VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.4% → 99.4%

Time: 6.5s

Alternatives: 23

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: 77.4% 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: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -1.65 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{-1}{F} \cdot F - t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 1.32 \cdot 10^{+54}:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{F} \cdot F - t\_0}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -1.65e+31)
     (/ (- (* (/ -1.0 F) F) t_0) (sin B))
     (if (<= F 1.32e+54)
       (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F (sin B))) (/ x (tan B)))
       (/ (- (* (/ 1.0 F) F) t_0) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -1.65e+31) {
		tmp = (((-1.0 / F) * F) - t_0) / sin(B);
	} else if (F <= 1.32e+54) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / sin(B))) - (x / tan(B));
	} else {
		tmp = (((1.0 / F) * F) - t_0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -1.65e+31)
		tmp = Float64(Float64(Float64(Float64(-1.0 / F) * F) - t_0) / sin(B));
	elseif (F <= 1.32e+54)
		tmp = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / sin(B))) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / F) * F) - t_0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -1.65e+31], N[(N[(N[(N[(-1.0 / F), $MachinePrecision] * F), $MachinePrecision] - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.32e+54], 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[(N[(N[(N[(1.0 / F), $MachinePrecision] * F), $MachinePrecision] - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.65e31

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. mult-flip-rev N/A

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

      11. lift-tan.f64 N/A

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

      12. tan-quot N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. div-flip-rev N/A

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

      16. associate-/l* N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in F around -inf

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

      1. lower-/.f64 55.4%

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

   8. Applied rewrites 55.4%

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

   if -1.65e31 < F < 1.3200000000000001e54

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 77.4%

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

   3. Applied rewrites 77.5%

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

   if 1.3200000000000001e54 < F

   1. Initial program 77.4%

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

   2. Taylor expanded in F around 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 47.4%

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

   4. Applied rewrites 47.4%

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

   6. Applied rewrites 55.5%

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

Alternative 2: 98.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -0.0225:\\ \;\;\;\;\frac{\frac{-1}{F} \cdot F - t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 39000:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{F} \cdot F - t\_0}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -0.0225)
     (/ (- (* (/ -1.0 F) F) t_0) (sin B))
     (if (<= F 39000.0)
       (fma F (/ (pow (fma x 2.0 2.0) -0.5) (sin B)) (/ (- x) (tan B)))
       (/ (- (* (/ 1.0 F) F) t_0) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -0.0225) {
		tmp = (((-1.0 / F) * F) - t_0) / sin(B);
	} else if (F <= 39000.0) {
		tmp = fma(F, (pow(fma(x, 2.0, 2.0), -0.5) / sin(B)), (-x / tan(B)));
	} else {
		tmp = (((1.0 / F) * F) - t_0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -0.0225)
		tmp = Float64(Float64(Float64(Float64(-1.0 / F) * F) - t_0) / sin(B));
	elseif (F <= 39000.0)
		tmp = fma(F, Float64((fma(x, 2.0, 2.0) ^ -0.5) / sin(B)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / F) * F) - t_0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -0.0225], N[(N[(N[(N[(-1.0 / F), $MachinePrecision] * F), $MachinePrecision] - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 39000.0], N[(F * N[(N[Power[N[(x * 2.0 + 2.0), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / F), $MachinePrecision] * F), $MachinePrecision] - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.022499999999999999

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. mult-flip-rev N/A

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

      11. lift-tan.f64 N/A

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

      12. tan-quot N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. div-flip-rev N/A

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

      16. associate-/l* N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in F around -inf

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

      1. lower-/.f64 55.4%

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

   8. Applied rewrites 55.4%

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

   if -0.022499999999999999 < F < 39000

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-lft-identity N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. lift-/.f64 N/A

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

      11. lift-neg.f64 N/A

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

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

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

      13. lift-*.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. lower-fma.f64 N/A

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

   5. Applied rewrites 85.7%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 56.1%

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

   if 39000 < F

   1. Initial program 77.4%

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

   2. Taylor expanded in F around 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 47.4%

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

   4. Applied rewrites 47.4%

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

   6. Applied rewrites 55.5%

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

Alternative 3: 98.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -0.0225:\\ \;\;\;\;\frac{\frac{-1}{F} \cdot F - t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{{\left(2 + 2 \cdot x\right)}^{-0.5} \cdot F - t\_0}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{F} \cdot F - t\_0}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -0.0225)
     (/ (- (* (/ -1.0 F) F) t_0) (sin B))
     (if (<= F 2e-9)
       (/ (- (* (pow (+ 2.0 (* 2.0 x)) -0.5) F) t_0) (sin B))
       (/ (- (* (/ 1.0 F) F) t_0) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -0.0225) {
		tmp = (((-1.0 / F) * F) - t_0) / sin(B);
	} else if (F <= 2e-9) {
		tmp = ((pow((2.0 + (2.0 * x)), -0.5) * F) - t_0) / sin(B);
	} else {
		tmp = (((1.0 / F) * F) - t_0) / sin(B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(b) * x
    if (f <= (-0.0225d0)) then
        tmp = ((((-1.0d0) / f) * f) - t_0) / sin(b)
    else if (f <= 2d-9) then
        tmp = ((((2.0d0 + (2.0d0 * x)) ** (-0.5d0)) * f) - t_0) / sin(b)
    else
        tmp = (((1.0d0 / f) * f) - t_0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.cos(B) * x;
	double tmp;
	if (F <= -0.0225) {
		tmp = (((-1.0 / F) * F) - t_0) / Math.sin(B);
	} else if (F <= 2e-9) {
		tmp = ((Math.pow((2.0 + (2.0 * x)), -0.5) * F) - t_0) / Math.sin(B);
	} else {
		tmp = (((1.0 / F) * F) - t_0) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.cos(B) * x
	tmp = 0
	if F <= -0.0225:
		tmp = (((-1.0 / F) * F) - t_0) / math.sin(B)
	elif F <= 2e-9:
		tmp = ((math.pow((2.0 + (2.0 * x)), -0.5) * F) - t_0) / math.sin(B)
	else:
		tmp = (((1.0 / F) * F) - t_0) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -0.0225)
		tmp = Float64(Float64(Float64(Float64(-1.0 / F) * F) - t_0) / sin(B));
	elseif (F <= 2e-9)
		tmp = Float64(Float64(Float64((Float64(2.0 + Float64(2.0 * x)) ^ -0.5) * F) - t_0) / sin(B));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / F) * F) - t_0) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = cos(B) * x;
	tmp = 0.0;
	if (F <= -0.0225)
		tmp = (((-1.0 / F) * F) - t_0) / sin(B);
	elseif (F <= 2e-9)
		tmp = ((((2.0 + (2.0 * x)) ^ -0.5) * F) - t_0) / sin(B);
	else
		tmp = (((1.0 / F) * F) - t_0) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -0.0225], N[(N[(N[(N[(-1.0 / F), $MachinePrecision] * F), $MachinePrecision] - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2e-9], N[(N[(N[(N[Power[N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / F), $MachinePrecision] * F), $MachinePrecision] - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.022499999999999999

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. mult-flip-rev N/A

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

      11. lift-tan.f64 N/A

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

      12. tan-quot N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. div-flip-rev N/A

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

      16. associate-/l* N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in F around -inf

