VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.3% → 99.7%

Time: 7.2s

Alternatives: 23

Speedup: 1.3×

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: 76.3% 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: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(-1, t\_0, t\_1\right)\\ \mathbf{elif}\;F \leq 5000000:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, F, \cos B \cdot \left(-x\right)\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, t\_0, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ (- x) (tan B))))
   (if (<= F -5e+39)
     (fma -1.0 t_0 t_1)
     (if (<= F 5000000.0)
       (/
        (fma (pow (fma 2.0 x (fma F F 2.0)) -0.5) F (* (cos B) (- x)))
        (sin B))
       (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 <= -5e+39) {
		tmp = fma(-1.0, t_0, t_1);
	} else if (F <= 5000000.0) {
		tmp = fma(pow(fma(2.0, x, fma(F, F, 2.0)), -0.5), F, (cos(B) * -x)) / sin(B);
	} 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 <= -5e+39)
		tmp = fma(-1.0, t_0, t_1);
	elseif (F <= 5000000.0)
		tmp = Float64(fma((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5), F, Float64(cos(B) * Float64(-x))) / sin(B));
	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, -5e+39], N[(-1.0 * t$95$0 + t$95$1), $MachinePrecision], If[LessEqual[F, 5000000.0], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F + N[(N[Cos[B], $MachinePrecision] * (-x)), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(1.0 * t$95$0 + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.00000000000000015e39

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 55.2%

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

   if -5.00000000000000015e39 < F < 5e6

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. sqr-pow N/A

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

      10. pow2 N/A

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

      11. lower-pow.f64 N/A

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

      12. lower-pow.f64 N/A

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

      13. lift-+.f64 N/A

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

      14. lift-+.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. lift-fma.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-fma.f64 N/A

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

      21. metadata-eval 84.6

          \[\leadsto \mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\color{blue}{-0.25}}\right)}^{2}, \frac{-x}{\tan B}\right) \]

   5. Applied rewrites 84.6%

      \[\leadsto \mathsf{fma}\left(F, \frac{1}{\sin B} \cdot \color{blue}{{\left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.25}\right)}^{2}}, \frac{-x}{\tan B}\right) \]

   7. Applied rewrites 76.4%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*r/ N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. div-add-rev N/A

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

   9. Applied rewrites 84.7%

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

   if 5e6 < F

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 55.2%

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

Alternative 2: 99.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\mathsf{fma}\left(-1, t\_0, t\_1\right)\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(2 + 2 \cdot x\right)}^{-0.5}}{\sin B}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, t\_0, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ (- x) (tan B))))
   (if (<= F -1.42)
     (fma -1.0 t_0 t_1)
     (if (<= F 1.4)
       (fma F (/ (pow (+ 2.0 (* 2.0 x)) -0.5) (sin B)) t_1)
       (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 <= -1.42) {
		tmp = fma(-1.0, t_0, t_1);
	} else if (F <= 1.4) {
		tmp = fma(F, (pow((2.0 + (2.0 * x)), -0.5) / sin(B)), t_1);
	} 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 <= -1.42)
		tmp = fma(-1.0, t_0, t_1);
	elseif (F <= 1.4)
		tmp = fma(F, Float64((Float64(2.0 + Float64(2.0 * x)) ^ -0.5) / sin(B)), t_1);
	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, -1.42], N[(-1.0 * t$95$0 + t$95$1), $MachinePrecision], If[LessEqual[F, 1.4], N[(F * N[(N[Power[N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(1.0 * t$95$0 + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4199999999999999

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 55.2%

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

   if -1.4199999999999999 < F < 1.3999999999999999

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.6%

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

   4. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-sin.f64 55.5

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

   6. Applied rewrites 55.5%

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

   if 1.3999999999999999 < F

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 55.2%

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

Alternative 3: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -1.35:\\ \;\;\;\;\mathsf{fma}\left(-1, t\_0, t\_1\right)\\ \mathbf{elif}\;F \leq 1.35:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {2}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, t\_0, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ (- x) (tan B))))
   (if (<= F -1.35)
     (fma -1.0 t_0 t_1)
     (if (<= F 1.35)
       (+ (- (* x (/ 1.0 (tan B)))) (* (/ F (sin B)) (pow 2.0 -0.5)))
       (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 <= -1.35) {
		tmp = fma(-1.0, t_0, t_1);
	} else if (F <= 1.35) {
		tmp = -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow(2.0, -0.5));
	} 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 <= -1.35)
		tmp = fma(-1.0, t_0, t_1);
	elseif (F <= 1.35)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / sin(B)) * (2.0 ^ -0.5)));
	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, -1.35], N[(-1.0 * t$95$0 + t$95$1), $MachinePrecision], If[LessEqual[F, 1.35], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 * t$95$0 + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3500000000000001

