VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.3% → 99.7%

Time: 22.3s

Alternatives: 21

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+16}:\\ \;\;\;\;-1 \cdot \left(F \cdot \frac{\frac{\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}}{F}\right)\\ \mathbf{elif}\;F \leq 1900000:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{1}{\sin B}}{F}, \frac{-x}{\tan B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1e+16)
   (* -1.0 (* F (/ (/ (fma (cos B) x 1.0) (sin B)) F)))
   (if (<= F 1900000.0)
     (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F (sin B))) (/ x (tan B)))
     (fma F (/ (/ 1.0 (sin B)) F) (/ (- x) (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e+16) {
		tmp = -1.0 * (F * ((fma(cos(B), x, 1.0) / sin(B)) / F));
	} else if (F <= 1900000.0) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / sin(B))) - (x / tan(B));
	} else {
		tmp = fma(F, ((1.0 / sin(B)) / F), (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1e+16)
		tmp = Float64(-1.0 * Float64(F * Float64(Float64(fma(cos(B), x, 1.0) / sin(B)) / F)));
	elseif (F <= 1900000.0)
		tmp = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / sin(B))) - Float64(x / tan(B)));
	else
		tmp = fma(F, Float64(Float64(1.0 / sin(B)) / F), Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1e+16], N[(-1.0 * N[(F * N[(N[(N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1900000.0], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e16

   1. Initial program 76.3%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.1%

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

   4. Applied rewrites 49.1%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-add-rev N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 49.2%

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

   6. Applied rewrites 49.2%

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

   if -1e16 < F < 1.9e6

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.3%

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

   3. Applied rewrites 76.5%

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

   if 1.9e6 < F

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   7. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.4%

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

   9. Applied rewrites 52.4%

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

       1. lift-/.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-/r* N/A

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

       6. rgt-mult-inverse N/A

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

       7. lift-/.f64 N/A

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

       8. associate-*r/ N/A

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

       9. lift-/.f64 N/A

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

       10. associate-/r* N/A

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

       11. *-commutative N/A

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

       12. lower-/.f64 N/A

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

       13. *-commutative N/A

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

       14. associate-/r* N/A

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

       15. lift-/.f64 N/A

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

       16. associate-*r/ N/A

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

       17. lift-/.f64 N/A

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

       18. rgt-mult-inverse N/A

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

       19. lower-/.f64 N/A

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

       20. lift-sin.f64 52.5%

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

   11. Applied rewrites 52.5%

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

Alternative 2: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -0.0105:\\ \;\;\;\;-1 \cdot \left(F \cdot \frac{\frac{\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}}{F}\right)\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {2}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{1}{\sin B}}{F}, \frac{-x}{\tan B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.0105)
   (* -1.0 (* F (/ (/ (fma (cos B) x 1.0) (sin B)) F)))
   (if (<= F 5.5e-6)
     (+ (- (* x (/ 1.0 (tan B)))) (* (/ F (sin B)) (pow 2.0 -0.5)))
     (fma F (/ (/ 1.0 (sin B)) F) (/ (- x) (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0105) {
		tmp = -1.0 * (F * ((fma(cos(B), x, 1.0) / sin(B)) / F));
	} else if (F <= 5.5e-6) {
		tmp = -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow(2.0, -0.5));
	} else {
		tmp = fma(F, ((1.0 / sin(B)) / F), (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.0105)
		tmp = Float64(-1.0 * Float64(F * Float64(Float64(fma(cos(B), x, 1.0) / sin(B)) / F)));
	elseif (F <= 5.5e-6)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / sin(B)) * (2.0 ^ -0.5)));
	else
		tmp = fma(F, Float64(Float64(1.0 / sin(B)) / F), Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.0105], N[(-1.0 * N[(F * N[(N[(N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.5e-6], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.0105000000000000007

   1. Initial program 76.3%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.1%

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

   4. Applied rewrites 49.1%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-add-rev N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 49.2%

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

   6. Applied rewrites 49.2%

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

   if -0.0105000000000000007 < F < 5.4999999999999999e-6

   1. Initial program 76.3%

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

   2. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval 54.5%

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

   4. Applied rewrites 54.5%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval 56.5%

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

   7. Applied rewrites 56.5%

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

   if 5.4999999999999999e-6 < F

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   7. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.4%

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

   9. Applied rewrites 52.4%

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

       1. lift-/.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-/r* N/A

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

       6. rgt-mult-inverse N/A

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

       7. lift-/.f64 N/A

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

       8. associate-*r/ N/A

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

       9. lift-/.f64 N/A

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

       10. associate-/r* N/A

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

       11. *-commutative N/A

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

       12. lower-/.f64 N/A

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

       13. *-commutative N/A

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

       14. associate-/r* N/A

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

       15. lift-/.f64 N/A

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

       16. associate-*r/ N/A

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

       17. lift-/.f64 N/A

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

       18. rgt-mult-inverse N/A

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

       19. lower-/.f64 N/A

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

       20. lift-sin.f64 52.5%

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

   11. Applied rewrites 52.5%

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

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -0.00072:\\ \;\;\;\;-1 \cdot \left(F \cdot \frac{\frac{\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}}{F}\right)\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(2 + 2 \cdot x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{1}{\sin B}}{F}, \frac{-x}{\tan B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.00072)
   (* -1.0 (* F (/ (/ (fma (cos B) x 1.0) (sin B)) F)))
   (if (<= F 5.5e-6)
     (+
      (- (* x (/ 1.0 (tan B))))
      (* (/ F (sin B)) (pow (+ 2.0 (* 2.0 x)) -0.5)))
     (fma F (/ (/ 1.0 (sin B)) F) (/ (- x) (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.00072) {
		tmp = -1.0 * (F * ((fma(cos(B), x, 1.0) / sin(B)) / F));
	} else if (F <= 5.5e-6) {
		tmp = -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow((2.0 + (2.0 * x)), -0.5));
	} else {
		tmp = fma(F, ((1.0 / sin(B)) / F), (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.00072)
		tmp = Float64(-1.0 * Float64(F * Float64(Float64(fma(cos(B), x, 1.0) / sin(B)) / F)));
	elseif (F <= 5.5e-6)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / sin(B)) * (Float64(2.0 + Float64(2.0 * x)) ^ -0.5)));
	else
		tmp = fma(F, Float64(Float64(1.0 / sin(B)) / F), Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.00072], N[(-1.0 * N[(F * N[(N[(N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.5e-6], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.20000000000000045e-4

   1. Initial program 76.3%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.1%

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

   4. Applied rewrites 49.1%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-add-rev N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 49.2%

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

   6. Applied rewrites 49.2%

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

   if -7.20000000000000045e-4 < F < 5.4999999999999999e-6

   1. Initial program 76.3%

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

   2. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval 54.5%

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

   4. Applied rewrites 54.5%

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

   if 5.4999999999999999e-6 < F

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   7. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.4%