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

      1. lower-/.f64 55.4%

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

   8. Applied rewrites 55.4%

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

   if -0.022499999999999999 < F < 2.0000000000000001e-9

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. mult-flip-rev N/A

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

      11. lift-tan.f64 N/A

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

      12. tan-quot N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. div-flip-rev N/A

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

      16. associate-/l* N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in F around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 55.7%

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

   8. Applied rewrites 55.7%

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

   if 2.0000000000000001e-9 < F

   1. Initial program 77.4%

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

   2. Taylor expanded in F around 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 47.4%

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

   4. Applied rewrites 47.4%

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

   6. Applied rewrites 55.5%

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

Alternative 4: 92.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -4.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{-1}{F} \cdot F - t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-95}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\tan B}, x, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B}\right)\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{F} \cdot F - t\_0}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -4.2e-13)
     (/ (- (* (/ -1.0 F) F) t_0) (sin B))
     (if (<= F 1.8e-95)
       (fma
        (/ -1.0 (tan B))
        x
        (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F B)))
       (if (<= F 2e-9)
         (/ (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F) x) (sin B))
         (/ (- (* (/ 1.0 F) F) t_0) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -4.2e-13) {
		tmp = (((-1.0 / F) * F) - t_0) / sin(B);
	} else if (F <= 1.8e-95) {
		tmp = fma((-1.0 / tan(B)), x, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / B)));
	} else if (F <= 2e-9) {
		tmp = ((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F) - x) / sin(B);
	} else {
		tmp = (((1.0 / F) * F) - t_0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -4.2e-13)
		tmp = Float64(Float64(Float64(Float64(-1.0 / F) * F) - t_0) / sin(B));
	elseif (F <= 1.8e-95)
		tmp = fma(Float64(-1.0 / tan(B)), x, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)));
	elseif (F <= 2e-9)
		tmp = Float64(Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F) - x) / sin(B));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / F) * F) - t_0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -4.2e-13], N[(N[(N[(N[(-1.0 / F), $MachinePrecision] * F), $MachinePrecision] - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.8e-95], N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x + N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2e-9], N[(N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / F), $MachinePrecision] * F), $MachinePrecision] - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.1999999999999998e-13

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. mult-flip-rev N/A

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

      11. lift-tan.f64 N/A

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

      12. tan-quot N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. div-flip-rev N/A

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

      16. associate-/l* N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in F around -inf

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

      1. lower-/.f64 55.4%

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

   8. Applied rewrites 55.4%

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

   if -4.1999999999999998e-13 < F < 1.8000000000000001e-95

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 77.4%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 77.4%

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

   3. Applied rewrites 77.4%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 62.2%

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

   if 1.8000000000000001e-95 < F < 2.0000000000000001e-9

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. mult-flip-rev N/A

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

      11. lift-tan.f64 N/A

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

      12. tan-quot N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. div-flip-rev N/A

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

      16. associate-/l* N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 60.1%

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

   if 2.0000000000000001e-9 < F

   1. Initial program 77.4%

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

   2. Taylor expanded in F around 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 47.4%

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

   4. Applied rewrites 47.4%

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

   6. Applied rewrites 55.5%

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

Alternative 5: 92.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -4.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{-1}{F} \cdot F - t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-95}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{F} \cdot F - t\_0}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -4.2e-13)
     (/ (- (* (/ -1.0 F) F) t_0) (sin B))
     (if (<= F 1.8e-95)
       (+
        (- (* x (/ 1.0 (tan B))))
        (* (/ F B) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
       (if (<= F 2e-9)
         (/ (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F) x) (sin B))
         (/ (- (* (/ 1.0 F) F) t_0) (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -4.2e-13) {
		tmp = (((-1.0 / F) * F) - t_0) / sin(B);
	} else if (F <= 1.8e-95) {
		tmp = -(x * (1.0 / tan(B))) + ((F / B) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else if (F <= 2e-9) {
		tmp = ((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F) - x) / sin(B);
	} else {
		tmp = (((1.0 / F) * F) - t_0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -4.2e-13)
		tmp = Float64(Float64(Float64(Float64(-1.0 / F) * F) - t_0) / sin(B));
	elseif (F <= 1.8e-95)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / B) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))));
	elseif (F <= 2e-9)
		tmp = Float64(Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F) - x) / sin(B));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / F) * F) - t_0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -4.2e-13], N[(N[(N[(N[(-1.0 / F), $MachinePrecision] * F), $MachinePrecision] - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.8e-95], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / B), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2e-9], N[(N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / F), $MachinePrecision] * F), $MachinePrecision] - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.1999999999999998e-13

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. mult-flip-rev N/A

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

      11. lift-tan.f64 N/A

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

      12. tan-quot N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. div-flip-rev N/A

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

      16. associate-/l* N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in F around -inf

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

      1. lower-/.f64 55.4%

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

   8. Applied rewrites 55.4%

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

   if -4.1999999999999998e-13 < F < 1.8000000000000001e-95

   1. Initial program 77.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.2%

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

   4. Applied rewrites 62.2%

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

   if 1.8000000000000001e-95 < F < 2.0000000000000001e-9

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. mult-flip-rev N/A

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

      11. lift-tan.f64 N/A

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

      12. tan-quot N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. div-flip-rev N/A

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

      16. associate-/l* N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 60.1%

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

   if 2.0000000000000001e-9 < F

   1. Initial program 77.4%

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

   2. Taylor expanded in F around 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 47.4%

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

   4. Applied rewrites 47.4%

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

   6. Applied rewrites 55.5%

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

Alternative 6: 86.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -4.2 \cdot 10^{-13}:\\ \;\;\;\;\frac{\frac{-1}{F} \cdot F - \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{-95}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\tan B}, x, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B}\right)\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{+157}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{1}{F}}{\sin B}, \frac{-x}{B \cdot \left(1 + 0.3333333333333333 \cdot {B}^{2}\right)}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.2e-13)
   (/ (- (* (/ -1.0 F) F) (* (cos B) x)) (sin B))
   (if (<= F 1.8e-95)
     (fma (/ -1.0 (tan B)) x (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F B)))
     (if (<= F 9.5e+157)
       (/ (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F) x) (sin B))
       (fma
        F
        (/ (/ 1.0 F) (sin B))
        (/ (- x) (* B (+ 1.0 (* 0.3333333333333333 (pow B 2.0))))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.2e-13) {
		tmp = (((-1.0 / F) * F) - (cos(B) * x)) / sin(B);
	} else if (F <= 1.8e-95) {
		tmp = fma((-1.0 / tan(B)), x, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / B)));
	} else if (F <= 9.5e+157) {
		tmp = ((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F) - x) / sin(B);
	} else {
		tmp = fma(F, ((1.0 / F) / sin(B)), (-x / (B * (1.0 + (0.3333333333333333 * pow(B, 2.0))))));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.2e-13)
		tmp = Float64(Float64(Float64(Float64(-1.0 / F) * F) - Float64(cos(B) * x)) / sin(B));
	elseif (F <= 1.8e-95)
		tmp = fma(Float64(-1.0 / tan(B)), x, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)));
	elseif (F <= 9.5e+157)
		tmp = Float64(Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F) - x) / sin(B));
	else
		tmp = fma(F, Float64(Float64(1.0 / F) / sin(B)), Float64(Float64(-x) / Float64(B * Float64(1.0 + Float64(0.3333333333333333 * (B ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.2e-13], N[(N[(N[(N[(-1.0 / F), $MachinePrecision] * F), $MachinePrecision] - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.8e-95], N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x + N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.5e+157], N[(N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[(B * N[(1.0 + N[(0.3333333333333333 * N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.1999999999999998e-13

