VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.2% → 99.6%

Time: 6.6s

Alternatives: 22

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{1}{\sin B}, t\_0\right)\\ \mathbf{elif}\;F \leq 5000:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{1}{F}}{\sin B}, \frac{-1}{\frac{\tan B}{x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -2e+19)
     (fma -1.0 (/ 1.0 (sin B)) t_0)
     (if (<= F 5000.0)
       (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B)) t_0)
       (fma F (/ (/ 1.0 F) (sin B)) (/ -1.0 (/ (tan B) x)))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -2e+19) {
		tmp = fma(-1.0, (1.0 / sin(B)), t_0);
	} else if (F <= 5000.0) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), t_0);
	} else {
		tmp = fma(F, ((1.0 / F) / sin(B)), (-1.0 / (tan(B) / x)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -2e+19)
		tmp = fma(-1.0, Float64(1.0 / sin(B)), t_0);
	elseif (F <= 5000.0)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), t_0);
	else
		tmp = fma(F, Float64(Float64(1.0 / F) / sin(B)), Float64(-1.0 / Float64(tan(B) / x)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2e+19], N[(-1.0 * N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, 5000.0], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2e19

   1. Initial program 76.2%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mult-flip-rev N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 84.7

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 56.6%

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

   if -2e19 < F < 5e3

   1. Initial program 76.2%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

   if 5e3 < F

   1. Initial program 76.2%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lift-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. associate-/r* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-neg-frac2 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-/r* N/A

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

      19. lift-sin.f64 N/A

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

      20. lift-cos.f64 N/A

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

      21. tan-quot N/A

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

      22. lift-tan.f64 N/A

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

      23. lower-/.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in F around inf

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

      1. lower-/.f64 52.3

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

   8. Applied rewrites 52.3%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -200000000:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{1}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 80000000000000:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{1}{F}}{\sin B}, \frac{-1}{\frac{\tan B}{x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -200000000.0)
   (fma -1.0 (/ 1.0 (sin B)) (/ (- x) (tan B)))
   (if (<= F 80000000000000.0)
     (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F (sin B))) (/ x (tan B)))
     (fma F (/ (/ 1.0 F) (sin B)) (/ -1.0 (/ (tan B) x))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -200000000.0) {
		tmp = fma(-1.0, (1.0 / sin(B)), (-x / tan(B)));
	} else if (F <= 80000000000000.0) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / sin(B))) - (x / tan(B));
	} else {
		tmp = fma(F, ((1.0 / F) / sin(B)), (-1.0 / (tan(B) / x)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -200000000.0)
		tmp = fma(-1.0, Float64(1.0 / sin(B)), Float64(Float64(-x) / tan(B)));
	elseif (F <= 80000000000000.0)
		tmp = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / sin(B))) - Float64(x / tan(B)));
	else
		tmp = fma(F, Float64(Float64(1.0 / F) / sin(B)), Float64(-1.0 / Float64(tan(B) / x)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -200000000.0], N[(-1.0 * N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 80000000000000.0], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2e8

   1. Initial program 76.2%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mult-flip-rev N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 84.7

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 56.6%

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

   if -2e8 < F < 8e13

   1. Initial program 76.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.2

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

   3. Applied rewrites 76.3%

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

   if 8e13 < F

   1. Initial program 76.2%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lift-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. associate-/r* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-neg-frac2 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-/r* N/A

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

      19. lift-sin.f64 N/A

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

      20. lift-cos.f64 N/A

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

      21. tan-quot N/A

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

      22. lift-tan.f64 N/A

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

      23. lower-/.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in F around inf

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

      1. lower-/.f64 52.3

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

   8. Applied rewrites 52.3%

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

Alternative 3: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{1}{\sin B}, t\_0\right)\\ \mathbf{elif}\;F \leq 2.6:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5}}{\sin B}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{1}{F}}{\sin B}, \frac{-1}{\frac{\tan B}{x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -1.42)
     (fma -1.0 (/ 1.0 (sin B)) t_0)
     (if (<= F 2.6)
       (fma F (/ (pow (fma 2.0 x 2.0) -0.5) (sin B)) t_0)
       (fma F (/ (/ 1.0 F) (sin B)) (/ -1.0 (/ (tan B) x)))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = fma(-1.0, (1.0 / sin(B)), t_0);
	} else if (F <= 2.6) {
		tmp = fma(F, (pow(fma(2.0, x, 2.0), -0.5) / sin(B)), t_0);
	} else {
		tmp = fma(F, ((1.0 / F) / sin(B)), (-1.0 / (tan(B) / x)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -1.42)
		tmp = fma(-1.0, Float64(1.0 / sin(B)), t_0);
	elseif (F <= 2.6)
		tmp = fma(F, Float64((fma(2.0, x, 2.0) ^ -0.5) / sin(B)), t_0);
	else
		tmp = fma(F, Float64(Float64(1.0 / F) / sin(B)), Float64(-1.0 / Float64(tan(B) / x)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.42], N[(-1.0 * N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, 2.6], N[(F * N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4199999999999999

   1. Initial program 76.2%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mult-flip-rev N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 84.7

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 56.6%

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

   if -1.4199999999999999 < F < 2.60000000000000009

   1. Initial program 76.2%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

   4. Taylor expanded in F around 0

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

   6. Applied rewrites 55.5%

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

   if 2.60000000000000009 < F

   1. Initial program 76.2%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lift-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. associate-/r* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-neg-frac2 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-/r* N/A

