64(1.0 / F))) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1e-58], N[(N[(N[(-1.0 / F), $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.8e-47], N[(N[(-1.0 * x), $MachinePrecision] / B), $MachinePrecision], N[(N[(F * N[(-1.0 * N[(x / F), $MachinePrecision] + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e-58

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 45.0%

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

   6. Applied rewrites 36.2%

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

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{-1}{F} \cdot \frac{F}{B} - \frac{x}{B} \]
   8. Step-by-step derivation

      1. lower-/.f64 21.8%

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

   9. Applied rewrites 21.8%

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

   if -1e-58 < F < 7.7999999999999996e-47

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 45.0%

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

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. lower-*.f64 30.0%

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

   7. Applied rewrites 30.0%

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

   if 7.7999999999999996e-47 < F

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 45.0%

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

   5. Taylor expanded in F around inf

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]
   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-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 29.1%

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

   7. Applied rewrites 29.1%

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

Alternative 20: 32.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{F} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{-47}:\\ \;\;\;\;\frac{-1 \cdot x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{F} \cdot \frac{F}{B} - \frac{x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -1e-58)
  (- (* (/ -1.0 F) (/ F B)) (/ x B))
  (if (<= F 7.8e-47)
    (/ (* -1.0 x) B)
    (- (* (/ 1.0 F) (/ F B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e-58) {
		tmp = ((-1.0 / F) * (F / B)) - (x / B);
	} else if (F <= 7.8e-47) {
		tmp = (-1.0 * x) / B;
	} else {
		tmp = ((1.0 / F) * (F / B)) - (x / B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1d-58)) then
        tmp = (((-1.0d0) / f) * (f / b)) - (x / b)
    else if (f <= 7.8d-47) then
        tmp = ((-1.0d0) * x) / b
    else
        tmp = ((1.0d0 / f) * (f / b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e-58) {
		tmp = ((-1.0 / F) * (F / B)) - (x / B);
	} else if (F <= 7.8e-47) {
		tmp = (-1.0 * x) / B;
	} else {
		tmp = ((1.0 / F) * (F / B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1e-58:
		tmp = ((-1.0 / F) * (F / B)) - (x / B)
	elif F <= 7.8e-47:
		tmp = (-1.0 * x) / B
	else:
		tmp = ((1.0 / F) * (F / B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1e-58)
		tmp = Float64(Float64(Float64(-1.0 / F) * Float64(F / B)) - Float64(x / B));
	elseif (F <= 7.8e-47)
		tmp = Float64(Float64(-1.0 * x) / B);
	else
		tmp = Float64(Float64(Float64(1.0 / F) * Float64(F / B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1e-58)
		tmp = ((-1.0 / F) * (F / B)) - (x / B);
	elseif (F <= 7.8e-47)
		tmp = (-1.0 * x) / B;
	else
		tmp = ((1.0 / F) * (F / B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1e-58], N[(N[(N[(-1.0 / F), $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.8e-47], N[(N[(-1.0 * x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(1.0 / F), $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e-58

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 45.0%

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

   6. Applied rewrites 36.2%

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

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{-1}{F} \cdot \frac{F}{B} - \frac{x}{B} \]
   8. Step-by-step derivation

      1. lower-/.f64 21.8%

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

   9. Applied rewrites 21.8%

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

   if -1e-58 < F < 7.7999999999999996e-47

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 45.0%

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

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. lower-*.f64 30.0%

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

   7. Applied rewrites 30.0%

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

   if 7.7999999999999996e-47 < F

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 45.0%

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

   6. Applied rewrites 36.2%

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

   7. Taylor expanded in F around inf

      \[\leadsto \frac{1}{F} \cdot \frac{F}{B} - \frac{x}{B} \]
   8. Step-by-step derivation