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. mult-flip-rev N/A

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

      11. lift-tan.f64 N/A

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

      12. tan-quot N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. div-flip-rev N/A

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

      16. associate-/l* N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in F around -inf

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

      1. lower-/.f64 55.4%

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

   8. Applied rewrites 55.4%

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

   if -4.1999999999999998e-13 < F < 1.8000000000000001e-95

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 77.4%

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 77.4%

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

   3. Applied rewrites 77.4%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 62.2%

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

   if 1.8000000000000001e-95 < F < 9.4999999999999996e157

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. mult-flip-rev N/A

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

      11. lift-tan.f64 N/A

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

      12. tan-quot N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. div-flip-rev N/A

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

      16. associate-/l* N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 60.1%

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

   if 9.4999999999999996e157 < F

   1. Initial program 77.4%

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

   2. Taylor expanded in F around 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 47.4%

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

   4. Applied rewrites 47.4%

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

      1. metadata-eval 47.4%

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

      2. metadata-eval 47.4%

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

   6. Applied rewrites 51.9%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{F}}{\sin B}, \frac{-x}{B \cdot \color{blue}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right)}}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{F}}{\sin B}, \frac{-x}{B \cdot \left(1 + \color{blue}{\frac{1}{3} \cdot {B}^{2}}\right)}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{F}}{\sin B}, \frac{-x}{B \cdot \left(1 + \frac{1}{3} \cdot \color{blue}{{B}^{2}}\right)}\right) \]

      4. lower-pow.f64 32.3%

         \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{F}}{\sin B}, \frac{-x}{B \cdot \left(1 + 0.3333333333333333 \cdot {B}^{\color{blue}{2}}\right)}\right) \]

   9. Applied rewrites 32.3%

      \[\leadsto \mathsf{fma}\left(F, \frac{\frac{1}{F}}{\sin B}, \frac{-x}{\color{blue}{B \cdot \left(1 + 0.3333333333333333 \cdot {B}^{2}\right)}}\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 7: 77.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{-x}{\frac{1}{\frac{1}{\tan B}}}\\ \mathbf{elif}\;x \leq 0.0135:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -8.2e-48)
   (/ (- x) (/ 1.0 (/ 1.0 (tan B))))
   (if (<= x 0.0135)
     (/ (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F) x) (sin B))
     (/ (- x) (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -8.2e-48) {
		tmp = -x / (1.0 / (1.0 / tan(B)));
	} else if (x <= 0.0135) {
		tmp = ((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F) - x) / sin(B);
	} else {
		tmp = -x / tan(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -8.2e-48)
		tmp = Float64(Float64(-x) / Float64(1.0 / Float64(1.0 / tan(B))));
	elseif (x <= 0.0135)
		tmp = Float64(Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F) - x) / sin(B));
	else
		tmp = Float64(Float64(-x) / tan(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -8.2e-48], N[((-x) / N[(1.0 / N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.0135], N[(N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.2000000000000003e-48

   1. Initial program 77.4%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.9%

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

   4. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. div-flip-rev N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. tan-quot N/A

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

      10. lift-tan.f64 N/A

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

      11. mult-flip-rev N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

      12. distribute-frac-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\color{blue}{\tan B}} \]

      13. lift-neg.f64 N/A

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

      14. lift-/.f64 56.0%

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

   6. Applied rewrites 56.0%

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

      1. lift-tan.f64 N/A

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

      2. tan-quot N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. div-flip N/A

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

      6. lower-unsound-/.f32 N/A

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

      7. lower-/.f32 N/A

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

      8. div-flip-rev N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. tan-quot N/A

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

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

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

      13. lower-/.f64 N/A

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

      14. lift-tan.f64 55.9%

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

   8. Applied rewrites 55.9%

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

   if -8.2000000000000003e-48 < x < 0.0135

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. distribute-neg-frac N/A

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

      8. lift-neg.f64 N/A

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

      9. remove-double-neg N/A

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

      10. mult-flip-rev N/A

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

      11. lift-tan.f64 N/A

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

      12. tan-quot N/A

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

      13. lift-sin.f64 N/A

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

      14. lift-cos.f64 N/A

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

      15. div-flip-rev N/A

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

      16. associate-/l* N/A

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

      17. lift-*.f64 N/A

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 60.1%

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

   if 0.0135 < x

   1. Initial program 77.4%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.9%

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

   4. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. div-flip-rev N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. tan-quot N/A

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

      10. lift-tan.f64 N/A

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

      11. mult-flip-rev N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

      12. distribute-frac-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\color{blue}{\tan B}} \]

      13. lift-neg.f64 N/A

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

      14. lift-/.f64 56.0%

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

   6. Applied rewrites 56.0%

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

Alternative 8: 73.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := -1 \cdot \frac{x}{B}\\ \mathbf{if}\;F \leq -2.2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, t\_1\right)\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{-152}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5}, \frac{F}{\sin B}, \frac{-x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 1, t\_1\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (* -1.0 (/ x B))))
   (if (<= F -2.2e-15)
     (fma t_0 -1.0 t_1)
     (if (<= F 2.9e-152)
       (/ (- x) (tan B))
       (if (<= F 2e-9)
         (fma (pow (fma 2.0 x 2.0) -0.5) (/ F (sin B)) (/ (- x) B))
         (fma t_0 1.0 t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = -1.0 * (x / B);
	double tmp;
	if (F <= -2.2e-15) {
		tmp = fma(t_0, -1.0, t_1);
	} else if (F <= 2.9e-152) {
		tmp = -x / tan(B);
	} else if (F <= 2e-9) {
		tmp = fma(pow(fma(2.0, x, 2.0), -0.5), (F / sin(B)), (-x / B));
	} else {
		tmp = fma(t_0, 1.0, t_1);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(-1.0 * Float64(x / B))
	tmp = 0.0
	if (F <= -2.2e-15)
		tmp = fma(t_0, -1.0, t_1);
	elseif (F <= 2.9e-152)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 2e-9)
		tmp = fma((fma(2.0, x, 2.0) ^ -0.5), Float64(F / sin(B)), Float64(Float64(-x) / B));
	else
		tmp = fma(t_0, 1.0, t_1);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.2e-15], N[(t$95$0 * -1.0 + t$95$1), $MachinePrecision], If[LessEqual[F, 2.9e-152], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2e-9], N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / B), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 1.0 + t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.1999999999999999e-15

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.0%

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

   6. Applied rewrites 58.0%

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

   7. Taylor expanded in F around -inf

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

   9. Applied rewrites 35.8%

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

   if -2.1999999999999999e-15 < F < 2.9000000000000001e-152

   1. Initial program 77.4%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.9%

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

   4. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. div-flip-rev N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. tan-quot N/A

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

      10. lift-tan.f64 N/A

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

      11. mult-flip-rev N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

      12. distribute-frac-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\color{blue}{\tan B}} \]