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

      19. lift-sin.f64 N/A

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

      20. lift-cos.f64 N/A

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

      21. tan-quot N/A

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

      22. lift-tan.f64 N/A

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

      23. lower-/.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in F around inf

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

      1. lower-/.f64 52.3

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

   8. Applied rewrites 52.3%

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

Alternative 4: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{1}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 2.6:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{1}{F}}{\sin B}, \frac{-1}{\frac{\tan B}{x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.42)
   (fma -1.0 (/ 1.0 (sin B)) (/ (- x) (tan B)))
   (if (<= F 2.6)
     (- (* (pow (fma 2.0 x 2.0) -0.5) (/ F (sin B))) (/ x (tan B)))
     (fma F (/ (/ 1.0 F) (sin B)) (/ -1.0 (/ (tan B) x))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.42) {
		tmp = fma(-1.0, (1.0 / sin(B)), (-x / tan(B)));
	} else if (F <= 2.6) {
		tmp = (pow(fma(2.0, x, 2.0), -0.5) * (F / sin(B))) - (x / tan(B));
	} else {
		tmp = fma(F, ((1.0 / F) / sin(B)), (-1.0 / (tan(B) / x)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.42)
		tmp = fma(-1.0, Float64(1.0 / sin(B)), Float64(Float64(-x) / tan(B)));
	elseif (F <= 2.6)
		tmp = Float64(Float64((fma(2.0, x, 2.0) ^ -0.5) * Float64(F / sin(B))) - Float64(x / tan(B)));
	else
		tmp = fma(F, Float64(Float64(1.0 / F) / sin(B)), Float64(-1.0 / Float64(tan(B) / x)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.42], N[(-1.0 * N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.6], N[(N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4199999999999999

   1. Initial program 76.2%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mult-flip-rev N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 84.7

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 56.6%

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

   if -1.4199999999999999 < F < 2.60000000000000009

   1. Initial program 76.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 76.9

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

   3. Applied rewrites 76.9%

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

   4. Taylor expanded in F around 0

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

   6. Applied rewrites 54.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. sub-flip-reverse N/A

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

      8. lower--.f64 54.2

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

   8. Applied rewrites 54.1%

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

   if 2.60000000000000009 < F

   1. Initial program 76.2%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lift-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. associate-/r* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-neg-frac2 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-/r* N/A

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

      19. lift-sin.f64 N/A

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

      20. lift-cos.f64 N/A

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

      21. tan-quot N/A

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

      22. lift-tan.f64 N/A

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

      23. lower-/.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in F around inf

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

      1. lower-/.f64 52.3

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

   8. Applied rewrites 52.3%

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

Alternative 5: 92.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.182:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{1}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 440:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{1}{F}}{\sin B}, \frac{-1}{\frac{\tan B}{x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.182)
   (fma -1.0 (/ 1.0 (sin B)) (/ (- x) (tan B)))
   (if (<= F 440.0)
     (+
      (- (* x (/ 1.0 (tan B))))
      (* (/ F B) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
     (fma F (/ (/ 1.0 F) (sin B)) (/ -1.0 (/ (tan B) x))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.182) {
		tmp = fma(-1.0, (1.0 / sin(B)), (-x / tan(B)));
	} else if (F <= 440.0) {
		tmp = -(x * (1.0 / tan(B))) + ((F / B) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = fma(F, ((1.0 / F) / sin(B)), (-1.0 / (tan(B) / x)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.182)
		tmp = fma(-1.0, Float64(1.0 / sin(B)), Float64(Float64(-x) / tan(B)));
	elseif (F <= 440.0)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / B) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))));
	else
		tmp = fma(F, Float64(Float64(1.0 / F) / sin(B)), Float64(-1.0 / Float64(tan(B) / x)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.182], N[(-1.0 * N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 440.0], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / B), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.182

   1. Initial program 76.2%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mult-flip-rev N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 84.7

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 56.6%

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

   if -0.182 < F < 440

   1. Initial program 76.2%

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

   2. Taylor expanded in 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 61.4

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

   4. Applied rewrites 61.4%

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

   if 440 < F

   1. Initial program 76.2%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lift-tan.f64 N/A

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

      6. tan-quot N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. associate-/r* N/A

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

      10. *-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. distribute-neg-frac2 N/A

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

      13. distribute-neg-frac N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. associate-/r* N/A

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

      19. lift-sin.f64 N/A

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

      20. lift-cos.f64 N/A

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

      21. tan-quot N/A

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

      22. lift-tan.f64 N/A

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

      23. lower-/.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in F around inf

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

      1. lower-/.f64 52.3

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

   8. Applied rewrites 52.3%

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

Alternative 6: 92.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -0.182:\\ \;\;\;\;\mathsf{fma}\left(-1, t\_0, t\_1\right)\\ \mathbf{elif}\;F \leq 440:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, t\_0, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ (- x) (tan B))))
   (if (<= F -0.182)
     (fma -1.0 t_0 t_1)
     (if (<= F 440.0)
       (+
        (- (* x (/ 1.0 (tan B))))
        (* (/ F B) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
       (fma 1.0 t_0 t_1)))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = -x / tan(B);
	double tmp;
	if (F <= -0.182) {
		tmp = fma(-1.0, t_0, t_1);
	} else if (F <= 440.0) {
		tmp = -(x * (1.0 / tan(B))) + ((F / B) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = fma(1.0, t_0, t_1);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -0.182)
		tmp = fma(-1.0, t_0, t_1);
	elseif (F <= 440.0)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / B) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))));
	else
		tmp = fma(1.0, t_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[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.182], N[(-1.0 * t$95$0 + t$95$1), $MachinePrecision], If[LessEqual[F, 440.0], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / B), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * t$95$0 + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.182

   1. Initial program 76.2%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mult-flip-rev N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 84.7

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 56.6%

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

   if -0.182 < F < 440

   1. Initial program 76.2%

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

   2. Taylor expanded in 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 61.4

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

   4. Applied rewrites 61.4%

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

   if 440 < F

   1. Initial program 76.2%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mult-flip-rev N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 84.7