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

   9. Applied rewrites 52.4%

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

       1. lift-/.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-/r* N/A

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

       6. rgt-mult-inverse N/A

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

       7. lift-/.f64 N/A

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

       8. associate-*r/ N/A

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

       9. lift-/.f64 N/A

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

       10. associate-/r* N/A

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

       11. *-commutative N/A

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

       12. lower-/.f64 N/A

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

       13. *-commutative N/A

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

       14. associate-/r* N/A

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

       15. lift-/.f64 N/A

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

       16. associate-*r/ N/A

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

       17. lift-/.f64 N/A

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

       18. rgt-mult-inverse N/A

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

       19. lower-/.f64 N/A

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

       20. lift-sin.f64 52.5%

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

   11. Applied rewrites 52.5%

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

Alternative 4: 91.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -24000:\\ \;\;\;\;-1 \cdot \left(F \cdot \frac{\frac{\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}}{F}\right)\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{1}{\sin B}}{F}, t\_0\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -24000.0)
     (* -1.0 (* F (/ (/ (fma (cos B) x 1.0) (sin B)) F)))
     (if (<= F 5.5e-6)
       (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) B) t_0)
       (fma F (/ (/ 1.0 (sin B)) F) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -24000.0) {
		tmp = -1.0 * (F * ((fma(cos(B), x, 1.0) / sin(B)) / F));
	} else if (F <= 5.5e-6) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / B), t_0);
	} else {
		tmp = fma(F, ((1.0 / sin(B)) / F), t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -24000.0)
		tmp = Float64(-1.0 * Float64(F * Float64(Float64(fma(cos(B), x, 1.0) / sin(B)) / F)));
	elseif (F <= 5.5e-6)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / B), t_0);
	else
		tmp = fma(F, Float64(Float64(1.0 / sin(B)) / F), 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, -24000.0], N[(-1.0 * N[(F * N[(N[(N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.5e-6], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / B), $MachinePrecision] + t$95$0), $MachinePrecision], N[(F * N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -24000

   1. Initial program 76.3%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.1%

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

   4. Applied rewrites 49.1%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-add-rev N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 49.2%

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

   6. Applied rewrites 49.2%

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

   if -24000 < F < 5.4999999999999999e-6

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   if 5.4999999999999999e-6 < F

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   7. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.4%

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

   9. Applied rewrites 52.4%

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

       1. lift-/.f64 N/A

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

       2. lift-sin.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-/r* N/A

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

       6. rgt-mult-inverse N/A

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

       7. lift-/.f64 N/A

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

       8. associate-*r/ N/A

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

       9. lift-/.f64 N/A

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

       10. associate-/r* N/A

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

       11. *-commutative N/A

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

       12. lower-/.f64 N/A

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

       13. *-commutative N/A

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

       14. associate-/r* N/A

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

       15. lift-/.f64 N/A

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

       16. associate-*r/ N/A

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

       17. lift-/.f64 N/A

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

       18. rgt-mult-inverse N/A

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

       19. lower-/.f64 N/A

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

       20. lift-sin.f64 52.5%

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

   11. Applied rewrites 52.5%

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

Alternative 5: 91.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -24000:\\ \;\;\;\;-1 \cdot \left(F \cdot \frac{\frac{\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}}{F}\right)\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B \cdot F} \cdot F - \frac{x}{\tan B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -24000.0)
   (* -1.0 (* F (/ (/ (fma (cos B) x 1.0) (sin B)) F)))
   (if (<= F 5.5e-6)
     (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) B) (/ (- x) (tan B)))
     (- (* (/ 1.0 (* (sin B) F)) F) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -24000.0) {
		tmp = -1.0 * (F * ((fma(cos(B), x, 1.0) / sin(B)) / F));
	} else if (F <= 5.5e-6) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / B), (-x / tan(B)));
	} else {
		tmp = ((1.0 / (sin(B) * F)) * F) - (x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -24000.0)
		tmp = Float64(-1.0 * Float64(F * Float64(Float64(fma(cos(B), x, 1.0) / sin(B)) / F)));
	elseif (F <= 5.5e-6)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / B), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(sin(B) * F)) * F) - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -24000.0], N[(-1.0 * N[(F * N[(N[(N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.5e-6], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / B), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -24000

   1. Initial program 76.3%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.1%

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

   4. Applied rewrites 49.1%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-add-rev N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 49.2%

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

   6. Applied rewrites 49.2%

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

   if -24000 < F < 5.4999999999999999e-6

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   if 5.4999999999999999e-6 < F

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   7. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.4%

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

   9. Applied rewrites 52.4%

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

       1. lift-fma.f64 N/A

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

       2. add-flip N/A

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

       3. lower--.f64 N/A

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

       4. *-commutative N/A

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

       5. lower-*.f64 N/A

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

       6. lift-sin.f64 N/A

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

       7. lift-*.f64 N/A

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

       8. *-commutative N/A

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

       9. lower-*.f64 N/A

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

       10. lift-sin.f64 N/A

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

       11. lift-sin.f64 N/A

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

   11. Applied rewrites 52.4%

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

Alternative 6: 85.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -24000:\\ \;\;\;\;-1 \cdot \left(F \cdot \frac{\frac{\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}}{F}\right)\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B}, t\_0\right)\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{+227}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{F \cdot \sin B}, -1 \cdot \frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{B \cdot F}, t\_0\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -24000.0)
     (* -1.0 (* F (/ (/ (fma (cos B) x 1.0) (sin B)) F)))
     (if (<= F 1.5e+26)
       (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) B) t_0)
       (if (<= F 5.8e+227)
         (fma F (/ 1.0 (* F (sin B))) (* -1.0 (/ x B)))
         (fma F (/ 1.0 (* B F)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -24000.0) {
		tmp = -1.0 * (F * ((fma(cos(B), x, 1.0) / sin(B)) / F));
	} else if (F <= 1.5e+26) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / B), t_0);
	} else if (F <= 5.8e+227) {
		tmp = fma(F, (1.0 / (F * sin(B))), (-1.0 * (x / B)));
	} else {
		tmp = fma(F, (1.0 / (B * F)), t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -24000.0)
		tmp = Float64(-1.0 * Float64(F * Float64(Float64(fma(cos(B), x, 1.0) / sin(B)) / F)));
	elseif (F <= 1.5e+26)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / B), t_0);
	elseif (F <= 5.8e+227)
		tmp = fma(F, Float64(1.0 / Float64(F * sin(B))), Float64(-1.0 * Float64(x / B)));
	else
		tmp = fma(F, Float64(1.0 / Float64(B * F)), 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, -24000.0], N[(-1.0 * N[(F * N[(N[(N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.5e+26], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / B), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, 5.8e+227], N[(F * N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(1.0 / N[(B * F), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -24000

   1. Initial program 76.3%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.1%

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

   4. Applied rewrites 49.1%

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

      1. lift-+.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-add-rev N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 49.2%

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

   6. Applied rewrites 49.2%

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

   if -24000 < F < 1.49999999999999999e26

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   if 1.49999999999999999e26 < F < 5.7999999999999997e227

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   7. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.4%

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

   9. Applied rewrites 52.4%

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

   10. Taylor expanded in B around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 32.6%

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

   12. Applied rewrites 32.6%

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

   if 5.7999999999999997e227 < F

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   7. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.4%

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

   9. Applied rewrites 52.4%

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

   10. Taylor expanded in B around 0

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

       1. lower-*.f64 50.0%

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

   12. Applied rewrites 50.0%

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

Alternative 7: 85.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -24000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B \cdot F} \cdot \left(-F\right)\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B}, t\_0\right)\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{+227}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{F \cdot \sin B}, -1 \cdot \frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{B \cdot F}, t\_0\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -24000.0)
     (* (/ (fma (cos B) x 1.0) (* (sin B) F)) (- F))
     (if (<= F 1.5e+26)
       (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) B) t_0)
       (if (<= F 5.8e+227)
         (fma F (/ 1.0 (* F (sin B))) (* -1.0 (/ x B)))
         (fma F (/ 1.0 (* B F)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -24000.0) {
		tmp = (fma(cos(B), x, 1.0) / (sin(B) * F)) * -F;
	} else if (F <= 1.5e+26) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / B), t_0);
	} else if (F <= 5.8e+227) {
		tmp = fma(F, (1.0 / (F * sin(B))), (-1.0 * (x / B)));
	} else {
		tmp = fma(F, (1.0 / (B * F)), t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -24000.0)
		tmp = Float64(Float64(fma(cos(B), x, 1.0) / Float64(sin(B) * F)) * Float64(-F));
	elseif (F <= 1.5e+26)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / B), t_0);
	elseif (F <= 5.8e+227)
		tmp = fma(F, Float64(1.0 / Float64(F * sin(B))), Float64(-1.0 * Float64(x / B)));
	else
		tmp = fma(F, Float64(1.0 / Float64(B * F)), 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, -24000.0], N[(N[(N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * (-F)), $MachinePrecision], If[LessEqual[F, 1.5e+26], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / B), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, 5.8e+227], N[(F * N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(1.0 / N[(B * F), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -24000