      13. lift-neg.f64 N/A

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

      14. lift-/.f64 56.0%

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

   6. Applied rewrites 56.0%

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

   if 2.9000000000000001e-152 < F < 2.0000000000000001e-9

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.0%

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

   6. Applied rewrites 58.0%

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

   7. Taylor expanded in F around 0

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

   9. Applied rewrites 36.6%

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

       1. lift-fma.f64 N/A

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

       2. *-commutative N/A

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

       3. lift-*.f64 N/A

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

       4. associate-*l* N/A

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

       5. lift-/.f64 N/A

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

       6. mult-flip N/A

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

       7. lift-sin.f64 N/A

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

       8. lower-fma.f64 N/A

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

       9. lift-sin.f64 N/A

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

       10. lower-/.f64 35.1%

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

       11. lift-*.f64 N/A

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

       12. mul-1-neg N/A

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

       13. lift-/.f64 N/A

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

       14. distribute-neg-frac N/A

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

   11. Applied rewrites 35.1%

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

   if 2.0000000000000001e-9 < F

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.0%

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

   6. Applied rewrites 58.0%

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

   7. Taylor expanded in F around inf

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

   9. Applied rewrites 35.9%

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

Alternative 9: 71.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := -1 \cdot \frac{x}{B}\\ \mathbf{if}\;F \leq -2.2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, t\_1\right)\\ \mathbf{elif}\;F \leq 1.82 \cdot 10^{-90}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 1, t\_1\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (* -1.0 (/ x B))))
   (if (<= F -2.2e-15)
     (fma t_0 -1.0 t_1)
     (if (<= F 1.82e-90)
       (/ (- x) (tan B))
       (if (<= F 2e-9)
         (/ (- (* F (pow (+ 2.0 (fma 2.0 x (pow F 2.0))) -0.5)) x) B)
         (fma t_0 1.0 t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = -1.0 * (x / B);
	double tmp;
	if (F <= -2.2e-15) {
		tmp = fma(t_0, -1.0, t_1);
	} else if (F <= 1.82e-90) {
		tmp = -x / tan(B);
	} else if (F <= 2e-9) {
		tmp = ((F * pow((2.0 + fma(2.0, x, pow(F, 2.0))), -0.5)) - x) / B;
	} else {
		tmp = fma(t_0, 1.0, t_1);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(-1.0 * Float64(x / B))
	tmp = 0.0
	if (F <= -2.2e-15)
		tmp = fma(t_0, -1.0, t_1);
	elseif (F <= 1.82e-90)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 2e-9)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + fma(2.0, x, (F ^ 2.0))) ^ -0.5)) - x) / B);
	else
		tmp = fma(t_0, 1.0, t_1);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.2e-15], N[(t$95$0 * -1.0 + t$95$1), $MachinePrecision], If[LessEqual[F, 1.82e-90], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2e-9], N[(N[(N[(F * N[Power[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(t$95$0 * 1.0 + t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.1999999999999999e-15

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.0%

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

   6. Applied rewrites 58.0%

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

   7. Taylor expanded in F around -inf

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

   9. Applied rewrites 35.8%

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

   if -2.1999999999999999e-15 < F < 1.8199999999999999e-90

   1. Initial program 77.4%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.9%

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

   4. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. div-flip-rev N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. tan-quot N/A

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

      10. lift-tan.f64 N/A

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

      11. mult-flip-rev N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

      12. distribute-frac-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\color{blue}{\tan B}} \]

      13. lift-neg.f64 N/A

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

      14. lift-/.f64 56.0%

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

   6. Applied rewrites 56.0%

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

   if 1.8199999999999999e-90 < F < 2.0000000000000001e-9

   1. Initial program 77.4%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 43.6%

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

   if 2.0000000000000001e-9 < F

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.0%

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

   6. Applied rewrites 58.0%

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

   7. Taylor expanded in F around inf

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

   9. Applied rewrites 35.9%

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

Alternative 10: 71.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := -1 \cdot \frac{x}{B}\\ \mathbf{if}\;F \leq -2.2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, t\_1\right)\\ \mathbf{elif}\;F \leq 1.82 \cdot 10^{-90}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{B}, {\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5} \cdot F, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 1, t\_1\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (* -1.0 (/ x B))))
   (if (<= F -2.2e-15)
     (fma t_0 -1.0 t_1)
     (if (<= F 1.82e-90)
       (/ (- x) (tan B))
       (if (<= F 2e-9)
         (fma (/ 1.0 B) (* (pow (fma 2.0 x 2.0) -0.5) F) t_1)
         (fma t_0 1.0 t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = -1.0 * (x / B);
	double tmp;
	if (F <= -2.2e-15) {
		tmp = fma(t_0, -1.0, t_1);
	} else if (F <= 1.82e-90) {
		tmp = -x / tan(B);
	} else if (F <= 2e-9) {
		tmp = fma((1.0 / B), (pow(fma(2.0, x, 2.0), -0.5) * F), t_1);
	} else {
		tmp = fma(t_0, 1.0, t_1);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(-1.0 * Float64(x / B))
	tmp = 0.0
	if (F <= -2.2e-15)
		tmp = fma(t_0, -1.0, t_1);
	elseif (F <= 1.82e-90)
		tmp = Float64(Float64(-x) / tan(B));
	elseif (F <= 2e-9)
		tmp = fma(Float64(1.0 / B), Float64((fma(2.0, x, 2.0) ^ -0.5) * F), t_1);
	else
		tmp = fma(t_0, 1.0, t_1);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.2e-15], N[(t$95$0 * -1.0 + t$95$1), $MachinePrecision], If[LessEqual[F, 1.82e-90], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2e-9], N[(N[(1.0 / B), $MachinePrecision] * N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] + t$95$1), $MachinePrecision], N[(t$95$0 * 1.0 + t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.1999999999999999e-15

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.0%

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

   6. Applied rewrites 58.0%

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

   7. Taylor expanded in F around -inf

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

   9. Applied rewrites 35.8%

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

   if -2.1999999999999999e-15 < F < 1.8199999999999999e-90

   1. Initial program 77.4%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.9%

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

   4. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. div-flip-rev N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. tan-quot N/A

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

      10. lift-tan.f64 N/A

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

      11. mult-flip-rev N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

      12. distribute-frac-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\color{blue}{\tan B}} \]