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 56.4%

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

Alternative 7: 89.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, -1 \cdot \frac{x}{B}\right)\\ t_1 := \frac{1}{\sin B}\\ t_2 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -1.55 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(-1, t\_1, t\_2\right)\\ \mathbf{elif}\;F \leq -3.3 \cdot 10^{-227}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 5.1 \cdot 10^{-138}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;F \leq 10800000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, t\_1, t\_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (fma
          F
          (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B))
          (* -1.0 (/ x B))))
        (t_1 (/ 1.0 (sin B)))
        (t_2 (/ (- x) (tan B))))
   (if (<= F -1.55e+19)
     (fma -1.0 t_1 t_2)
     (if (<= F -3.3e-227)
       t_0
       (if (<= F 5.1e-138)
         t_2
         (if (<= F 10800000.0) t_0 (fma 1.0 t_1 t_2)))))))

C

double code(double F, double B, double x) {
	double t_0 = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), (-1.0 * (x / B)));
	double t_1 = 1.0 / sin(B);
	double t_2 = -x / tan(B);
	double tmp;
	if (F <= -1.55e+19) {
		tmp = fma(-1.0, t_1, t_2);
	} else if (F <= -3.3e-227) {
		tmp = t_0;
	} else if (F <= 5.1e-138) {
		tmp = t_2;
	} else if (F <= 10800000.0) {
		tmp = t_0;
	} else {
		tmp = fma(1.0, t_1, t_2);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), Float64(-1.0 * Float64(x / B)))
	t_1 = Float64(1.0 / sin(B))
	t_2 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -1.55e+19)
		tmp = fma(-1.0, t_1, t_2);
	elseif (F <= -3.3e-227)
		tmp = t_0;
	elseif (F <= 5.1e-138)
		tmp = t_2;
	elseif (F <= 10800000.0)
		tmp = t_0;
	else
		tmp = fma(1.0, t_1, t_2);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.55e+19], N[(-1.0 * t$95$1 + t$95$2), $MachinePrecision], If[LessEqual[F, -3.3e-227], t$95$0, If[LessEqual[F, 5.1e-138], t$95$2, If[LessEqual[F, 10800000.0], t$95$0, N[(1.0 * t$95$1 + t$95$2), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.55e19

   1. Initial program 76.2%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mult-flip-rev N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 84.7

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 56.6%

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

   if -1.55e19 < F < -3.2999999999999999e-227 or 5.1000000000000002e-138 < F < 1.08e7

   1. Initial program 76.2%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 57.3

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

   6. Applied rewrites 57.3%

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

   if -3.2999999999999999e-227 < F < 5.1000000000000002e-138

   1. Initial program 76.2%

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

   2. Taylor expanded in F around 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 55.4

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

   4. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-flip N/A

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

      5. mul-1-neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. tan-quot N/A

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

      14. lift-tan.f64 N/A

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

      15. div-flip N/A

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

      16. lift-/.f64 55.5

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

   6. Applied rewrites 55.5%

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

   if 1.08e7 < F

   1. Initial program 76.2%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mult-flip-rev N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 84.7

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 56.4%

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

Alternative 8: 76.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(-1, t\_0, t\_1\right)\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F, t\_0, -1 \cdot \frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ (- x) (tan B))))
   (if (<= x -2.9e-52)
     (fma -1.0 t_0 t_1)
     (if (<= x 1.85e-34)
       (fma (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F) t_0 (* -1.0 (/ x B)))
       t_1))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = -x / tan(B);
	double tmp;
	if (x <= -2.9e-52) {
		tmp = fma(-1.0, t_0, t_1);
	} else if (x <= 1.85e-34) {
		tmp = fma((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F), t_0, (-1.0 * (x / B)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (x <= -2.9e-52)
		tmp = fma(-1.0, t_0, t_1);
	elseif (x <= 1.85e-34)
		tmp = fma(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F), t_0, Float64(-1.0 * Float64(x / B)));
	else
		tmp = 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[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.9e-52], N[(-1.0 * t$95$0 + t$95$1), $MachinePrecision], If[LessEqual[x, 1.85e-34], N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] * t$95$0 + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.9000000000000002e-52

   1. Initial program 76.2%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mult-flip-rev N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 84.7

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 56.6%

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

   if -2.9000000000000002e-52 < x < 1.84999999999999994e-34

   1. Initial program 76.2%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mult-flip-rev N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 84.7

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 57.3

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

   8. Applied rewrites 57.3%

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

   if 1.84999999999999994e-34 < x

   1. Initial program 76.2%

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

   2. Taylor expanded in F around 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 55.4

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

   4. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-flip N/A

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

      5. mul-1-neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. tan-quot N/A

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

      14. lift-tan.f64 N/A

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

      15. div-flip N/A

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

      16. lift-/.f64 55.5

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

   6. Applied rewrites 55.5%

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

Alternative 9: 76.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;x \leq -6.3 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{F \cdot B}, t\_0\right)\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F, \frac{1}{\sin B}, -1 \cdot \frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= x -6.3e-12)
     (fma F (/ 1.0 (* F B)) t_0)
     (if (<= x 1.85e-34)
       (fma
        (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F)
        (/ 1.0 (sin B))
        (* -1.0 (/ x B)))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (x <= -6.3e-12) {
		tmp = fma(F, (1.0 / (F * B)), t_0);
	} else if (x <= 1.85e-34) {
		tmp = fma((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F), (1.0 / sin(B)), (-1.0 * (x / B)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (x <= -6.3e-12)
		tmp = fma(F, Float64(1.0 / Float64(F * B)), t_0);
	elseif (x <= 1.85e-34)
		tmp = fma(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F), Float64(1.0 / sin(B)), Float64(-1.0 * Float64(x / 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[x, -6.3e-12], N[(F * N[(1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[x, 1.85e-34], N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] * N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.3000000000000002e-12

   1. Initial program 76.2%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.4

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

   6. Applied rewrites 52.4%

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

   7. Taylor expanded in B around 0

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

   9. Applied rewrites 49.7%

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

   if -6.3000000000000002e-12 < x < 1.84999999999999994e-34

   1. Initial program 76.2%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-neg.f64 N/A

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

      7. distribute-frac-neg N/A

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

      8. mult-flip-rev N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. *-commutative N/A