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 55.2%

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

   if -1.3500000000000001 < F < 1.3500000000000001

   1. Initial program 76.3%

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

   2. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval 54.1

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

   4. Applied rewrites 54.1%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval 56.4

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

   7. Applied rewrites 56.4%

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

   if 1.3500000000000001 < F

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 55.2%

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

Alternative 4: 91.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}\\ t_1 := \frac{1}{\sin B}\\ t_2 := -\frac{x}{B}\\ t_3 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -86000000000000:\\ \;\;\;\;\mathsf{fma}\left(-1, t\_1, t\_3\right)\\ \mathbf{elif}\;F \leq -1 \cdot 10^{-168}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{t\_0}{\sin B}, t\_2\right)\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-169}:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4500000:\\ \;\;\;\;\mathsf{fma}\left(t\_1, t\_0 \cdot F, t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1, t\_1, t\_3\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (fma 2.0 x (fma F F 2.0)) -0.5))
        (t_1 (/ 1.0 (sin B)))
        (t_2 (- (/ x B)))
        (t_3 (/ (- x) (tan B))))
   (if (<= F -86000000000000.0)
     (fma -1.0 t_1 t_3)
     (if (<= F -1e-168)
       (fma F (/ t_0 (sin B)) t_2)
       (if (<= F 1.65e-169)
         (- (* (pow (fma 2.0 x 2.0) -0.5) (/ F B)) (/ x (tan B)))
         (if (<= F 4500000.0) (fma t_1 (* t_0 F) t_2) (fma 1.0 t_1 t_3)))))))

C

double code(double F, double B, double x) {
	double t_0 = pow(fma(2.0, x, fma(F, F, 2.0)), -0.5);
	double t_1 = 1.0 / sin(B);
	double t_2 = -(x / B);
	double t_3 = -x / tan(B);
	double tmp;
	if (F <= -86000000000000.0) {
		tmp = fma(-1.0, t_1, t_3);
	} else if (F <= -1e-168) {
		tmp = fma(F, (t_0 / sin(B)), t_2);
	} else if (F <= 1.65e-169) {
		tmp = (pow(fma(2.0, x, 2.0), -0.5) * (F / B)) - (x / tan(B));
	} else if (F <= 4500000.0) {
		tmp = fma(t_1, (t_0 * F), t_2);
	} else {
		tmp = fma(1.0, t_1, t_3);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(2.0, x, fma(F, F, 2.0)) ^ -0.5
	t_1 = Float64(1.0 / sin(B))
	t_2 = Float64(-Float64(x / B))
	t_3 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -86000000000000.0)
		tmp = fma(-1.0, t_1, t_3);
	elseif (F <= -1e-168)
		tmp = fma(F, Float64(t_0 / sin(B)), t_2);
	elseif (F <= 1.65e-169)
		tmp = Float64(Float64((fma(2.0, x, 2.0) ^ -0.5) * Float64(F / B)) - Float64(x / tan(B)));
	elseif (F <= 4500000.0)
		tmp = fma(t_1, Float64(t_0 * F), t_2);
	else
		tmp = fma(1.0, t_1, t_3);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-N[(x / B), $MachinePrecision])}, Block[{t$95$3 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -86000000000000.0], N[(-1.0 * t$95$1 + t$95$3), $MachinePrecision], If[LessEqual[F, -1e-168], N[(F * N[(t$95$0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[F, 1.65e-169], N[(N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4500000.0], N[(t$95$1 * N[(t$95$0 * F), $MachinePrecision] + t$95$2), $MachinePrecision], N[(1.0 * t$95$1 + t$95$3), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -8.6e13

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 55.2%

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

   if -8.6e13 < F < -1e-168

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   6. Applied rewrites 58.3%

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

   if -1e-168 < F < 1.65000000000000013e-169

   1. Initial program 76.3%

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

   2. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval 54.1

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

   4. Applied rewrites 54.1%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 48.1

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

   7. Applied rewrites 48.1%

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

      1. metadata-eval 48.1

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

      2. metadata-eval 48.1

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. sub-flip-reverse N/A

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

   9. Applied rewrites 48.2%

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

   if 1.65000000000000013e-169 < F < 4.5e6

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   6. Applied rewrites 58.3%