   1. Initial program 76.3%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.1%

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

   4. Applied rewrites 49.1%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-rgt-neg-in N/A

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 49.1%

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

   if -24000 < F < 1.49999999999999999e26

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   if 1.49999999999999999e26 < F < 5.7999999999999997e227

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   7. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.4%

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

   9. Applied rewrites 52.4%

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

   10. Taylor expanded in B around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 32.6%

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

   12. Applied rewrites 32.6%

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

   if 5.7999999999999997e227 < F

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   7. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.4%

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

   9. Applied rewrites 52.4%

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

   10. Taylor expanded in B around 0

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

       1. lower-*.f64 50.0%

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

   12. Applied rewrites 50.0%

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

Alternative 8: 78.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ t_1 := \frac{1}{F \cdot \sin B}\\ \mathbf{if}\;F \leq -1.45 \cdot 10^{+189}:\\ \;\;\;\;-1 \cdot \left(F \cdot \left(t\_1 + \frac{x}{B \cdot F}\right)\right)\\ \mathbf{elif}\;F \leq -6.1 \cdot 10^{+29}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{-1}{F}}{B}, t\_0\right)\\ \mathbf{elif}\;F \leq -1.28 \cdot 10^{-101}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5}}{B}, t\_0\right)\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{+227}:\\ \;\;\;\;\mathsf{fma}\left(F, t\_1, -1 \cdot \frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{B \cdot F}, t\_0\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))) (t_1 (/ 1.0 (* F (sin B)))))
   (if (<= F -1.45e+189)
     (* -1.0 (* F (+ t_1 (/ x (* B F)))))
     (if (<= F -6.1e+29)
       (fma F (/ (/ -1.0 F) B) t_0)
       (if (<= F -1.28e-101)
         (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) (/ F (sin B))) (/ x B))
         (if (<= F 4.2e+19)
           (fma F (/ (pow (fma 2.0 x 2.0) -0.5) B) t_0)
           (if (<= F 5.8e+227)
             (fma F t_1 (* -1.0 (/ x B)))
             (fma F (/ 1.0 (* B F)) t_0))))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double t_1 = 1.0 / (F * sin(B));
	double tmp;
	if (F <= -1.45e+189) {
		tmp = -1.0 * (F * (t_1 + (x / (B * F))));
	} else if (F <= -6.1e+29) {
		tmp = fma(F, ((-1.0 / F) / B), t_0);
	} else if (F <= -1.28e-101) {
		tmp = (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * (F / sin(B))) - (x / B);
	} else if (F <= 4.2e+19) {
		tmp = fma(F, (pow(fma(2.0, x, 2.0), -0.5) / B), t_0);
	} else if (F <= 5.8e+227) {
		tmp = fma(F, t_1, (-1.0 * (x / B)));
	} else {
		tmp = fma(F, (1.0 / (B * F)), t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	t_1 = Float64(1.0 / Float64(F * sin(B)))
	tmp = 0.0
	if (F <= -1.45e+189)
		tmp = Float64(-1.0 * Float64(F * Float64(t_1 + Float64(x / Float64(B * F)))));
	elseif (F <= -6.1e+29)
		tmp = fma(F, Float64(Float64(-1.0 / F) / B), t_0);
	elseif (F <= -1.28e-101)
		tmp = Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * Float64(F / sin(B))) - Float64(x / B));
	elseif (F <= 4.2e+19)
		tmp = fma(F, Float64((fma(2.0, x, 2.0) ^ -0.5) / B), t_0);
	elseif (F <= 5.8e+227)
		tmp = fma(F, t_1, Float64(-1.0 * Float64(x / B)));
	else
		tmp = fma(F, Float64(1.0 / Float64(B * F)), t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.45e+189], N[(-1.0 * N[(F * N[(t$95$1 + N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -6.1e+29], N[(F * N[(N[(-1.0 / F), $MachinePrecision] / B), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, -1.28e-101], N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.2e+19], N[(F * N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] / B), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, 5.8e+227], N[(F * t$95$1 + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(1.0 / N[(B * F), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -1.4500000000000001e189

   1. Initial program 76.3%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.1%

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

   4. Applied rewrites 49.1%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 32.9%

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

   7. Applied rewrites 32.9%

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

   if -1.4500000000000001e189 < F < -6.0999999999999998e29

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 52.1%

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

   9. Applied rewrites 52.1%

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

   if -6.0999999999999998e29 < F < -1.27999999999999995e-101

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 49.0%

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

   4. Applied rewrites 49.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 49.0%

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

   6. Applied rewrites 49.0%

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

   if -1.27999999999999995e-101 < F < 4.2e19

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   7. Taylor expanded in F around 0

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

   9. Applied rewrites 50.4%

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

   if 4.2e19 < F < 5.7999999999999997e227

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   7. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.4%

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

   9. Applied rewrites 52.4%

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

   10. Taylor expanded in B around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 32.6%

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

   12. Applied rewrites 32.6%

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

   if 5.7999999999999997e227 < F

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   7. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.4%

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

   9. Applied rewrites 52.4%

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

   10. Taylor expanded in B around 0

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

       1. lower-*.f64 50.0%

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

   12. Applied rewrites 50.0%

       \[\leadsto \mathsf{fma}\left(F, \frac{1}{B \cdot \color{blue}{F}}, \frac{-x}{\tan B}\right) \]
3. Recombined 6 regimes into one program.
4. Add Preprocessing

Alternative 9: 78.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{F \cdot \sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -1.9 \cdot 10^{+156}:\\ \;\;\;\;-1 \cdot \left(F \cdot \left(t\_0 + \frac{x}{B \cdot F}\right)\right)\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{+26}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B}, t\_1\right)\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{+227}:\\ \;\;\;\;\mathsf{fma}\left(F, t\_0, -1 \cdot \frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{B \cdot F}, t\_1\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* F (sin B)))) (t_1 (/ (- x) (tan B))))
   (if (<= F -1.9e+156)
     (* -1.0 (* F (+ t_0 (/ x (* B F)))))
     (if (<= F 1.5e+26)
       (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) B) t_1)
       (if (<= F 5.8e+227)
         (fma F t_0 (* -1.0 (/ x B)))
         (fma F (/ 1.0 (* B F)) t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / (F * sin(B));
	double t_1 = -x / tan(B);
	double tmp;
	if (F <= -1.9e+156) {
		tmp = -1.0 * (F * (t_0 + (x / (B * F))));
	} else if (F <= 1.5e+26) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / B), t_1);
	} else if (F <= 5.8e+227) {
		tmp = fma(F, t_0, (-1.0 * (x / B)));
	} else {
		tmp = fma(F, (1.0 / (B * F)), t_1);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / Float64(F * sin(B)))
	t_1 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -1.9e+156)
		tmp = Float64(-1.0 * Float64(F * Float64(t_0 + Float64(x / Float64(B * F)))));
	elseif (F <= 1.5e+26)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / B), t_1);
	elseif (F <= 5.8e+227)
		tmp = fma(F, t_0, Float64(-1.0 * Float64(x / B)));
	else
		tmp = fma(F, Float64(1.0 / Float64(B * F)), t_1);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.9e+156], N[(-1.0 * N[(F * N[(t$95$0 + N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.5e+26], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / B), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[F, 5.8e+227], N[(F * t$95$0 + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(1.0 / N[(B * F), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.90000000000000012e156