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

      17. lower-fma.f64 N/A

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

      18. lower-/.f64 84.7

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 57.3

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

   8. Applied rewrites 57.3%

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

   if 1.84999999999999994e-34 < x

   1. Initial program 76.2%

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

   2. Taylor expanded in F around 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 55.4

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

   4. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-flip N/A

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

      5. mul-1-neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. tan-quot N/A

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

      14. lift-tan.f64 N/A

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

      15. div-flip N/A

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

      16. lift-/.f64 55.5

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

   6. Applied rewrites 55.5%

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

Alternative 10: 75.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, -1 \cdot \frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= x -2.1e+16)
     t_0
     (if (<= x 1.85e-34)
       (fma
        F
        (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B))
        (* -1.0 (/ x B)))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (x <= -2.1e+16) {
		tmp = t_0;
	} else if (x <= 1.85e-34) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), (-1.0 * (x / B)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (x <= -2.1e+16)
		tmp = t_0;
	elseif (x <= 1.85e-34)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), Float64(-1.0 * Float64(x / 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[x, -2.1e+16], t$95$0, If[LessEqual[x, 1.85e-34], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1e16 or 1.84999999999999994e-34 < x

   1. Initial program 76.2%

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

   2. Taylor expanded in F around 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 55.4

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

   4. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-flip N/A

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

      5. mul-1-neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. tan-quot N/A

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

      14. lift-tan.f64 N/A

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

      15. div-flip N/A

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

      16. lift-/.f64 55.5

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

   6. Applied rewrites 55.5%

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

   if -2.1e16 < x < 1.84999999999999994e-34

   1. Initial program 76.2%

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

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 57.3

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

   6. Applied rewrites 57.3%

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 61.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 1.25 \cdot 10^{-5}:\\ \;\;\;\;\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 1.25e-5)
   (/ (- (* F (pow (+ 2.0 (fma 2.0 x (pow F 2.0))) -0.5)) x) B)
   (/ (- x) (tan B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 1.25e-5) {
		tmp = ((F * pow((2.0 + fma(2.0, x, pow(F, 2.0))), -0.5)) - x) / B;
	} else {
		tmp = -x / tan(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 1.25e-5)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + fma(2.0, x, (F ^ 2.0))) ^ -0.5)) - x) / B);
	else
		tmp = Float64(Float64(-x) / tan(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 1.25e-5], 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[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.25000000000000006e-5

   1. Initial program 76.2%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 43.3%

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

   if 1.25000000000000006e-5 < B

   1. Initial program 76.2%

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

   2. Taylor expanded in F around 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 55.4

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

   4. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-flip N/A

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

      5. mul-1-neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. tan-quot N/A

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

      14. lift-tan.f64 N/A

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

      15. div-flip N/A

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

      16. lift-/.f64 55.5

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

   6. Applied rewrites 55.5%

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

Alternative 12: 61.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-126}:\\ \;\;\;\;\left(-x \cdot \frac{1}{B}\right) + \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\frac{B}{F}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= x -2.1e+16)
     t_0
     (if (<= x 3e-126)
       (+ (- (* x (/ 1.0 B))) (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ B F)))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (x <= -2.1e+16) {
		tmp = t_0;
	} else if (x <= 3e-126) {
		tmp = -(x * (1.0 / B)) + (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / (B / F));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (x <= -2.1e+16)
		tmp = t_0;
	elseif (x <= 3e-126)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / B))) + Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / Float64(B / F)));
	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[x, -2.1e+16], t$95$0, If[LessEqual[x, 3e-126], N[((-N[(x * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]) + N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[(B / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1e16 or 3.0000000000000002e-126 < x

   1. Initial program 76.2%

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

   2. Taylor expanded in F around 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 55.4

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

   4. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-flip N/A

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

      5. mul-1-neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. tan-quot N/A

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

      14. lift-tan.f64 N/A

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

      15. div-flip N/A

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

      16. lift-/.f64 55.5

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

   6. Applied rewrites 55.5%

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

   if -2.1e16 < x < 3.0000000000000002e-126

   1. Initial program 76.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 76.9

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

   3. Applied rewrites 76.9%

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

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 49.5

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

   6. Applied rewrites 49.5%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 35.5

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

   9. Applied rewrites 35.5%

      \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\color{blue}{\frac{B}{F}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 57.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-126}:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{1}{B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= x -1.4e-8)
     t_0
     (if (<= x 3e-126)
       (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F B)) (* (/ 1.0 B) x))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (x <= -1.4e-8) {
		tmp = t_0;
	} else if (x <= 3e-126) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / B)) - ((1.0 / B) * x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (x <= -1.4e-8)
		tmp = t_0;
	elseif (x <= 3e-126)
		tmp = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - Float64(Float64(1.0 / B) * x));
	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[x, -1.4e-8], t$95$0, If[LessEqual[x, 3e-126], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / B), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4e-8 or 3.0000000000000002e-126 < x

   1. Initial program 76.2%

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

   2. Taylor expanded in F around 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 55.4

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

   4. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-flip N/A

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

      5. mul-1-neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. tan-quot N/A

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

      14. lift-tan.f64 N/A

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

      15. div-flip N/A

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

      16. lift-/.f64 55.5

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

   6. Applied rewrites 55.5%

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

   if -1.4e-8 < x < 3.0000000000000002e-126

   1. Initial program 76.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-flip N/A

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

      5. mult-flip-rev N/A

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

      6. lower-/.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. lift-/.f64 N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 76.9