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

   if 4.5e6 < F

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 55.2%

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

Alternative 5: 84.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}\\ t_1 := -\frac{x}{B}\\ \mathbf{if}\;F \leq -86000000000000:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{1}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq -1 \cdot 10^{-168}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{t\_0}{\sin B}, t\_1\right)\\ \mathbf{elif}\;F \leq 2.12 \cdot 10^{-165}:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{+140}:\\ \;\;\;\;t\_1 + \frac{1}{\frac{\sin B}{t\_0 \cdot F}}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{+147}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot {2}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{1}{F}}{\sin B}, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (fma 2.0 x (fma F F 2.0)) -0.5)) (t_1 (- (/ x B))))
   (if (<= F -86000000000000.0)
     (fma -1.0 (/ 1.0 (sin B)) (/ (- x) (tan B)))
     (if (<= F -1e-168)
       (fma F (/ t_0 (sin B)) t_1)
       (if (<= F 2.12e-165)
         (- (* (pow (fma 2.0 x 2.0) -0.5) (/ F B)) (/ x (tan B)))
         (if (<= F 3.4e+140)
           (+ t_1 (/ 1.0 (/ (sin B) (* t_0 F))))
           (if (<= F 1.75e+147)
             (+ (- (* x (/ 1.0 (tan B)))) (* (/ F B) (pow 2.0 -0.5)))
             (fma F (/ (/ 1.0 F) (sin B)) t_1))))))))

C

double code(double F, double B, double x) {
	double t_0 = pow(fma(2.0, x, fma(F, F, 2.0)), -0.5);
	double t_1 = -(x / B);
	double tmp;
	if (F <= -86000000000000.0) {
		tmp = fma(-1.0, (1.0 / sin(B)), (-x / tan(B)));
	} else if (F <= -1e-168) {
		tmp = fma(F, (t_0 / sin(B)), t_1);
	} else if (F <= 2.12e-165) {
		tmp = (pow(fma(2.0, x, 2.0), -0.5) * (F / B)) - (x / tan(B));
	} else if (F <= 3.4e+140) {
		tmp = t_1 + (1.0 / (sin(B) / (t_0 * F)));
	} else if (F <= 1.75e+147) {
		tmp = -(x * (1.0 / tan(B))) + ((F / B) * pow(2.0, -0.5));
	} else {
		tmp = fma(F, ((1.0 / F) / sin(B)), t_1);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(2.0, x, fma(F, F, 2.0)) ^ -0.5
	t_1 = Float64(-Float64(x / B))
	tmp = 0.0
	if (F <= -86000000000000.0)
		tmp = fma(-1.0, Float64(1.0 / sin(B)), Float64(Float64(-x) / tan(B)));
	elseif (F <= -1e-168)
		tmp = fma(F, Float64(t_0 / sin(B)), t_1);
	elseif (F <= 2.12e-165)
		tmp = Float64(Float64((fma(2.0, x, 2.0) ^ -0.5) * Float64(F / B)) - Float64(x / tan(B)));
	elseif (F <= 3.4e+140)
		tmp = Float64(t_1 + Float64(1.0 / Float64(sin(B) / Float64(t_0 * F))));
	elseif (F <= 1.75e+147)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / B) * (2.0 ^ -0.5)));
	else
		tmp = fma(F, Float64(Float64(1.0 / F) / sin(B)), t_1);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$1 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[F, -86000000000000.0], N[(-1.0 * N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1e-168], N[(F * N[(t$95$0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[F, 2.12e-165], N[(N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.4e+140], N[(t$95$1 + N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.75e+147], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / B), $MachinePrecision] * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -8.6e13

   1. Initial program 76.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 55.2%

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

   if -8.6e13 < F < -1e-168

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   6. Applied rewrites 58.3%

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

   if -1e-168 < F < 2.11999999999999998e-165

   1. Initial program 76.3%

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

   2. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval 54.1

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

   4. Applied rewrites 54.1%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 48.1

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

   7. Applied rewrites 48.1%

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

      1. metadata-eval 48.1

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

      2. metadata-eval 48.1

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. sub-flip-reverse N/A

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

   9. Applied rewrites 48.2%

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

   if 2.11999999999999998e-165 < F < 3.4e140

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. div-flip N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 58.3

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

   6. Applied rewrites 58.4%

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

   if 3.4e140 < F < 1.74999999999999987e147

   1. Initial program 76.3%

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

   2. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval 54.1

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

   4. Applied rewrites 54.1%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 48.1

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

   7. Applied rewrites 48.1%

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

   8. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval 53.4

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

   10. Applied rewrites 53.4%

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

   if 1.74999999999999987e147 < F

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 28.0

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

   7. Applied rewrites 28.0%

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

      1. metadata-eval 28.0

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

      2. metadata-eval 28.0

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

   9. Applied rewrites 32.4%

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

Alternative 6: 76.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ t_1 := \frac{F}{\sin B}\\ t_2 := \left(-x \cdot \frac{1}{\tan B}\right) + t\_1 \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_2 \leq 40:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot t\_1 - \frac{x}{B}\\ \mathbf{elif}\;t\_2 \leq 10^{+305}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{1}{F}}{\sin B}, -\frac{x}{B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) (/ F B)) (/ x (tan B))))
        (t_1 (/ F (sin B)))
        (t_2
         (+
          (- (* x (/ 1.0 (tan B))))
          (* t_1 (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))))
   (if (<= t_2 -1e+16)
     t_0
     (if (<= t_2 40.0)
       (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) t_1) (/ x B))
       (if (<= t_2 1e+305) t_0 (fma F (/ (/ 1.0 F) (sin B)) (- (/ x B))))))))