   1. Initial program 76.3%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.1%

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

   4. Applied rewrites 49.1%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 32.9%

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

   7. Applied rewrites 32.9%

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

   if -1.90000000000000012e156 < F < 1.49999999999999999e26

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   if 1.49999999999999999e26 < F < 5.7999999999999997e227

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   7. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.4%

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

   9. Applied rewrites 52.4%

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

   10. Taylor expanded in B around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 32.6%

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

   12. Applied rewrites 32.6%

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

   if 5.7999999999999997e227 < F

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   7. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.4%

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

   9. Applied rewrites 52.4%

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

   10. Taylor expanded in B around 0

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

       1. lower-*.f64 50.0%

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

   12. Applied rewrites 50.0%

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

Alternative 10: 78.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{F \cdot \sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -0.0105:\\ \;\;\;\;-1 \cdot \left(F \cdot \left(t\_0 + \frac{x}{B \cdot F}\right)\right)\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5}}{B}, t\_1\right)\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{+227}:\\ \;\;\;\;\mathsf{fma}\left(F, t\_0, -1 \cdot \frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{B \cdot F}, t\_1\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* F (sin B)))) (t_1 (/ (- x) (tan B))))
   (if (<= F -0.0105)
     (* -1.0 (* F (+ t_0 (/ x (* B F)))))
     (if (<= F 4.2e+19)
       (fma F (/ (pow (fma 2.0 x 2.0) -0.5) B) t_1)
       (if (<= F 5.8e+227)
         (fma F t_0 (* -1.0 (/ x B)))
         (fma F (/ 1.0 (* B F)) t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / (F * sin(B));
	double t_1 = -x / tan(B);
	double tmp;
	if (F <= -0.0105) {
		tmp = -1.0 * (F * (t_0 + (x / (B * F))));
	} else if (F <= 4.2e+19) {
		tmp = fma(F, (pow(fma(2.0, x, 2.0), -0.5) / B), t_1);
	} else if (F <= 5.8e+227) {
		tmp = fma(F, t_0, (-1.0 * (x / B)));
	} else {
		tmp = fma(F, (1.0 / (B * F)), t_1);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / Float64(F * sin(B)))
	t_1 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -0.0105)
		tmp = Float64(-1.0 * Float64(F * Float64(t_0 + Float64(x / Float64(B * F)))));
	elseif (F <= 4.2e+19)
		tmp = fma(F, Float64((fma(2.0, x, 2.0) ^ -0.5) / B), t_1);
	elseif (F <= 5.8e+227)
		tmp = fma(F, t_0, Float64(-1.0 * Float64(x / B)));
	else
		tmp = fma(F, Float64(1.0 / Float64(B * F)), t_1);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.0105], N[(-1.0 * N[(F * N[(t$95$0 + N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.2e+19], N[(F * N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] / B), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[F, 5.8e+227], N[(F * t$95$0 + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(1.0 / N[(B * F), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.0105000000000000007

   1. Initial program 76.3%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.1%

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

   4. Applied rewrites 49.1%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 32.9%

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

   7. Applied rewrites 32.9%

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

   if -0.0105000000000000007 < F < 4.2e19

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   7. Taylor expanded in F around 0

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

   9. Applied rewrites 50.4%

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

   if 4.2e19 < F < 5.7999999999999997e227

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   7. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.4%

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

   9. Applied rewrites 52.4%

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

   10. Taylor expanded in B around 0

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

       1. lower-*.f64 N/A

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

       2. lower-/.f64 32.6%

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

   12. Applied rewrites 32.6%

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

   if 5.7999999999999997e227 < F

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   7. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.4%

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

   9. Applied rewrites 52.4%

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

   10. Taylor expanded in B around 0

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

       1. lower-*.f64 50.0%

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

   12. Applied rewrites 50.0%

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

Alternative 11: 68.4% accurate, 1.7× speedup

Math

\[\mathsf{copysign}\left(1, B\right) \cdot \begin{array}{l} \mathbf{if}\;\left|B\right| \leq 1.08 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\left|B\right|}, -1 \cdot \frac{x}{\left|B\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan \left(\left|B\right|\right)}\right) + \frac{F}{\left|B\right|} \cdot \frac{1}{F}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (*
  (copysign 1.0 B)
  (if (<= (fabs B) 1.08e-10)
    (fma
     F
     (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (fabs B))
     (* -1.0 (/ x (fabs B))))
    (+ (- (* x (/ 1.0 (tan (fabs B))))) (* (/ F (fabs B)) (/ 1.0 F))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (fabs(B) <= 1.08e-10) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / fabs(B)), (-1.0 * (x / fabs(B))));
	} else {
		tmp = -(x * (1.0 / tan(fabs(B)))) + ((F / fabs(B)) * (1.0 / F));
	}
	return copysign(1.0, B) * tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (abs(B) <= 1.08e-10)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / abs(B)), Float64(-1.0 * Float64(x / abs(B))));
	else
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(abs(B))))) + Float64(Float64(F / abs(B)) * Float64(1.0 / F)));
	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], 1.08e-10], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Abs[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(x / N[Abs[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[(x * N[(1.0 / N[Tan[N[Abs[B], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Abs[B], $MachinePrecision]), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if B < 1.08000000000000002e-10

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 43.4%

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

   9. Applied rewrites 43.4%

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

   if 1.08000000000000002e-10 < B

   1. Initial program 76.3%

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

   2. Taylor expanded in F around inf

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

      1. lower-/.f64 47.5%

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

   4. Applied rewrites 47.5%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 46.1%

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

   7. Applied rewrites 46.1%

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

Alternative 12: 66.8% accurate, 1.8× speedup

Math

\[\mathsf{copysign}\left(1, B\right) \cdot \begin{array}{l} \mathbf{if}\;\left|B\right| \leq 10^{-24}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\left|B\right|}, -1 \cdot \frac{x}{\left|B\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{\left|B\right| \cdot F}, \frac{-x}{\tan \left(\left|B\right|\right)}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (*
  (copysign 1.0 B)
  (if (<= (fabs B) 1e-24)
    (fma
     F
     (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (fabs B))
     (* -1.0 (/ x (fabs B))))
    (fma F (/ 1.0 (* (fabs B) F)) (/ (- x) (tan (fabs B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (fabs(B) <= 1e-24) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / fabs(B)), (-1.0 * (x / fabs(B))));
	} else {
		tmp = fma(F, (1.0 / (fabs(B) * F)), (-x / tan(fabs(B))));
	}
	return copysign(1.0, B) * tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (abs(B) <= 1e-24)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / abs(B)), Float64(-1.0 * Float64(x / abs(B))));
	else
		tmp = fma(F, Float64(1.0 / Float64(abs(B) * F)), Float64(Float64(-x) / tan(abs(B))));
	end
	return Float64(copysign(1.0, B) * tmp)
end

Wolfram

code[F_, B_, x_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[B]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[B], $MachinePrecision], 1e-24], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Abs[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(x / N[Abs[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(1.0 / N[(N[Abs[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[N[Abs[B], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if B < 9.99999999999999924e-25

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right)} \]

   3. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\color{blue}{B}}, \frac{-x}{\tan B}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 70.3%

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\color{blue}{B}}, \frac{-x}{\tan B}\right) \]