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

   3. Applied rewrites 76.9%

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

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 49.5

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

   6. Applied rewrites 49.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 49.5

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

      6. lift-/.f64 N/A

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

      7. mult-flip N/A

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

      8. lift-/.f64 N/A

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

      9. div-flip N/A

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

      10. lift-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lift-sin.f64 N/A

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

      13. lower-/.f64 48.8

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 48.8

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

   8. Applied rewrites 48.8%

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

   9. Taylor expanded in B around 0

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

   11. Applied rewrites 34.8%

       \[\leadsto {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\color{blue}{B}} - \frac{1}{B} \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 56.8% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;x \leq -1.6 \cdot 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.05 \cdot 10^{-227}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= x -1.6e-13) t_0 (if (<= x 4.05e-227) (/ -1.0 (sin B)) t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (x <= -1.6e-13) {
		tmp = t_0;
	} else if (x <= 4.05e-227) {
		tmp = -1.0 / sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x / tan(b)
    if (x <= (-1.6d-13)) then
        tmp = t_0
    else if (x <= 4.05d-227) then
        tmp = (-1.0d0) / sin(b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -x / Math.tan(B);
	double tmp;
	if (x <= -1.6e-13) {
		tmp = t_0;
	} else if (x <= 4.05e-227) {
		tmp = -1.0 / Math.sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -x / math.tan(B)
	tmp = 0
	if x <= -1.6e-13:
		tmp = t_0
	elif x <= 4.05e-227:
		tmp = -1.0 / math.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 (x <= -1.6e-13)
		tmp = t_0;
	elseif (x <= 4.05e-227)
		tmp = Float64(-1.0 / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -x / tan(B);
	tmp = 0.0;
	if (x <= -1.6e-13)
		tmp = t_0;
	elseif (x <= 4.05e-227)
		tmp = -1.0 / sin(B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.6e-13], t$95$0, If[LessEqual[x, 4.05e-227], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6e-13 or 4.04999999999999993e-227 < x

   1. Initial program 76.2%

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

   2. Taylor expanded in F around 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 55.4

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

   4. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. div-flip N/A

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

      5. mul-1-neg N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. associate-/l/ N/A

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

      11. lift-sin.f64 N/A

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

      12. lift-cos.f64 N/A

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

      13. tan-quot N/A

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

      14. lift-tan.f64 N/A

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

      15. div-flip N/A

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

      16. lift-/.f64 55.5

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

   6. Applied rewrites 55.5%

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

   if -1.6e-13 < x < 4.04999999999999993e-227

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.7

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 46.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 0.00145:\\ \;\;\;\;-1 \cdot \frac{x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.8e-13)
   (/ -1.0 (sin B))
   (if (<= F 0.00145) (* -1.0 (/ x (sin B))) (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.8e-13) {
		tmp = -1.0 / sin(B);
	} else if (F <= 0.00145) {
		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 <= (-1.8d-13)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 0.00145d0) 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 <= -1.8e-13) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 0.00145) {
		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 <= -1.8e-13:
		tmp = -1.0 / math.sin(B)
	elif F <= 0.00145:
		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 <= -1.8e-13)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 0.00145)
		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 <= -1.8e-13)
		tmp = -1.0 / sin(B);
	elseif (F <= 0.00145)
		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, -1.8e-13], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.00145], 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 < -1.7999999999999999e-13

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.7

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -1.7999999999999999e-13 < F < 0.00145

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 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 55.4

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{x}{\sin \color{blue}{B}} \]
   6. Step-by-step derivation

   7. Applied rewrites 31.1%

      \[\leadsto -1 \cdot \frac{x}{\sin \color{blue}{B}} \]

   if 0.00145 < F

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 16.9

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 16.9%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 44.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{-13}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 5.9 \cdot 10^{-28}:\\ \;\;\;\;-1 \cdot \frac{1}{\frac{B \cdot \left(1 + 0.3333333333333333 \cdot {B}^{2}\right)}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.7e-13)
   (/ -1.0 (sin B))
   (if (<= F 5.9e-28)
     (* -1.0 (/ 1.0 (/ (* B (+ 1.0 (* 0.3333333333333333 (pow B 2.0)))) x)))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.7e-13) {
		tmp = -1.0 / sin(B);
	} else if (F <= 5.9e-28) {
		tmp = -1.0 * (1.0 / ((B * (1.0 + (0.3333333333333333 * pow(B, 2.0)))) / x));
	} 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 <= (-1.7d-13)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 5.9d-28) then
        tmp = (-1.0d0) * (1.0d0 / ((b * (1.0d0 + (0.3333333333333333d0 * (b ** 2.0d0)))) / x))
    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 <= -1.7e-13) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 5.9e-28) {
		tmp = -1.0 * (1.0 / ((B * (1.0 + (0.3333333333333333 * Math.pow(B, 2.0)))) / x));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.7e-13:
		tmp = -1.0 / math.sin(B)
	elif F <= 5.9e-28:
		tmp = -1.0 * (1.0 / ((B * (1.0 + (0.3333333333333333 * math.pow(B, 2.0)))) / x))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.7e-13)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 5.9e-28)
		tmp = Float64(-1.0 * Float64(1.0 / Float64(Float64(B * Float64(1.0 + Float64(0.3333333333333333 * (B ^ 2.0)))) / x)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.7e-13)
		tmp = -1.0 / sin(B);
	elseif (F <= 5.9e-28)
		tmp = -1.0 * (1.0 / ((B * (1.0 + (0.3333333333333333 * (B ^ 2.0)))) / x));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.7e-13], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.9e-28], N[(-1.0 * N[(1.0 / N[(N[(B * N[(1.0 + N[(0.3333333333333333 * N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.70000000000000008e-13

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.7

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -1.70000000000000008e-13 < F < 5.9000000000000002e-28

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 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 55.4

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      2. div-flip N/A

         \[\leadsto -1 \cdot \frac{1}{\color{blue}{\frac{\sin B}{x \cdot \cos B}}} \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{1}{\color{blue}{\frac{\sin B}{x \cdot \cos B}}} \]

      4. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{\sin B}{x \cdot \color{blue}{\cos B}}} \]

      5. *-commutative N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{\sin B}{\cos B \cdot \color{blue}{x}}} \]

      6. associate-/r* N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{\frac{\sin B}{\cos B}}{\color{blue}{x}}} \]

      7. lift-sin.f64 N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{\frac{\sin B}{\cos B}}{x}} \]

      8. lift-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{\frac{\sin B}{\cos B}}{x}} \]

      9. tan-quot N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{\tan B}{x}} \]