      13. lift-neg.f64 N/A

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

      14. lift-/.f64 56.0%

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

   6. Applied rewrites 56.0%

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

   if 1.8199999999999999e-90 < F < 2.0000000000000001e-9

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.0%

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

   6. Applied rewrites 58.0%

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

   7. Taylor expanded in F around 0

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

   9. Applied rewrites 36.6%

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

   10. Taylor expanded in B around 0

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

       1. lower-/.f64 29.9%

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

   12. Applied rewrites 29.9%

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

   if 2.0000000000000001e-9 < F

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.0%

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

   6. Applied rewrites 58.0%

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

   7. Taylor expanded in F around inf

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

   9. Applied rewrites 35.9%

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

Alternative 11: 64.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ t_1 := \frac{1}{\sin B}\\ t_2 := -1 \cdot \frac{x}{B}\\ \mathbf{if}\;F \leq -2.2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, -1, t\_2\right)\\ \mathbf{elif}\;F \leq 1.82 \cdot 10^{-90}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 480000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{B}, {\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5} \cdot F, t\_2\right)\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{+186}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))) (t_1 (/ 1.0 (sin B))) (t_2 (* -1.0 (/ x B))))
   (if (<= F -2.2e-15)
     (fma t_1 -1.0 t_2)
     (if (<= F 1.82e-90)
       t_0
       (if (<= F 480000000000.0)
         (fma (/ 1.0 B) (* (pow (fma 2.0 x 2.0) -0.5) F) t_2)
         (if (<= F 6.2e+186) t_1 t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double t_1 = 1.0 / sin(B);
	double t_2 = -1.0 * (x / B);
	double tmp;
	if (F <= -2.2e-15) {
		tmp = fma(t_1, -1.0, t_2);
	} else if (F <= 1.82e-90) {
		tmp = t_0;
	} else if (F <= 480000000000.0) {
		tmp = fma((1.0 / B), (pow(fma(2.0, x, 2.0), -0.5) * F), t_2);
	} else if (F <= 6.2e+186) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	t_1 = Float64(1.0 / sin(B))
	t_2 = Float64(-1.0 * Float64(x / B))
	tmp = 0.0
	if (F <= -2.2e-15)
		tmp = fma(t_1, -1.0, t_2);
	elseif (F <= 1.82e-90)
		tmp = t_0;
	elseif (F <= 480000000000.0)
		tmp = fma(Float64(1.0 / B), Float64((fma(2.0, x, 2.0) ^ -0.5) * F), t_2);
	elseif (F <= 6.2e+186)
		tmp = t_1;
	else
		tmp = 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[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.2e-15], N[(t$95$1 * -1.0 + t$95$2), $MachinePrecision], If[LessEqual[F, 1.82e-90], t$95$0, If[LessEqual[F, 480000000000.0], N[(N[(1.0 / B), $MachinePrecision] * N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[F, 6.2e+186], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.1999999999999999e-15

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.0%

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

   6. Applied rewrites 58.0%

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

   7. Taylor expanded in F around -inf

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

   9. Applied rewrites 35.8%

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

   if -2.1999999999999999e-15 < F < 1.8199999999999999e-90 or 6.2000000000000002e186 < F

   1. Initial program 77.4%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.9%

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

   4. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. div-flip-rev N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. tan-quot N/A

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

      10. lift-tan.f64 N/A

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

      11. mult-flip-rev N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

      12. distribute-frac-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\color{blue}{\tan B}} \]

      13. lift-neg.f64 N/A

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

      14. lift-/.f64 56.0%

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

   6. Applied rewrites 56.0%

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

   if 1.8199999999999999e-90 < F < 4.8e11

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.0%

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

   6. Applied rewrites 58.0%

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

   7. Taylor expanded in F around 0

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

   9. Applied rewrites 36.6%

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

   10. Taylor expanded in B around 0

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

       1. lower-/.f64 29.9%

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

   12. Applied rewrites 29.9%

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

   if 4.8e11 < F < 6.2000000000000002e186

   1. Initial program 77.4%

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

   2. Taylor expanded in F around 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 16.8%

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

   4. Applied rewrites 16.8%

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

Alternative 12: 59.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -3.4 \cdot 10^{+202}:\\ \;\;\;\;\frac{1}{\frac{\tan B}{-x}}\\ \mathbf{elif}\;F \leq -1.35 \cdot 10^{+18}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.82 \cdot 10^{-90}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 480000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{B}, {\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5} \cdot F, -1 \cdot \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{+186}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -3.4e+202)
     (/ 1.0 (/ (tan B) (- x)))
     (if (<= F -1.35e+18)
       (/ -1.0 (sin B))
       (if (<= F 1.82e-90)
         t_0
         (if (<= F 480000000000.0)
           (fma (/ 1.0 B) (* (pow (fma 2.0 x 2.0) -0.5) F) (* -1.0 (/ x B)))
           (if (<= F 6.2e+186) (/ 1.0 (sin B)) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -3.4e+202) {
		tmp = 1.0 / (tan(B) / -x);
	} else if (F <= -1.35e+18) {
		tmp = -1.0 / sin(B);
	} else if (F <= 1.82e-90) {
		tmp = t_0;
	} else if (F <= 480000000000.0) {
		tmp = fma((1.0 / B), (pow(fma(2.0, x, 2.0), -0.5) * F), (-1.0 * (x / B)));
	} else if (F <= 6.2e+186) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -3.4e+202)
		tmp = Float64(1.0 / Float64(tan(B) / Float64(-x)));
	elseif (F <= -1.35e+18)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 1.82e-90)
		tmp = t_0;
	elseif (F <= 480000000000.0)
		tmp = fma(Float64(1.0 / B), Float64((fma(2.0, x, 2.0) ^ -0.5) * F), Float64(-1.0 * Float64(x / B)));
	elseif (F <= 6.2e+186)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = 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, -3.4e+202], N[(1.0 / N[(N[Tan[B], $MachinePrecision] / (-x)), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.35e+18], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.82e-90], t$95$0, If[LessEqual[F, 480000000000.0], N[(N[(1.0 / B), $MachinePrecision] * N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.2e+186], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -3.4000000000000001e202

   1. Initial program 77.4%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.9%

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

   4. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. div-flip-rev N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. tan-quot N/A

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

      10. lift-tan.f64 N/A

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

      11. mult-flip-rev N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

      12. distribute-frac-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\color{blue}{\tan B}} \]

      13. lift-neg.f64 N/A

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

      14. lift-/.f64 56.0%

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

   6. Applied rewrites 56.0%

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

      1. lift-tan.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip N/A

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

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

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

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

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

      6. lift-tan.f64 55.9%

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

   8. Applied rewrites 55.9%

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

   if -3.4000000000000001e202 < F < -1.35e18

   1. Initial program 77.4%

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

   2. Taylor expanded in F around -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 17.2%

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

   4. Applied rewrites 17.2%

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

   if -1.35e18 < F < 1.8199999999999999e-90 or 6.2000000000000002e186 < F

   1. Initial program 77.4%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.9%

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

   4. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. div-flip-rev N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. tan-quot N/A

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

      10. lift-tan.f64 N/A

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

      11. mult-flip-rev N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

      12. distribute-frac-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\color{blue}{\tan B}} \]

      13. lift-neg.f64 N/A

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

      14. lift-/.f64 56.0%

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

   6. Applied rewrites 56.0%

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

   if 1.8199999999999999e-90 < F < 4.8e11

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.0%

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

   6. Applied rewrites 58.0%

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

   7. Taylor expanded in F around 0

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

   9. Applied rewrites 36.6%

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

   10. Taylor expanded in B around 0

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

       1. lower-/.f64 29.9%

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

   12. Applied rewrites 29.9%

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

   if 4.8e11 < F < 6.2000000000000002e186

   1. Initial program 77.4%

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

   2. Taylor expanded in F around 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 16.8%

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

   4. Applied rewrites 16.8%

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

Alternative 13: 59.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -3.4 \cdot 10^{+202}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq -1.35 \cdot 10^{+18}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.82 \cdot 10^{-90}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 480000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{B}, {\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5} \cdot F, -1 \cdot \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 6.2 \cdot 10^{+186}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -3.4e+202)
     t_0
     (if (<= F -1.35e+18)
       (/ -1.0 (sin B))
       (if (<= F 1.82e-90)
         t_0
         (if (<= F 480000000000.0)
           (fma (/ 1.0 B) (* (pow (fma 2.0 x 2.0) -0.5) F) (* -1.0 (/ x B)))
           (if (<= F 6.2e+186) (/ 1.0 (sin B)) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -3.4e+202) {
		tmp = t_0;
	} else if (F <= -1.35e+18) {
		tmp = -1.0 / sin(B);
	} else if (F <= 1.82e-90) {
		tmp = t_0;
	} else if (F <= 480000000000.0) {
		tmp = fma((1.0 / B), (pow(fma(2.0, x, 2.0), -0.5) * F), (-1.0 * (x / B)));
	} else if (F <= 6.2e+186) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -3.4e+202)
		tmp = t_0;
	elseif (F <= -1.35e+18)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 1.82e-90)
		tmp = t_0;
	elseif (F <= 480000000000.0)
		tmp = fma(Float64(1.0 / B), Float64((fma(2.0, x, 2.0) ^ -0.5) * F), Float64(-1.0 * Float64(x / B)));
	elseif (F <= 6.2e+186)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = 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, -3.4e+202], t$95$0, If[LessEqual[F, -1.35e+18], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.82e-90], t$95$0, If[LessEqual[F, 480000000000.0], N[(N[(1.0 / B), $MachinePrecision] * N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.2e+186], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.4000000000000001e202 or -1.35e18 < F < 1.8199999999999999e-90 or 6.2000000000000002e186 < F