   7. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B}, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{B}, -1 \cdot \color{blue}{\frac{x}{B}}\right) \]

      2. lower-/.f64 43.4%

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B}, -1 \cdot \frac{x}{\color{blue}{B}}\right) \]

   9. Applied rewrites 43.4%

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B}, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]

   if 9.99999999999999924e-25 < B

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. mult-flip N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right)} \]

   3. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\color{blue}{B}}, \frac{-x}{\tan B}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 70.3%

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\color{blue}{B}}, \frac{-x}{\tan B}\right) \]

   7. Taylor expanded in F around inf

      \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{1}{F \cdot \sin B}}, \frac{-x}{\tan B}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{1}{\color{blue}{F \cdot \sin B}}, \frac{-x}{\tan B}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{1}{F \cdot \color{blue}{\sin B}}, \frac{-x}{\tan B}\right) \]

      3. lower-sin.f64 52.4%

         \[\leadsto \mathsf{fma}\left(F, \frac{1}{F \cdot \sin B}, \frac{-x}{\tan B}\right) \]

   9. Applied rewrites 52.4%

      \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{1}{F \cdot \sin B}}, \frac{-x}{\tan B}\right) \]

   10. Taylor expanded in B around 0

       \[\leadsto \mathsf{fma}\left(F, \frac{1}{B \cdot \color{blue}{F}}, \frac{-x}{\tan B}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 50.0%

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{B \cdot F}, \frac{-x}{\tan B}\right) \]

   12. Applied rewrites 50.0%

       \[\leadsto \mathsf{fma}\left(F, \frac{1}{B \cdot \color{blue}{F}}, \frac{-x}{\tan B}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 58.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -7600000000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{-6}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{F \cdot \sin B}, -1 \cdot \frac{x}{B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7600000000000.0)
   (/ -1.0 (sin B))
   (if (<= F 5.5e-6)
     (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) (/ F B)) (/ x B))
     (fma F (/ 1.0 (* F (sin B))) (* -1.0 (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7600000000000.0) {
		tmp = -1.0 / sin(B);
	} else if (F <= 5.5e-6) {
		tmp = (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = fma(F, (1.0 / (F * sin(B))), (-1.0 * (x / B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7600000000000.0)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 5.5e-6)
		tmp = Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	else
		tmp = fma(F, Float64(1.0 / Float64(F * sin(B))), Float64(-1.0 * Float64(x / B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7600000000000.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.5e-6], N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(F * N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.6e12

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.0%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.0%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -7.6e12 < F < 5.4999999999999999e-6

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.0%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.0%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-\frac{x}{B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \frac{x}{B}} \]

      5. lower--.f64 49.0%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \frac{x}{B}} \]

   6. Applied rewrites 49.0%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{B}} \]

   7. Taylor expanded in B around 0

      \[\leadsto {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \color{blue}{\frac{F}{B}} - \frac{x}{B} \]
   8. Step-by-step derivation

      1. lower-/.f64 35.2%

         \[\leadsto {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\color{blue}{B}} - \frac{x}{B} \]

   9. Applied rewrites 35.2%

      \[\leadsto {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \color{blue}{\frac{F}{B}} - \frac{x}{B} \]

   if 5.4999999999999999e-6 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. mult-flip N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right)} \]

   3. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\color{blue}{B}}, \frac{-x}{\tan B}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 70.3%

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\color{blue}{B}}, \frac{-x}{\tan B}\right) \]

   7. Taylor expanded in F around inf

      \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{1}{F \cdot \sin B}}, \frac{-x}{\tan B}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{1}{\color{blue}{F \cdot \sin B}}, \frac{-x}{\tan B}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{1}{F \cdot \color{blue}{\sin B}}, \frac{-x}{\tan B}\right) \]

      3. lower-sin.f64 52.4%

         \[\leadsto \mathsf{fma}\left(F, \frac{1}{F \cdot \sin B}, \frac{-x}{\tan B}\right) \]

   9. Applied rewrites 52.4%

      \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{1}{F \cdot \sin B}}, \frac{-x}{\tan B}\right) \]

   10. Taylor expanded in B around 0

       \[\leadsto \mathsf{fma}\left(F, \frac{1}{F \cdot \sin B}, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{F \cdot \sin B}, -1 \cdot \color{blue}{\frac{x}{B}}\right) \]

       2. lower-/.f64 32.6%

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{F \cdot \sin B}, -1 \cdot \frac{x}{\color{blue}{B}}\right) \]

   12. Applied rewrites 32.6%

       \[\leadsto \mathsf{fma}\left(F, \frac{1}{F \cdot \sin B}, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 51.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -7600000000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{+174}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B}, -1 \cdot \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 1.02 \cdot 10^{+260}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{F} \cdot \frac{F}{\sin B} - \frac{x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7600000000000.0)
   (/ -1.0 (sin B))
   (if (<= F 5.2e+174)
     (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) B) (* -1.0 (/ x B)))
     (if (<= F 1.02e+260)
       (/ 1.0 (sin B))
       (- (* (/ -1.0 F) (/ F (sin B))) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7600000000000.0) {
		tmp = -1.0 / sin(B);
	} else if (F <= 5.2e+174) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / B), (-1.0 * (x / B)));
	} else if (F <= 1.02e+260) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = ((-1.0 / F) * (F / sin(B))) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7600000000000.0)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 5.2e+174)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / B), Float64(-1.0 * Float64(x / B)));
	elseif (F <= 1.02e+260)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(Float64(-1.0 / F) * Float64(F / sin(B))) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7600000000000.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.2e+174], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / B), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.02e+260], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(N[(-1.0 / F), $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7.6e12

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.0%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.0%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -7.6e12 < F < 5.1999999999999997e174

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. mult-flip N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right)} \]

   3. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\color{blue}{B}}, \frac{-x}{\tan B}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 70.3%

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\color{blue}{B}}, \frac{-x}{\tan B}\right) \]

   7. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B}, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{B}, -1 \cdot \color{blue}{\frac{x}{B}}\right) \]

      2. lower-/.f64 43.4%

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B}, -1 \cdot \frac{x}{\color{blue}{B}}\right) \]

   9. Applied rewrites 43.4%

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B}, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]

   if 5.1999999999999997e174 < F < 1.02e260

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.5%

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 17.5%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]

   if 1.02e260 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.0%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.0%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-\frac{x}{B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \frac{x}{B}} \]

      5. lower--.f64 49.0%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \frac{x}{B}} \]

   6. Applied rewrites 49.0%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{B}} \]

   7. Taylor expanded in F around 0

      \[\leadsto {\left(\mathsf{fma}\left(x, 2, \color{blue}{2}\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 35.3%

      \[\leadsto {\left(\mathsf{fma}\left(x, 2, \color{blue}{2}\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{B} \]

   10. Taylor expanded in F around -inf

       \[\leadsto \color{blue}{\frac{-1}{F}} \cdot \frac{F}{\sin B} - \frac{x}{B} \]
   11. Step-by-step derivation

       1. lower-/.f64 27.3%

          \[\leadsto \frac{-1}{\color{blue}{F}} \cdot \frac{F}{\sin B} - \frac{x}{B} \]