      10. lift-tan.f64 N/A

          \[\leadsto -1 \cdot \frac{1}{\frac{\tan B}{x}} \]

      11. lower-/.f64 55.4

          \[\leadsto -1 \cdot \frac{1}{\frac{\tan B}{\color{blue}{x}}} \]

   6. Applied rewrites 55.4%

      \[\leadsto -1 \cdot \frac{1}{\color{blue}{\frac{\tan B}{x}}} \]

   7. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{1}{\frac{B \cdot \left(1 + \frac{1}{3} \cdot {B}^{2}\right)}{x}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{B \cdot \left(1 + \frac{1}{3} \cdot {B}^{2}\right)}{x}} \]

      2. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{B \cdot \left(1 + \frac{1}{3} \cdot {B}^{2}\right)}{x}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{B \cdot \left(1 + \frac{1}{3} \cdot {B}^{2}\right)}{x}} \]

      4. lower-pow.f64 28.7

         \[\leadsto -1 \cdot \frac{1}{\frac{B \cdot \left(1 + 0.3333333333333333 \cdot {B}^{2}\right)}{x}} \]

   9. Applied rewrites 28.7%

      \[\leadsto -1 \cdot \frac{1}{\frac{B \cdot \left(1 + 0.3333333333333333 \cdot {B}^{2}\right)}{x}} \]

   if 5.9000000000000002e-28 < F

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 16.9

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 16.9%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 36.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{-13}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{1}{\frac{B \cdot \left(1 + 0.3333333333333333 \cdot {B}^{2}\right)}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.7e-13)
   (/ -1.0 (sin B))
   (* -1.0 (/ 1.0 (/ (* B (+ 1.0 (* 0.3333333333333333 (pow B 2.0)))) x)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.7e-13) {
		tmp = -1.0 / sin(B);
	} else {
		tmp = -1.0 * (1.0 / ((B * (1.0 + (0.3333333333333333 * pow(B, 2.0)))) / x));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.7d-13)) then
        tmp = (-1.0d0) / sin(b)
    else
        tmp = (-1.0d0) * (1.0d0 / ((b * (1.0d0 + (0.3333333333333333d0 * (b ** 2.0d0)))) / x))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.7e-13) {
		tmp = -1.0 / Math.sin(B);
	} else {
		tmp = -1.0 * (1.0 / ((B * (1.0 + (0.3333333333333333 * Math.pow(B, 2.0)))) / x));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.7e-13:
		tmp = -1.0 / math.sin(B)
	else:
		tmp = -1.0 * (1.0 / ((B * (1.0 + (0.3333333333333333 * math.pow(B, 2.0)))) / x))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.7e-13)
		tmp = Float64(-1.0 / sin(B));
	else
		tmp = Float64(-1.0 * Float64(1.0 / Float64(Float64(B * Float64(1.0 + Float64(0.3333333333333333 * (B ^ 2.0)))) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.7e-13)
		tmp = -1.0 / sin(B);
	else
		tmp = -1.0 * (1.0 / ((B * (1.0 + (0.3333333333333333 * (B ^ 2.0)))) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.7e-13], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(1.0 / N[(N[(B * N[(1.0 + N[(0.3333333333333333 * N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.70000000000000008e-13

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.7

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -1.70000000000000008e-13 < F

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 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 55.4

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      2. div-flip N/A

         \[\leadsto -1 \cdot \frac{1}{\color{blue}{\frac{\sin B}{x \cdot \cos B}}} \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{1}{\color{blue}{\frac{\sin B}{x \cdot \cos B}}} \]

      4. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{\sin B}{x \cdot \color{blue}{\cos B}}} \]

      5. *-commutative N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{\sin B}{\cos B \cdot \color{blue}{x}}} \]

      6. associate-/r* N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{\frac{\sin B}{\cos B}}{\color{blue}{x}}} \]

      7. lift-sin.f64 N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{\frac{\sin B}{\cos B}}{x}} \]

      8. lift-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{\frac{\sin B}{\cos B}}{x}} \]

      9. tan-quot N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{\tan B}{x}} \]

      10. lift-tan.f64 N/A

          \[\leadsto -1 \cdot \frac{1}{\frac{\tan B}{x}} \]

      11. lower-/.f64 55.4

          \[\leadsto -1 \cdot \frac{1}{\frac{\tan B}{\color{blue}{x}}} \]

   6. Applied rewrites 55.4%

      \[\leadsto -1 \cdot \frac{1}{\color{blue}{\frac{\tan B}{x}}} \]

   7. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{1}{\frac{B \cdot \left(1 + \frac{1}{3} \cdot {B}^{2}\right)}{x}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{B \cdot \left(1 + \frac{1}{3} \cdot {B}^{2}\right)}{x}} \]

      2. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{B \cdot \left(1 + \frac{1}{3} \cdot {B}^{2}\right)}{x}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{B \cdot \left(1 + \frac{1}{3} \cdot {B}^{2}\right)}{x}} \]

      4. lower-pow.f64 28.7

         \[\leadsto -1 \cdot \frac{1}{\frac{B \cdot \left(1 + 0.3333333333333333 \cdot {B}^{2}\right)}{x}} \]

   9. Applied rewrites 28.7%

      \[\leadsto -1 \cdot \frac{1}{\frac{B \cdot \left(1 + 0.3333333333333333 \cdot {B}^{2}\right)}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 30.6% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{-25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.9 \cdot 10^{-225}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x B))))
   (if (<= x -2.9e-25) t_0 (if (<= x 4.9e-225) (/ -1.0 B) t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (x <= -2.9e-25) {
		tmp = t_0;
	} else if (x <= 4.9e-225) {
		tmp = -1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -(x / b)
    if (x <= (-2.9d-25)) then
        tmp = t_0
    else if (x <= 4.9d-225) then
        tmp = (-1.0d0) / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (x <= -2.9e-25) {
		tmp = t_0;
	} else if (x <= 4.9e-225) {
		tmp = -1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -(x / B)
	tmp = 0
	if x <= -2.9e-25:
		tmp = t_0
	elif x <= 4.9e-225:
		tmp = -1.0 / B
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	tmp = 0.0
	if (x <= -2.9e-25)
		tmp = t_0;
	elseif (x <= 4.9e-225)
		tmp = Float64(-1.0 / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -(x / B);
	tmp = 0.0;
	if (x <= -2.9e-25)
		tmp = t_0;
	elseif (x <= 4.9e-225)
		tmp = -1.0 / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[x, -2.9e-25], t$95$0, If[LessEqual[x, 4.9e-225], N[(-1.0 / B), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.9000000000000001e-25 or 4.89999999999999971e-225 < x