      1. lower-/.f64 21.9%

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

   9. Applied rewrites 21.9%

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

Alternative 21: 32.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-58}:\\ \;\;\;\;\frac{-1}{F} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9 \cdot 10^{+136}:\\ \;\;\;\;\frac{-1 \cdot x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -1e-58)
  (- (* (/ -1.0 F) (/ F B)) (/ x B))
  (if (<= F 9e+136) (/ (* -1.0 x) B) (/ 1.0 B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e-58) {
		tmp = ((-1.0 / F) * (F / B)) - (x / B);
	} else if (F <= 9e+136) {
		tmp = (-1.0 * x) / 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 <= (-1d-58)) then
        tmp = (((-1.0d0) / f) * (f / b)) - (x / b)
    else if (f <= 9d+136) then
        tmp = ((-1.0d0) * x) / 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 <= -1e-58) {
		tmp = ((-1.0 / F) * (F / B)) - (x / B);
	} else if (F <= 9e+136) {
		tmp = (-1.0 * x) / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1e-58:
		tmp = ((-1.0 / F) * (F / B)) - (x / B)
	elif F <= 9e+136:
		tmp = (-1.0 * x) / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1e-58)
		tmp = Float64(Float64(Float64(-1.0 / F) * Float64(F / B)) - Float64(x / B));
	elseif (F <= 9e+136)
		tmp = Float64(Float64(-1.0 * x) / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1e-58)
		tmp = ((-1.0 / F) * (F / B)) - (x / B);
	elseif (F <= 9e+136)
		tmp = (-1.0 * x) / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1e-58], N[(N[(N[(-1.0 / F), $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9e+136], N[(N[(-1.0 * x), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e-58

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 45.0%

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

   6. Applied rewrites 36.2%

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

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{-1}{F} \cdot \frac{F}{B} - \frac{x}{B} \]
   8. Step-by-step derivation

      1. lower-/.f64 21.8%

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

   9. Applied rewrites 21.8%

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

   if -1e-58 < F < 8.9999999999999999e136

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 45.0%

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

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. lower-*.f64 30.0%

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

   7. Applied rewrites 30.0%

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

   if 8.9999999999999999e136 < F

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 45.0%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 10.5%

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

   7. Applied rewrites 10.5%

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

   8. Taylor expanded in F around inf

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

      1. lower-/.f64 10.1%

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

   10. Applied rewrites 10.1%

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

Alternative 22: 30.9% accurate, 7.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{-1 \cdot x}{B}\\ \mathbf{if}\;x \leq -6.2 \cdot 10^{-99}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-186}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (/ (* -1.0 x) B)))
  (if (<= x -6.2e-99) t_0 (if (<= x 6e-186) (/ -1.0 B) t_0))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 * x) / B;
	double tmp;
	if (x <= -6.2e-99) {
		tmp = t_0;
	} else if (x <= 6e-186) {
		tmp = -1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) * x) / b
    if (x <= (-6.2d-99)) then
        tmp = t_0
    else if (x <= 6d-186) then
        tmp = (-1.0d0) / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 * x) / B;
	double tmp;
	if (x <= -6.2e-99) {
		tmp = t_0;
	} else if (x <= 6e-186) {
		tmp = -1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 * x) / B
	tmp = 0
	if x <= -6.2e-99:
		tmp = t_0
	elif x <= 6e-186:
		tmp = -1.0 / B
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 * x) / B)
	tmp = 0.0
	if (x <= -6.2e-99)
		tmp = t_0;
	elseif (x <= 6e-186)
		tmp = Float64(-1.0 / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 * x) / B;
	tmp = 0.0;
	if (x <= -6.2e-99)
		tmp = t_0;
	elseif (x <= 6e-186)
		tmp = -1.0 / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 * x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[x, -6.2e-99], t$95$0, If[LessEqual[x, 6e-186], N[(-1.0 / B), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -6.1999999999999997e-99 or 6.0000000000000003e-186 < x

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 45.0%

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

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. lower-*.f64 30.0%

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

   7. Applied rewrites 30.0%

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

   if -6.1999999999999997e-99 < x < 6.0000000000000003e-186

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 45.0%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 10.5%

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

   7. Applied rewrites 10.5%

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

Alternative 23: 17.7% accurate, 14.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq 4.2 \cdot 10^{-175}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F 4.2e-175) (/ -1.0 B) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 4.2e-175) {
		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 <= 4.2d-175) 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 <= 4.2e-175) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 4.2e-175:
		tmp = -1.0 / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 4.2e-175)
		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 <= 4.2e-175)
		tmp = -1.0 / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 4.2e-175], N[(-1.0 / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 4.2e-175

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 45.0%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 10.5%

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

   7. Applied rewrites 10.5%

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

   if 4.2e-175 < F

   1. Initial program 76.6%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 45.0%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 10.5%

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

   7. Applied rewrites 10.5%

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

   8. Taylor expanded in F around inf

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

      1. lower-/.f64 10.1%

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

   10. Applied rewrites 10.1%

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

Alternative 24: 10.5% accurate, 25.4× 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 76.6%

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

2. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
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 45.0%

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

5. Taylor expanded in F around -inf

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

   1. lower-/.f64 10.5%

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

7. Applied rewrites 10.5%

   \[\leadsto \frac{-1}{\color{blue}{B}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025210 
(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))))))