   12. Applied rewrites 27.3%

       \[\leadsto \color{blue}{\frac{-1}{F}} \cdot \frac{F}{\sin B} - \frac{x}{B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 50.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -7600000000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{+174}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B}, -1 \cdot \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 1.85 \cdot 10^{+256}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{1 + x}{F \cdot B} \cdot F\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7600000000000.0)
   (/ -1.0 (sin B))
   (if (<= F 5.2e+174)
     (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) B) (* -1.0 (/ x B)))
     (if (<= F 1.85e+256)
       (/ 1.0 (sin B))
       (* -1.0 (* (/ (+ 1.0 x) (* F B)) F))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7600000000000.0) {
		tmp = -1.0 / sin(B);
	} else if (F <= 5.2e+174) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / B), (-1.0 * (x / B)));
	} else if (F <= 1.85e+256) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = -1.0 * (((1.0 + x) / (F * B)) * F);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7600000000000.0)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 5.2e+174)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / B), Float64(-1.0 * Float64(x / B)));
	elseif (F <= 1.85e+256)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(-1.0 * Float64(Float64(Float64(1.0 + x) / Float64(F * B)) * F));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7600000000000.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.2e+174], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / B), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.85e+256], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(N[(1.0 + x), $MachinePrecision] / N[(F * B), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7.6e12

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.0%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.0%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -7.6e12 < F < 5.1999999999999997e174

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. mult-flip N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right)} \]

   3. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\color{blue}{B}}, \frac{-x}{\tan B}\right) \]
   5. Step-by-step derivation

   6. Applied rewrites 70.3%

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\color{blue}{B}}, \frac{-x}{\tan B}\right) \]

   7. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B}, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{B}, -1 \cdot \color{blue}{\frac{x}{B}}\right) \]

      2. lower-/.f64 43.4%

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B}, -1 \cdot \frac{x}{\color{blue}{B}}\right) \]

   9. Applied rewrites 43.4%

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B}, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]

   if 5.1999999999999997e174 < F < 1.85000000000000016e256

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.5%

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 17.5%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]

   if 1.85000000000000016e256 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \color{blue}{\left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{\color{blue}{x \cdot \cos B}}{F \cdot \sin B}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \color{blue}{\cos B}}{F \cdot \sin B}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}\right)\right) \]

      11. lower-sin.f64 49.1%

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

   4. Applied rewrites 49.1%

      \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      5. lower-/.f64 27.9%

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

   7. Applied rewrites 27.9%

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      2. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      3. associate-/l* N/A

         \[\leadsto -1 \cdot \left(F \cdot \frac{\frac{1}{F} + \frac{x}{F}}{\color{blue}{B}}\right) \]

      4. *-commutative N/A

         \[\leadsto -1 \cdot \left(\frac{\frac{1}{F} + \frac{x}{F}}{B} \cdot F\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\frac{1}{F} + \frac{x}{F}}{B} \cdot F\right) \]

      6. lift-+.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\frac{1}{F} + \frac{x}{F}}{B} \cdot F\right) \]

      7. lift-/.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\frac{1}{F} + \frac{x}{F}}{B} \cdot F\right) \]

      8. lift-/.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\frac{1}{F} + \frac{x}{F}}{B} \cdot F\right) \]

      9. div-add-rev N/A

         \[\leadsto -1 \cdot \left(\frac{\frac{1 + x}{F}}{B} \cdot F\right) \]

      10. associate-/l/ N/A

          \[\leadsto -1 \cdot \left(\frac{1 + x}{F \cdot B} \cdot F\right) \]

      11. lower-/.f64 N/A

          \[\leadsto -1 \cdot \left(\frac{1 + x}{F \cdot B} \cdot F\right) \]

      12. lower-+.f64 N/A

          \[\leadsto -1 \cdot \left(\frac{1 + x}{F \cdot B} \cdot F\right) \]

      13. lower-*.f64 26.6%

          \[\leadsto -1 \cdot \left(\frac{1 + x}{F \cdot B} \cdot F\right) \]

   9. Applied rewrites 26.6%

      \[\leadsto -1 \cdot \left(\frac{1 + x}{F \cdot B} \cdot F\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 16: 49.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -7600000000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1420:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.85 \cdot 10^{+256}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{1 + x}{F \cdot B} \cdot F\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7600000000000.0)
   (/ -1.0 (sin B))
   (if (<= F 1420.0)
     (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) (/ F B)) (/ x B))
     (if (<= F 1.85e+256)
       (/ 1.0 (sin B))
       (* -1.0 (* (/ (+ 1.0 x) (* F B)) F))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7600000000000.0) {
		tmp = -1.0 / sin(B);
	} else if (F <= 1420.0) {
		tmp = (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 1.85e+256) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = -1.0 * (((1.0 + x) / (F * B)) * F);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7600000000000.0)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 1420.0)
		tmp = Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	elseif (F <= 1.85e+256)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(-1.0 * Float64(Float64(Float64(1.0 + x) / Float64(F * B)) * F));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7600000000000.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1420.0], N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.85e+256], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(N[(1.0 + x), $MachinePrecision] / N[(F * B), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7.6e12

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.0%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.0%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -7.6e12 < F < 1420

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.0%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.0%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-\frac{x}{B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \frac{x}{B}} \]

      5. lower--.f64 49.0%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \frac{x}{B}} \]

   6. Applied rewrites 49.0%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{B}} \]

   7. Taylor expanded in B around 0

      \[\leadsto {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \color{blue}{\frac{F}{B}} - \frac{x}{B} \]
   8. Step-by-step derivation

      1. lower-/.f64 35.2%

         \[\leadsto {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\color{blue}{B}} - \frac{x}{B} \]

   9. Applied rewrites 35.2%

      \[\leadsto {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \color{blue}{\frac{F}{B}} - \frac{x}{B} \]

   if 1420 < F < 1.85000000000000016e256

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.5%

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 17.5%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]

   if 1.85000000000000016e256 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \color{blue}{\left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{\color{blue}{x \cdot \cos B}}{F \cdot \sin B}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \color{blue}{\cos B}}{F \cdot \sin B}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}\right)\right) \]

      11. lower-sin.f64 49.1%

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

   4. Applied rewrites 49.1%

      \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      5. lower-/.f64 27.9%

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

   7. Applied rewrites 27.9%

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      2. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      3. associate-/l* N/A

         \[\leadsto -1 \cdot \left(F \cdot \frac{\frac{1}{F} + \frac{x}{F}}{\color{blue}{B}}\right) \]

      4. *-commutative N/A

         \[\leadsto -1 \cdot \left(\frac{\frac{1}{F} + \frac{x}{F}}{B} \cdot F\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\frac{1}{F} + \frac{x}{F}}{B} \cdot F\right) \]

      6. lift-+.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\frac{1}{F} + \frac{x}{F}}{B} \cdot F\right) \]

      7. lift-/.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\frac{1}{F} + \frac{x}{F}}{B} \cdot F\right) \]

      8. lift-/.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\frac{1}{F} + \frac{x}{F}}{B} \cdot F\right) \]

      9. div-add-rev N/A

         \[\leadsto -1 \cdot \left(\frac{\frac{1 + x}{F}}{B} \cdot F\right) \]

      10. associate-/l/ N/A

          \[\leadsto -1 \cdot \left(\frac{1 + x}{F \cdot B} \cdot F\right) \]

      11. lower-/.f64 N/A

          \[\leadsto -1 \cdot \left(\frac{1 + x}{F \cdot B} \cdot F\right) \]

      12. lower-+.f64 N/A

          \[\leadsto -1 \cdot \left(\frac{1 + x}{F \cdot B} \cdot F\right) \]

      13. lower-*.f64 26.6%

          \[\leadsto -1 \cdot \left(\frac{1 + x}{F \cdot B} \cdot F\right) \]