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 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 55.4

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 28.8

         \[\leadsto -1 \cdot \frac{x}{B} \]

   7. Applied rewrites 28.8%

      \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]

   9. Applied rewrites 28.8%

      \[\leadsto \color{blue}{-\frac{x}{B}} \]

   if -2.9000000000000001e-25 < x < 4.89999999999999971e-225

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.7

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-1}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 10.8%

      \[\leadsto \frac{-1}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 29.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot {B}^{2} - 1}{B}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{1}{\frac{B \cdot \left(1 + 0.3333333333333333 \cdot {B}^{2}\right)}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.8e-13)
   (/ (- (* -0.16666666666666666 (pow B 2.0)) 1.0) B)
   (* -1.0 (/ 1.0 (/ (* B (+ 1.0 (* 0.3333333333333333 (pow B 2.0)))) x)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.8e-13) {
		tmp = ((-0.16666666666666666 * pow(B, 2.0)) - 1.0) / B;
	} else {
		tmp = -1.0 * (1.0 / ((B * (1.0 + (0.3333333333333333 * pow(B, 2.0)))) / x));
	}
	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 <= (-1.8d-13)) then
        tmp = (((-0.16666666666666666d0) * (b ** 2.0d0)) - 1.0d0) / b
    else
        tmp = (-1.0d0) * (1.0d0 / ((b * (1.0d0 + (0.3333333333333333d0 * (b ** 2.0d0)))) / x))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.8e-13) {
		tmp = ((-0.16666666666666666 * Math.pow(B, 2.0)) - 1.0) / B;
	} else {
		tmp = -1.0 * (1.0 / ((B * (1.0 + (0.3333333333333333 * Math.pow(B, 2.0)))) / x));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.8e-13:
		tmp = ((-0.16666666666666666 * math.pow(B, 2.0)) - 1.0) / B
	else:
		tmp = -1.0 * (1.0 / ((B * (1.0 + (0.3333333333333333 * math.pow(B, 2.0)))) / x))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.8e-13)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * (B ^ 2.0)) - 1.0) / B);
	else
		tmp = Float64(-1.0 * Float64(1.0 / Float64(Float64(B * Float64(1.0 + Float64(0.3333333333333333 * (B ^ 2.0)))) / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.8e-13)
		tmp = ((-0.16666666666666666 * (B ^ 2.0)) - 1.0) / B;
	else
		tmp = -1.0 * (1.0 / ((B * (1.0 + (0.3333333333333333 * (B ^ 2.0)))) / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.8e-13], N[(N[(N[(-0.16666666666666666 * N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / B), $MachinePrecision], N[(-1.0 * N[(1.0 / N[(N[(B * N[(1.0 + N[(0.3333333333333333 * N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.7999999999999999e-13

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.7

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   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 10.6

         \[\leadsto \frac{-0.16666666666666666 \cdot {B}^{2} - 1}{B} \]

   7. Applied rewrites 10.6%

      \[\leadsto \frac{-0.16666666666666666 \cdot {B}^{2} - 1}{\color{blue}{B}} \]

   if -1.7999999999999999e-13 < F

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 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 55.4

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      2. div-flip N/A

         \[\leadsto -1 \cdot \frac{1}{\color{blue}{\frac{\sin B}{x \cdot \cos B}}} \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{1}{\color{blue}{\frac{\sin B}{x \cdot \cos B}}} \]

      4. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{\sin B}{x \cdot \color{blue}{\cos B}}} \]

      5. *-commutative N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{\sin B}{\cos B \cdot \color{blue}{x}}} \]

      6. associate-/r* N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{\frac{\sin B}{\cos B}}{\color{blue}{x}}} \]

      7. lift-sin.f64 N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{\frac{\sin B}{\cos B}}{x}} \]

      8. lift-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{\frac{\sin B}{\cos B}}{x}} \]

      9. tan-quot N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{\tan B}{x}} \]

      10. lift-tan.f64 N/A

          \[\leadsto -1 \cdot \frac{1}{\frac{\tan B}{x}} \]

      11. lower-/.f64 55.4

          \[\leadsto -1 \cdot \frac{1}{\frac{\tan B}{\color{blue}{x}}} \]

   6. Applied rewrites 55.4%

      \[\leadsto -1 \cdot \frac{1}{\color{blue}{\frac{\tan B}{x}}} \]

   7. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{1}{\frac{B \cdot \left(1 + \frac{1}{3} \cdot {B}^{2}\right)}{x}} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{B \cdot \left(1 + \frac{1}{3} \cdot {B}^{2}\right)}{x}} \]

      2. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{B \cdot \left(1 + \frac{1}{3} \cdot {B}^{2}\right)}{x}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{1}{\frac{B \cdot \left(1 + \frac{1}{3} \cdot {B}^{2}\right)}{x}} \]

      4. lower-pow.f64 28.7

         \[\leadsto -1 \cdot \frac{1}{\frac{B \cdot \left(1 + 0.3333333333333333 \cdot {B}^{2}\right)}{x}} \]