   1. Initial program 77.4%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.9%

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

   4. Applied rewrites 55.9%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. div-flip-rev N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. tan-quot N/A

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

      10. lift-tan.f64 N/A

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

      11. mult-flip-rev N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

      12. distribute-frac-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\color{blue}{\tan B}} \]

      13. lift-neg.f64 N/A

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

      14. lift-/.f64 56.0%

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

   6. Applied rewrites 56.0%

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

   if -3.4000000000000001e202 < F < -1.35e18

   1. Initial program 77.4%

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

   2. Taylor expanded in F around -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 17.2%

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

   4. Applied rewrites 17.2%

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

   if 1.8199999999999999e-90 < F < 4.8e11

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.0%

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

   6. Applied rewrites 58.0%

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

   7. Taylor expanded in F around 0

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

   9. Applied rewrites 36.6%

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

   10. Taylor expanded in B around 0

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

       1. lower-/.f64 29.9%

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

   12. Applied rewrites 29.9%

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

   if 4.8e11 < F < 6.2000000000000002e186

   1. Initial program 77.4%

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

   2. Taylor expanded in F around 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 16.8%

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

   4. Applied rewrites 16.8%

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

Alternative 14: 51.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -2.15:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 480000000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{B}, {\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5} \cdot F, -1 \cdot \frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.15)
   (/ -1.0 (sin B))
   (if (<= F 480000000000.0)
     (fma (/ 1.0 B) (* (pow (fma 2.0 x 2.0) -0.5) F) (* -1.0 (/ x B)))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.15) {
		tmp = -1.0 / sin(B);
	} else if (F <= 480000000000.0) {
		tmp = fma((1.0 / B), (pow(fma(2.0, x, 2.0), -0.5) * F), (-1.0 * (x / B)));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.15)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 480000000000.0)
		tmp = fma(Float64(1.0 / B), Float64((fma(2.0, x, 2.0) ^ -0.5) * F), Float64(-1.0 * Float64(x / B)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.15], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 480000000000.0], N[(N[(1.0 / B), $MachinePrecision] * N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.1499999999999999

   1. Initial program 77.4%

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

   2. Taylor expanded in F around -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 17.2%

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

   4. Applied rewrites 17.2%

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

   if -2.1499999999999999 < F < 4.8e11

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.0%

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

   6. Applied rewrites 58.0%

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

   7. Taylor expanded in F around 0

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

   9. Applied rewrites 36.6%

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

   10. Taylor expanded in B around 0

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

       1. lower-/.f64 29.9%

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

   12. Applied rewrites 29.9%

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

   if 4.8e11 < F

   1. Initial program 77.4%

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

   2. Taylor expanded in F around 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 16.8%

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

   4. Applied rewrites 16.8%

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

Alternative 15: 45.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.06 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 480000000000:\\ \;\;\;\;-1 \cdot \frac{x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.06e+15)
   (/ -1.0 (sin B))
   (if (<= F 480000000000.0) (* -1.0 (/ x (sin B))) (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.06e+15) {
		tmp = -1.0 / sin(B);
	} else if (F <= 480000000000.0) {
		tmp = -1.0 * (x / sin(B));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.06d+15)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 480000000000.0d0) then
        tmp = (-1.0d0) * (x / sin(b))
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.06e+15) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 480000000000.0) {
		tmp = -1.0 * (x / Math.sin(B));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.06e+15:
		tmp = -1.0 / math.sin(B)
	elif F <= 480000000000.0:
		tmp = -1.0 * (x / math.sin(B))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.06e+15)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 480000000000.0)
		tmp = Float64(-1.0 * Float64(x / sin(B)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.06e+15)
		tmp = -1.0 / sin(B);
	elseif (F <= 480000000000.0)
		tmp = -1.0 * (x / sin(B));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.06e+15], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 480000000000.0], N[(-1.0 * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.06e15

   1. Initial program 77.4%

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

   2. Taylor expanded in F around -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 17.2%

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

   4. Applied rewrites 17.2%

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

   if -1.06e15 < F < 4.8e11

   1. Initial program 77.4%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.9%

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{x}{\sin \color{blue}{B}} \]
   6. Step-by-step derivation

   7. Applied rewrites 31.4%

      \[\leadsto -1 \cdot \frac{x}{\sin \color{blue}{B}} \]

   if 4.8e11 < F

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 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 16.8%

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 16.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 44.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -40000000000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 480000000000:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -40000000000000.0)
   (/ -1.0 (sin B))
   (if (<= F 480000000000.0) (- (/ x B)) (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -40000000000000.0) {
		tmp = -1.0 / sin(B);
	} else if (F <= 480000000000.0) {
		tmp = -(x / B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-40000000000000.0d0)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 480000000000.0d0) then
        tmp = -(x / b)
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -40000000000000.0) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 480000000000.0) {
		tmp = -(x / B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -40000000000000.0:
		tmp = -1.0 / math.sin(B)
	elif F <= 480000000000.0:
		tmp = -(x / B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -40000000000000.0)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 480000000000.0)
		tmp = Float64(-Float64(x / B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -40000000000000.0)
		tmp = -1.0 / sin(B);
	elseif (F <= 480000000000.0)
		tmp = -(x / B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -40000000000000.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 480000000000.0], (-N[(x / B), $MachinePrecision]), N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4e13

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -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 17.2%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -4e13 < F < 4.8e11

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.9%

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 29.1%

         \[\leadsto -1 \cdot \frac{x}{B} \]

   7. Applied rewrites 29.1%

      \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x}{B}\right) \]

      3. lower-neg.f64 29.1%

         \[\leadsto -\frac{x}{B} \]

   9. Applied rewrites 29.1%

      \[\leadsto \color{blue}{-\frac{x}{B}} \]

   if 4.8e11 < F

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 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 16.8%

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 16.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 36.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -40000000000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, 0.3333333333333333 \cdot \left({B}^{2} \cdot x\right)\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -40000000000000.0)
   (/ -1.0 (sin B))
   (/ (fma -1.0 x (* 0.3333333333333333 (* (pow B 2.0) x))) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -40000000000000.0) {
		tmp = -1.0 / sin(B);
	} else {
		tmp = fma(-1.0, x, (0.3333333333333333 * (pow(B, 2.0) * x))) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -40000000000000.0)
		tmp = Float64(-1.0 / sin(B));
	else
		tmp = Float64(fma(-1.0, x, Float64(0.3333333333333333 * Float64((B ^ 2.0) * x))) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -40000000000000.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * x + N[(0.3333333333333333 * N[(N[Power[B, 2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -4e13