   9. Applied rewrites 26.6%

      \[\leadsto -1 \cdot \left(\frac{1 + x}{F \cdot B} \cdot F\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 17: 49.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -0.0105:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1350:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.85 \cdot 10^{+256}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(\frac{1 + x}{F \cdot B} \cdot F\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.0105)
   (/ -1.0 (sin B))
   (if (<= F 1350.0)
     (- (* (pow (fma x 2.0 2.0) -0.5) (/ F B)) (/ x B))
     (if (<= F 1.85e+256)
       (/ 1.0 (sin B))
       (* -1.0 (* (/ (+ 1.0 x) (* F B)) F))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0105) {
		tmp = -1.0 / sin(B);
	} else if (F <= 1350.0) {
		tmp = (pow(fma(x, 2.0, 2.0), -0.5) * (F / B)) - (x / B);
	} else if (F <= 1.85e+256) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = -1.0 * (((1.0 + x) / (F * B)) * F);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.0105)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 1350.0)
		tmp = Float64(Float64((fma(x, 2.0, 2.0) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	elseif (F <= 1.85e+256)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(-1.0 * Float64(Float64(Float64(1.0 + x) / Float64(F * B)) * F));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.0105], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1350.0], N[(N[(N[Power[N[(x * 2.0 + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.85e+256], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(N[(1.0 + x), $MachinePrecision] / N[(F * B), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.0105000000000000007

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.0%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.0%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -0.0105000000000000007 < F < 1350

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.0%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.0%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-\frac{x}{B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \frac{x}{B}} \]

      5. lower--.f64 49.0%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \frac{x}{B}} \]

   6. Applied rewrites 49.0%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{B}} \]

   7. Taylor expanded in F around 0

      \[\leadsto {\left(\mathsf{fma}\left(x, 2, \color{blue}{2}\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 35.3%

      \[\leadsto {\left(\mathsf{fma}\left(x, 2, \color{blue}{2}\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{B} \]

   10. Taylor expanded in B around 0

       \[\leadsto {\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5} \cdot \color{blue}{\frac{F}{B}} - \frac{x}{B} \]
   11. Step-by-step derivation

       1. lower-/.f64 28.1%

          \[\leadsto {\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5} \cdot \frac{F}{\color{blue}{B}} - \frac{x}{B} \]

   12. Applied rewrites 28.1%

       \[\leadsto {\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5} \cdot \color{blue}{\frac{F}{B}} - \frac{x}{B} \]

   if 1350 < F < 1.85000000000000016e256

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.5%

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 17.5%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]

   if 1.85000000000000016e256 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \color{blue}{\left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{\color{blue}{x \cdot \cos B}}{F \cdot \sin B}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \color{blue}{\cos B}}{F \cdot \sin B}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}\right)\right) \]

      11. lower-sin.f64 49.1%

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

   4. Applied rewrites 49.1%

      \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      5. lower-/.f64 27.9%

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

   7. Applied rewrites 27.9%

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      2. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      3. associate-/l* N/A

         \[\leadsto -1 \cdot \left(F \cdot \frac{\frac{1}{F} + \frac{x}{F}}{\color{blue}{B}}\right) \]

      4. *-commutative N/A

         \[\leadsto -1 \cdot \left(\frac{\frac{1}{F} + \frac{x}{F}}{B} \cdot F\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\frac{1}{F} + \frac{x}{F}}{B} \cdot F\right) \]

      6. lift-+.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\frac{1}{F} + \frac{x}{F}}{B} \cdot F\right) \]

      7. lift-/.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\frac{1}{F} + \frac{x}{F}}{B} \cdot F\right) \]

      8. lift-/.f64 N/A

         \[\leadsto -1 \cdot \left(\frac{\frac{1}{F} + \frac{x}{F}}{B} \cdot F\right) \]

      9. div-add-rev N/A

         \[\leadsto -1 \cdot \left(\frac{\frac{1 + x}{F}}{B} \cdot F\right) \]

      10. associate-/l/ N/A

          \[\leadsto -1 \cdot \left(\frac{1 + x}{F \cdot B} \cdot F\right) \]

      11. lower-/.f64 N/A

          \[\leadsto -1 \cdot \left(\frac{1 + x}{F \cdot B} \cdot F\right) \]

      12. lower-+.f64 N/A

          \[\leadsto -1 \cdot \left(\frac{1 + x}{F \cdot B} \cdot F\right) \]

      13. lower-*.f64 26.6%

          \[\leadsto -1 \cdot \left(\frac{1 + x}{F \cdot B} \cdot F\right) \]

   9. Applied rewrites 26.6%

      \[\leadsto -1 \cdot \left(\frac{1 + x}{F \cdot B} \cdot F\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 18: 43.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -0.0105:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 6.3 \cdot 10^{+38}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{F \cdot \left(\frac{F}{F \cdot F} + \frac{x}{F}\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.0105)
   (/ -1.0 (sin B))
   (if (<= F 6.3e+38)
     (- (* (pow (fma x 2.0 2.0) -0.5) (/ F B)) (/ x B))
     (* -1.0 (/ (* F (+ (/ F (* F F)) (/ x F))) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0105) {
		tmp = -1.0 / sin(B);
	} else if (F <= 6.3e+38) {
		tmp = (pow(fma(x, 2.0, 2.0), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = -1.0 * ((F * ((F / (F * F)) + (x / F))) / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.0105)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 6.3e+38)
		tmp = Float64(Float64((fma(x, 2.0, 2.0) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	else
		tmp = Float64(-1.0 * Float64(Float64(F * Float64(Float64(F / Float64(F * F)) + Float64(x / F))) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.0105], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.3e+38], N[(N[(N[Power[N[(x * 2.0 + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(F * N[(N[(F / N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.0105000000000000007

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.0%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.0%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -0.0105000000000000007 < F < 6.30000000000000003e38

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.0%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.0%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-\frac{x}{B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \frac{x}{B}} \]

      5. lower--.f64 49.0%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \frac{x}{B}} \]

   6. Applied rewrites 49.0%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{B}} \]

   7. Taylor expanded in F around 0

      \[\leadsto {\left(\mathsf{fma}\left(x, 2, \color{blue}{2}\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 35.3%

      \[\leadsto {\left(\mathsf{fma}\left(x, 2, \color{blue}{2}\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{B} \]

   10. Taylor expanded in B around 0

       \[\leadsto {\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5} \cdot \color{blue}{\frac{F}{B}} - \frac{x}{B} \]
   11. Step-by-step derivation

       1. lower-/.f64 28.1%

          \[\leadsto {\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5} \cdot \frac{F}{\color{blue}{B}} - \frac{x}{B} \]

   12. Applied rewrites 28.1%

       \[\leadsto {\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5} \cdot \color{blue}{\frac{F}{B}} - \frac{x}{B} \]

   if 6.30000000000000003e38 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \color{blue}{\left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{\color{blue}{x \cdot \cos B}}{F \cdot \sin B}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \color{blue}{\cos B}}{F \cdot \sin B}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}\right)\right) \]

      11. lower-sin.f64 49.1%

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

   4. Applied rewrites 49.1%

      \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      5. lower-/.f64 27.9%

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

   7. Applied rewrites 27.9%

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      2. mult-flip N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(1 \cdot \frac{1}{F} + \frac{x}{F}\right)}{B} \]

      3. rgt-mult-inverse N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\left(F \cdot \frac{1}{F}\right) \cdot \frac{1}{F} + \frac{x}{F}\right)}{B} \]

      4. mult-flip-rev N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{F}{F} \cdot \frac{1}{F} + \frac{x}{F}\right)}{B} \]

      5. frac-times N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{F \cdot 1}{F \cdot F} + \frac{x}{F}\right)}{B} \]

      6. *-commutative N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1 \cdot F}{F \cdot F} + \frac{x}{F}\right)}{B} \]

      7. *-lft-identity N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{F}{F \cdot F} + \frac{x}{F}\right)}{B} \]

      8. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{F}{F \cdot F} + \frac{x}{F}\right)}{B} \]