   9. Applied rewrites 28.7%

      \[\leadsto -1 \cdot \frac{1}{\frac{B \cdot \left(1 + 0.3333333333333333 \cdot {B}^{2}\right)}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 29.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot {B}^{2} - 1}{B}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{x + -0.3333333333333333 \cdot \left({B}^{2} \cdot x\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.8e-13)
   (/ (- (* -0.16666666666666666 (pow B 2.0)) 1.0) B)
   (* -1.0 (/ (+ x (* -0.3333333333333333 (* (pow B 2.0) x))) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.8e-13) {
		tmp = ((-0.16666666666666666 * pow(B, 2.0)) - 1.0) / B;
	} else {
		tmp = -1.0 * ((x + (-0.3333333333333333 * (pow(B, 2.0) * 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 <= (-1.8d-13)) then
        tmp = (((-0.16666666666666666d0) * (b ** 2.0d0)) - 1.0d0) / b
    else
        tmp = (-1.0d0) * ((x + ((-0.3333333333333333d0) * ((b ** 2.0d0) * x))) / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.8e-13) {
		tmp = ((-0.16666666666666666 * Math.pow(B, 2.0)) - 1.0) / B;
	} else {
		tmp = -1.0 * ((x + (-0.3333333333333333 * (Math.pow(B, 2.0) * x))) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.8e-13:
		tmp = ((-0.16666666666666666 * math.pow(B, 2.0)) - 1.0) / B
	else:
		tmp = -1.0 * ((x + (-0.3333333333333333 * (math.pow(B, 2.0) * x))) / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.8e-13)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * (B ^ 2.0)) - 1.0) / B);
	else
		tmp = Float64(-1.0 * Float64(Float64(x + Float64(-0.3333333333333333 * Float64((B ^ 2.0) * x))) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.8e-13)
		tmp = ((-0.16666666666666666 * (B ^ 2.0)) - 1.0) / B;
	else
		tmp = -1.0 * ((x + (-0.3333333333333333 * ((B ^ 2.0) * x))) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.8e-13], N[(N[(N[(-0.16666666666666666 * N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / B), $MachinePrecision], N[(-1.0 * N[(N[(x + N[(-0.3333333333333333 * N[(N[Power[B, 2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.7999999999999999e-13

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.7

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   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 10.6

         \[\leadsto \frac{-0.16666666666666666 \cdot {B}^{2} - 1}{B} \]

   7. Applied rewrites 10.6%

      \[\leadsto \frac{-0.16666666666666666 \cdot {B}^{2} - 1}{\color{blue}{B}} \]

   if -1.7999999999999999e-13 < F

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 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 55.4

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      2. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      3. associate-/l* N/A

         \[\leadsto -1 \cdot \left(x \cdot \color{blue}{\frac{\cos B}{\sin B}}\right) \]

      4. div-flip-rev N/A

         \[\leadsto -1 \cdot \left(x \cdot \frac{1}{\color{blue}{\frac{\sin B}{\cos B}}}\right) \]

      5. lift-sin.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos \color{blue}{B}}}\right) \]

      6. lift-cos.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \frac{1}{\frac{\sin B}{\cos B}}\right) \]

      7. tan-quot N/A

         \[\leadsto -1 \cdot \left(x \cdot \frac{1}{\tan B}\right) \]

      8. lift-tan.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \frac{1}{\tan B}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto -1 \cdot \left(x \cdot \frac{1}{\color{blue}{\tan B}}\right) \]

      10. *-commutative N/A

          \[\leadsto -1 \cdot \left(\frac{1}{\tan B} \cdot \color{blue}{x}\right) \]

      11. lower-*.f64 55.3

          \[\leadsto -1 \cdot \left(\frac{1}{\tan B} \cdot \color{blue}{x}\right) \]

   6. Applied rewrites 55.3%

      \[\leadsto -1 \cdot \left(\frac{1}{\tan B} \cdot \color{blue}{x}\right) \]

   7. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{x + \frac{-1}{3} \cdot \left({B}^{2} \cdot x\right)}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x + \frac{-1}{3} \cdot \left({B}^{2} \cdot x\right)}{B} \]

      2. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{x + \frac{-1}{3} \cdot \left({B}^{2} \cdot x\right)}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x + \frac{-1}{3} \cdot \left({B}^{2} \cdot x\right)}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x + \frac{-1}{3} \cdot \left({B}^{2} \cdot x\right)}{B} \]

      5. lower-pow.f64 28.7

         \[\leadsto -1 \cdot \frac{x + -0.3333333333333333 \cdot \left({B}^{2} \cdot x\right)}{B} \]

   9. Applied rewrites 28.7%

      \[\leadsto -1 \cdot \frac{x + -0.3333333333333333 \cdot \left({B}^{2} \cdot x\right)}{\color{blue}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 29.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.8 \cdot 10^{-13}:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot {B}^{2} - 1}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.8e-13)
   (/ (- (* -0.16666666666666666 (pow B 2.0)) 1.0) B)
   (- (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.8e-13) {
		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 <= (-1.8d-13)) 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 <= -1.8e-13) {
		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 <= -1.8e-13:
		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 <= -1.8e-13)
		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 <= -1.8e-13)
		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, -1.8e-13], 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 < -1.7999999999999999e-13

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.7

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   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 10.6

         \[\leadsto \frac{-0.16666666666666666 \cdot {B}^{2} - 1}{B} \]

   7. Applied rewrites 10.6%

      \[\leadsto \frac{-0.16666666666666666 \cdot {B}^{2} - 1}{\color{blue}{B}} \]

   if -1.7999999999999999e-13 < F

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 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 55.4

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 28.8

         \[\leadsto -1 \cdot \frac{x}{B} \]

   7. Applied rewrites 28.8%

      \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]

   9. Applied rewrites 28.8%

      \[\leadsto \color{blue}{-\frac{x}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 10.8% accurate, 26.5× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 76.2%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in F around -inf

   \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

   2. lower-sin.f64 17.7

      \[\leadsto \frac{-1}{\sin B} \]

4. Applied rewrites 17.7%

   \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

5. Taylor expanded in B around 0

   \[\leadsto \frac{-1}{B} \]
6. Step-by-step derivation

7. Applied rewrites 10.8%

   \[\leadsto \frac{-1}{B} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(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))))))