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -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 17.2%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -4e13 < F

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.9%

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \frac{\cos B}{\sin B}\right) \]

      6. div-flip-rev N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right) \]

      8. lift-cos.f64 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right) \]

      9. tan-quot N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right) \]

      10. lift-tan.f64 N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right) \]

      11. mult-flip-rev N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

      12. distribute-frac-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\color{blue}{\tan B}} \]

      13. lift-neg.f64 N/A

          \[\leadsto \frac{-x}{\tan \color{blue}{B}} \]

      14. lift-/.f64 56.0%

          \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

   6. Applied rewrites 56.0%

      \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{-1 \cdot x + \frac{1}{3} \cdot \left({B}^{2} \cdot x\right)}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot x + \frac{1}{3} \cdot \left({B}^{2} \cdot x\right)}{B} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{1}{3} \cdot \left({B}^{2} \cdot x\right)\right)}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{1}{3} \cdot \left({B}^{2} \cdot x\right)\right)}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{1}{3} \cdot \left({B}^{2} \cdot x\right)\right)}{B} \]

      5. lower-pow.f64 29.1%

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, 0.3333333333333333 \cdot \left({B}^{2} \cdot x\right)\right)}{B} \]

   9. Applied rewrites 29.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1, x, 0.3333333333333333 \cdot \left({B}^{2} \cdot x\right)\right)}{\color{blue}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 29.9% accurate, 3.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.26 \cdot 10^{+18}:\\ \;\;\;\;\frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, 0.3333333333333333 \cdot \left({B}^{2} \cdot x\right)\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.26e+18)
   (/ -1.0 (* B (+ 1.0 (* -0.16666666666666666 (pow B 2.0)))))
   (/ (fma -1.0 x (* 0.3333333333333333 (* (pow B 2.0) x))) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.26e+18) {
		tmp = -1.0 / (B * (1.0 + (-0.16666666666666666 * pow(B, 2.0))));
	} else {
		tmp = fma(-1.0, x, (0.3333333333333333 * (pow(B, 2.0) * x))) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.26e+18)
		tmp = Float64(-1.0 / Float64(B * Float64(1.0 + Float64(-0.16666666666666666 * (B ^ 2.0)))));
	else
		tmp = Float64(fma(-1.0, x, Float64(0.3333333333333333 * Float64((B ^ 2.0) * x))) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.26e+18], N[(-1.0 / N[(B * N[(1.0 + N[(-0.16666666666666666 * N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * x + N[(0.3333333333333333 * N[(N[Power[B, 2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.26e18

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -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 17.2%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-1}{B \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {B}^{2}\right)}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{B \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {B}^{2}}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{B}^{2}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{\color{blue}{2}}\right)} \]

      4. lower-pow.f64 10.0%

         \[\leadsto \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{2}\right)} \]

   7. Applied rewrites 10.0%

      \[\leadsto \frac{-1}{B \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {B}^{2}\right)}} \]

   if -1.26e18 < F

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.9%

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \frac{\cos B}{\sin B}\right) \]

      6. div-flip-rev N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right) \]

      8. lift-cos.f64 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right) \]

      9. tan-quot N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right) \]

      10. lift-tan.f64 N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right) \]

      11. mult-flip-rev N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

      12. distribute-frac-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\color{blue}{\tan B}} \]

      13. lift-neg.f64 N/A

          \[\leadsto \frac{-x}{\tan \color{blue}{B}} \]

      14. lift-/.f64 56.0%

          \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

   6. Applied rewrites 56.0%

      \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{-1 \cdot x + \frac{1}{3} \cdot \left({B}^{2} \cdot x\right)}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot x + \frac{1}{3} \cdot \left({B}^{2} \cdot x\right)}{B} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{1}{3} \cdot \left({B}^{2} \cdot x\right)\right)}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{1}{3} \cdot \left({B}^{2} \cdot x\right)\right)}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{1}{3} \cdot \left({B}^{2} \cdot x\right)\right)}{B} \]

      5. lower-pow.f64 29.1%

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, 0.3333333333333333 \cdot \left({B}^{2} \cdot x\right)\right)}{B} \]

   9. Applied rewrites 29.1%

      \[\leadsto \frac{\mathsf{fma}\left(-1, x, 0.3333333333333333 \cdot \left({B}^{2} \cdot x\right)\right)}{\color{blue}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 29.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.26 \cdot 10^{+18}:\\ \;\;\;\;\frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B \cdot \left(1 + 0.3333333333333333 \cdot {B}^{2}\right)}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.26e+18)
   (/ -1.0 (* B (+ 1.0 (* -0.16666666666666666 (pow B 2.0)))))
   (/ (- x) (* B (+ 1.0 (* 0.3333333333333333 (pow B 2.0)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.26e+18) {
		tmp = -1.0 / (B * (1.0 + (-0.16666666666666666 * pow(B, 2.0))));
	} else {
		tmp = -x / (B * (1.0 + (0.3333333333333333 * pow(B, 2.0))));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.26d+18)) then
        tmp = (-1.0d0) / (b * (1.0d0 + ((-0.16666666666666666d0) * (b ** 2.0d0))))
    else
        tmp = -x / (b * (1.0d0 + (0.3333333333333333d0 * (b ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.26e+18) {
		tmp = -1.0 / (B * (1.0 + (-0.16666666666666666 * Math.pow(B, 2.0))));
	} else {
		tmp = -x / (B * (1.0 + (0.3333333333333333 * Math.pow(B, 2.0))));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.26e+18:
		tmp = -1.0 / (B * (1.0 + (-0.16666666666666666 * math.pow(B, 2.0))))
	else:
		tmp = -x / (B * (1.0 + (0.3333333333333333 * math.pow(B, 2.0))))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.26e+18)
		tmp = Float64(-1.0 / Float64(B * Float64(1.0 + Float64(-0.16666666666666666 * (B ^ 2.0)))));
	else
		tmp = Float64(Float64(-x) / Float64(B * Float64(1.0 + Float64(0.3333333333333333 * (B ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.26e+18)
		tmp = -1.0 / (B * (1.0 + (-0.16666666666666666 * (B ^ 2.0))));
	else
		tmp = -x / (B * (1.0 + (0.3333333333333333 * (B ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.26e+18], N[(-1.0 / N[(B * N[(1.0 + N[(-0.16666666666666666 * N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-x) / N[(B * N[(1.0 + N[(0.3333333333333333 * N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.26e18

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -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 17.2%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-1}{B \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {B}^{2}\right)}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{B \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {B}^{2}}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{B}^{2}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{\color{blue}{2}}\right)} \]

      4. lower-pow.f64 10.0%

         \[\leadsto \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{2}\right)} \]

   7. Applied rewrites 10.0%

      \[\leadsto \frac{-1}{B \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {B}^{2}\right)}} \]

   if -1.26e18 < F

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.9%

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      3. lift-/.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      5. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \frac{\cos B}{\sin B}\right) \]

      6. div-flip-rev N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right) \]

      7. lift-sin.f64 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right) \]

      8. lift-cos.f64 N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right) \]

      9. tan-quot N/A

         \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right) \]

      10. lift-tan.f64 N/A

          \[\leadsto \mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right) \]