      9. lower-*.f64 23.2%

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{F}{F \cdot F} + \frac{x}{F}\right)}{B} \]

   9. Applied rewrites 23.2%

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{F}{F \cdot F} + \frac{x}{F}\right)}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 43.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -0.00072:\\ \;\;\;\;-\frac{1 + x}{B}\\ \mathbf{elif}\;F \leq 6.3 \cdot 10^{+38}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{F \cdot \left(\frac{F}{F \cdot F} + \frac{x}{F}\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.00072)
   (- (/ (+ 1.0 x) B))
   (if (<= F 6.3e+38)
     (- (* (pow (fma x 2.0 2.0) -0.5) (/ F B)) (/ x B))
     (* -1.0 (/ (* F (+ (/ F (* F F)) (/ x F))) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.00072) {
		tmp = -((1.0 + x) / B);
	} else if (F <= 6.3e+38) {
		tmp = (pow(fma(x, 2.0, 2.0), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = -1.0 * ((F * ((F / (F * F)) + (x / F))) / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.00072)
		tmp = Float64(-Float64(Float64(1.0 + x) / B));
	elseif (F <= 6.3e+38)
		tmp = Float64(Float64((fma(x, 2.0, 2.0) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	else
		tmp = Float64(-1.0 * Float64(Float64(F * Float64(Float64(F / Float64(F * F)) + Float64(x / F))) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.00072], (-N[(N[(1.0 + x), $MachinePrecision] / B), $MachinePrecision]), If[LessEqual[F, 6.3e+38], N[(N[(N[Power[N[(x * 2.0 + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(F * N[(N[(F / N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.20000000000000045e-4

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \color{blue}{\left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{\color{blue}{x \cdot \cos B}}{F \cdot \sin B}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \color{blue}{\cos B}}{F \cdot \sin B}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}\right)\right) \]

      11. lower-sin.f64 49.1%

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

   4. Applied rewrites 49.1%

      \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      5. lower-/.f64 27.9%

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

   7. Applied rewrites 27.9%

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B}\right) \]

      3. lower-neg.f64 27.9%

         \[\leadsto -\frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

   9. Applied rewrites 29.0%

      \[\leadsto -\frac{1 + x}{B} \]

   if -7.20000000000000045e-4 < F < 6.30000000000000003e38

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 49.0%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 49.0%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-\frac{x}{B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \frac{x}{B}} \]

      5. lower--.f64 49.0%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \frac{x}{B}} \]

   6. Applied rewrites 49.0%

      \[\leadsto \color{blue}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{B}} \]

   7. Taylor expanded in F around 0

      \[\leadsto {\left(\mathsf{fma}\left(x, 2, \color{blue}{2}\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 35.3%

      \[\leadsto {\left(\mathsf{fma}\left(x, 2, \color{blue}{2}\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{B} \]

   10. Taylor expanded in B around 0

       \[\leadsto {\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5} \cdot \color{blue}{\frac{F}{B}} - \frac{x}{B} \]
   11. Step-by-step derivation

       1. lower-/.f64 28.1%

          \[\leadsto {\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5} \cdot \frac{F}{\color{blue}{B}} - \frac{x}{B} \]

   12. Applied rewrites 28.1%

       \[\leadsto {\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5} \cdot \color{blue}{\frac{F}{B}} - \frac{x}{B} \]

   if 6.30000000000000003e38 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \color{blue}{\left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{\color{blue}{x \cdot \cos B}}{F \cdot \sin B}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \color{blue}{\cos B}}{F \cdot \sin B}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}\right)\right) \]

      11. lower-sin.f64 49.1%

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

   4. Applied rewrites 49.1%

      \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      5. lower-/.f64 27.9%

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

   7. Applied rewrites 27.9%

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      2. mult-flip N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(1 \cdot \frac{1}{F} + \frac{x}{F}\right)}{B} \]

      3. rgt-mult-inverse N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\left(F \cdot \frac{1}{F}\right) \cdot \frac{1}{F} + \frac{x}{F}\right)}{B} \]

      4. mult-flip-rev N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{F}{F} \cdot \frac{1}{F} + \frac{x}{F}\right)}{B} \]

      5. frac-times N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{F \cdot 1}{F \cdot F} + \frac{x}{F}\right)}{B} \]

      6. *-commutative N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1 \cdot F}{F \cdot F} + \frac{x}{F}\right)}{B} \]

      7. *-lft-identity N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{F}{F \cdot F} + \frac{x}{F}\right)}{B} \]

      8. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{F}{F \cdot F} + \frac{x}{F}\right)}{B} \]

      9. lower-*.f64 23.2%

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{F}{F \cdot F} + \frac{x}{F}\right)}{B} \]

   9. Applied rewrites 23.2%

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{F}{F \cdot F} + \frac{x}{F}\right)}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 29.0% accurate, 14.3× speedup

Math

\[-\frac{1 + x}{B} \]

FPCore

(FPCore (F B x) :precision binary64 (- (/ (+ 1.0 x) B)))

C

double code(double F, double B, double x) {
	return -((1.0 + x) / B);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -((1.0d0 + x) / b)
end function

Java

public static double code(double F, double B, double x) {
	return -((1.0 + x) / B);
}

Python

def code(F, B, x):
	return -((1.0 + x) / B)

Julia

function code(F, B, x)
	return Float64(-Float64(Float64(1.0 + x) / B))
end

MATLAB

function tmp = code(F, B, x)
	tmp = -((1.0 + x) / B);
end

Wolfram

code[F_, B_, x_] := (-N[(N[(1.0 + x), $MachinePrecision] / B), $MachinePrecision])

Derivation

1. Initial program 76.3%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in F around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]
3. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto -1 \cdot \left(F \cdot \color{blue}{\left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)}\right) \]

   3. lower-+.f64 N/A

      \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}\right)\right) \]

   4. lower-/.f64 N/A

      \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{\color{blue}{x \cdot \cos B}}{F \cdot \sin B}\right)\right) \]

   5. lower-*.f64 N/A

      \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \color{blue}{\cos B}}{F \cdot \sin B}\right)\right) \]

   6. lower-sin.f64 N/A

      \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

   7. lower-/.f64 N/A

      \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}\right)\right) \]

   8. lower-*.f64 N/A

      \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}\right)\right) \]

   9. lower-cos.f64 N/A

      \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

   10. lower-*.f64 N/A

       \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}\right)\right) \]

   11. lower-sin.f64 49.1%

       \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

4. Applied rewrites 49.1%

   \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

5. Taylor expanded in B around 0

   \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{\color{blue}{B}} \]
6. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

   2. lower-*.f64 N/A

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

   3. lower-+.f64 N/A

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

   4. lower-/.f64 N/A

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

   5. lower-/.f64 27.9%

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

7. Applied rewrites 27.9%

   \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{\color{blue}{B}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto -1 \cdot \color{blue}{\frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B}} \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B}\right) \]

   3. lower-neg.f64 27.9%

      \[\leadsto -\frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

9. Applied rewrites 29.0%

   \[\leadsto -\frac{1 + x}{B} \]
10. Add Preprocessing

Alternative 21: 10.7% accurate, 26.5× speedup

Math

\[\frac{-1}{B} \]

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 76.3%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in F around -inf

   \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

   2. lower-sin.f64 17.0%

      \[\leadsto \frac{-1}{\sin B} \]

4. Applied rewrites 17.0%

   \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

5. Taylor expanded in B around 0

   \[\leadsto \frac{-1}{B} \]
6. Step-by-step derivation

7. Applied rewrites 10.7%

   \[\leadsto \frac{-1}{B} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025189 
(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))))))