      11. mult-flip-rev N/A

          \[\leadsto \mathsf{neg}\left(\frac{x}{\tan B}\right) \]

      12. distribute-frac-neg N/A

          \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\color{blue}{\tan B}} \]

      13. lift-neg.f64 N/A

          \[\leadsto \frac{-x}{\tan \color{blue}{B}} \]

      14. lift-/.f64 56.0%

          \[\leadsto \frac{-x}{\color{blue}{\tan B}} \]

   6. Applied rewrites 56.0%

      \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{-x}{B \cdot \color{blue}{\left(1 + \frac{1}{3} \cdot {B}^{2}\right)}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-x}{B \cdot \left(1 + \color{blue}{\frac{1}{3} \cdot {B}^{2}}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{-x}{B \cdot \left(1 + \frac{1}{3} \cdot \color{blue}{{B}^{2}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-x}{B \cdot \left(1 + \frac{1}{3} \cdot {B}^{\color{blue}{2}}\right)} \]

      4. lower-pow.f64 29.0%

         \[\leadsto \frac{-x}{B \cdot \left(1 + 0.3333333333333333 \cdot {B}^{2}\right)} \]

   9. Applied rewrites 29.0%

      \[\leadsto \frac{-x}{B \cdot \color{blue}{\left(1 + 0.3333333333333333 \cdot {B}^{2}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 29.8% accurate, 3.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.26 \cdot 10^{+18}:\\ \;\;\;\;\frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.26e+18)
   (/ -1.0 (* B (+ 1.0 (* -0.16666666666666666 (pow B 2.0)))))
   (- (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.26e+18) {
		tmp = -1.0 / (B * (1.0 + (-0.16666666666666666 * pow(B, 2.0))));
	} else {
		tmp = -(x / B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.26d+18)) then
        tmp = (-1.0d0) / (b * (1.0d0 + ((-0.16666666666666666d0) * (b ** 2.0d0))))
    else
        tmp = -(x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.26e+18) {
		tmp = -1.0 / (B * (1.0 + (-0.16666666666666666 * Math.pow(B, 2.0))));
	} else {
		tmp = -(x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.26e+18:
		tmp = -1.0 / (B * (1.0 + (-0.16666666666666666 * math.pow(B, 2.0))))
	else:
		tmp = -(x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.26e+18)
		tmp = Float64(-1.0 / Float64(B * Float64(1.0 + Float64(-0.16666666666666666 * (B ^ 2.0)))));
	else
		tmp = Float64(-Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.26e+18)
		tmp = -1.0 / (B * (1.0 + (-0.16666666666666666 * (B ^ 2.0))));
	else
		tmp = -(x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.26e+18], N[(-1.0 / N[(B * N[(1.0 + N[(-0.16666666666666666 * N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(x / B), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if F < -1.26e18

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -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 17.2%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-1}{B \cdot \color{blue}{\left(1 + \frac{-1}{6} \cdot {B}^{2}\right)}} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{B \cdot \left(1 + \color{blue}{\frac{-1}{6} \cdot {B}^{2}}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \color{blue}{{B}^{2}}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot {B}^{\color{blue}{2}}\right)} \]

      4. lower-pow.f64 10.0%

         \[\leadsto \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{2}\right)} \]

   7. Applied rewrites 10.0%

      \[\leadsto \frac{-1}{B \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot {B}^{2}\right)}} \]

   if -1.26e18 < F

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.9%

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 29.1%

         \[\leadsto -1 \cdot \frac{x}{B} \]

   7. Applied rewrites 29.1%

      \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x}{B}\right) \]

      3. lower-neg.f64 29.1%

         \[\leadsto -\frac{x}{B} \]

   9. Applied rewrites 29.1%

      \[\leadsto \color{blue}{-\frac{x}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 29.8% accurate, 3.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.26 \cdot 10^{+18}:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot {B}^{2} - 1}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.26e+18)
   (/ (- (* -0.16666666666666666 (pow B 2.0)) 1.0) B)
   (- (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.26e+18) {
		tmp = ((-0.16666666666666666 * pow(B, 2.0)) - 1.0) / B;
	} else {
		tmp = -(x / B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.26d+18)) then
        tmp = (((-0.16666666666666666d0) * (b ** 2.0d0)) - 1.0d0) / b
    else
        tmp = -(x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.26e+18) {
		tmp = ((-0.16666666666666666 * Math.pow(B, 2.0)) - 1.0) / B;
	} else {
		tmp = -(x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.26e+18:
		tmp = ((-0.16666666666666666 * math.pow(B, 2.0)) - 1.0) / B
	else:
		tmp = -(x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.26e+18)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * (B ^ 2.0)) - 1.0) / B);
	else
		tmp = Float64(-Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.26e+18)
		tmp = ((-0.16666666666666666 * (B ^ 2.0)) - 1.0) / B;
	else
		tmp = -(x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.26e+18], N[(N[(N[(-0.16666666666666666 * N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / B), $MachinePrecision], (-N[(x / B), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if F < -1.26e18

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -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 17.2%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]

      4. lower-pow.f64 9.8%

         \[\leadsto \frac{-0.16666666666666666 \cdot {B}^{2} - 1}{B} \]

   7. Applied rewrites 9.8%

      \[\leadsto \frac{-0.16666666666666666 \cdot {B}^{2} - 1}{\color{blue}{B}} \]

   if -1.26e18 < F

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.9%

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 29.1%

         \[\leadsto -1 \cdot \frac{x}{B} \]

   7. Applied rewrites 29.1%

      \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x}{B}\right) \]

      3. lower-neg.f64 29.1%

         \[\leadsto -\frac{x}{B} \]

   9. Applied rewrites 29.1%

      \[\leadsto \color{blue}{-\frac{x}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 29.8% accurate, 12.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.26 \cdot 10^{+18}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.26e+18) (/ -1.0 B) (- (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.26e+18) {
		tmp = -1.0 / B;
	} else {
		tmp = -(x / B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.26d+18)) then
        tmp = (-1.0d0) / b
    else
        tmp = -(x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.26e+18) {
		tmp = -1.0 / B;
	} else {
		tmp = -(x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.26e+18:
		tmp = -1.0 / B
	else:
		tmp = -(x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.26e+18)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(-Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.26e+18)
		tmp = -1.0 / B;
	else
		tmp = -(x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.26e+18], N[(-1.0 / B), $MachinePrecision], (-N[(x / B), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if F < -1.26e18

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -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 17.2%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-1}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 10.1%

      \[\leadsto \frac{-1}{B} \]

   if -1.26e18 < F

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.9%

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 29.1%

         \[\leadsto -1 \cdot \frac{x}{B} \]

   7. Applied rewrites 29.1%

      \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x}{B}\right) \]

      3. lower-neg.f64 29.1%

         \[\leadsto -\frac{x}{B} \]

   9. Applied rewrites 29.1%

      \[\leadsto \color{blue}{-\frac{x}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 10.1% accurate, 26.5× speedup

Math

\[\frac{-1}{B} \]

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 77.4%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in F around -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 17.2%

      \[\leadsto \frac{-1}{\sin B} \]

4. Applied rewrites 17.2%

   \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

5. Taylor expanded in B around 0

   \[\leadsto \frac{-1}{B} \]
6. Step-by-step derivation

7. Applied rewrites 10.1%

   \[\leadsto \frac{-1}{B} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025191 
(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))))))