C

double code(double F, double B, double x) {
	double t_0 = (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * (F / B)) - (x / tan(B));
	double t_1 = F / sin(B);
	double t_2 = -(x * (1.0 / tan(B))) + (t_1 * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	double tmp;
	if (t_2 <= -1e+16) {
		tmp = t_0;
	} else if (t_2 <= 40.0) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * t_1) - (x / B);
	} else if (t_2 <= 1e+305) {
		tmp = t_0;
	} else {
		tmp = fma(F, ((1.0 / F) / sin(B)), -(x / B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / tan(B)))
	t_1 = Float64(F / sin(B))
	t_2 = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(t_1 * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))))
	tmp = 0.0
	if (t_2 <= -1e+16)
		tmp = t_0;
	elseif (t_2 <= 40.0)
		tmp = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * t_1) - Float64(x / B));
	elseif (t_2 <= 1e+305)
		tmp = t_0;
	else
		tmp = fma(F, Float64(Float64(1.0 / F) / sin(B)), Float64(-Float64(x / B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(t$95$1 * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+16], t$95$0, If[LessEqual[t$95$2, 40.0], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * t$95$1), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+305], t$95$0, N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + (-N[(x / B), $MachinePrecision])), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < -1e16 or 40 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 9.9999999999999994e304

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. distribute-frac-neg N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower--.f64 N/A

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

   6. Applied rewrites 61.5%

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

   if -1e16 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 40

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 50.1

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

   6. Applied rewrites 50.1%

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

   if 9.9999999999999994e304 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))))))

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 28.0

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

   7. Applied rewrites 28.0%

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

      1. metadata-eval 28.0

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

      2. metadata-eval 28.0

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

   9. Applied rewrites 32.4%

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

Alternative 7: 76.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{B}\\ t_1 := \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, t\_0\right)\\ \mathbf{if}\;F \leq -1.75 \cdot 10^{+165}:\\ \;\;\;\;-\mathsf{fma}\left(F, \frac{\frac{-1}{F \cdot \sin B}}{t\_0}, 1\right) \cdot \frac{x}{B}\\ \mathbf{elif}\;F \leq -1 \cdot 10^{-168}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 2.12 \cdot 10^{-165}:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{+140}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{+147}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot {2}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{1}{F}}{\sin B}, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x B)))
        (t_1 (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B)) t_0)))
   (if (<= F -1.75e+165)
     (- (* (fma F (/ (/ -1.0 (* F (sin B))) t_0) 1.0) (/ x B)))
     (if (<= F -1e-168)
       t_1
       (if (<= F 2.12e-165)
         (- (* (pow (fma 2.0 x 2.0) -0.5) (/ F B)) (/ x (tan B)))
         (if (<= F 3.4e+140)
           t_1
           (if (<= F 1.75e+147)
             (+ (- (* x (/ 1.0 (tan B)))) (* (/ F B) (pow 2.0 -0.5)))
             (fma F (/ (/ 1.0 F) (sin B)) t_0))))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double t_1 = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), t_0);
	double tmp;
	if (F <= -1.75e+165) {
		tmp = -(fma(F, ((-1.0 / (F * sin(B))) / t_0), 1.0) * (x / B));
	} else if (F <= -1e-168) {
		tmp = t_1;
	} else if (F <= 2.12e-165) {
		tmp = (pow(fma(2.0, x, 2.0), -0.5) * (F / B)) - (x / tan(B));
	} else if (F <= 3.4e+140) {
		tmp = t_1;
	} else if (F <= 1.75e+147) {
		tmp = -(x * (1.0 / tan(B))) + ((F / B) * pow(2.0, -0.5));
	} else {
		tmp = fma(F, ((1.0 / F) / sin(B)), t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	t_1 = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), t_0)
	tmp = 0.0
	if (F <= -1.75e+165)
		tmp = Float64(-Float64(fma(F, Float64(Float64(-1.0 / Float64(F * sin(B))) / t_0), 1.0) * Float64(x / B)));
	elseif (F <= -1e-168)
		tmp = t_1;
	elseif (F <= 2.12e-165)
		tmp = Float64(Float64((fma(2.0, x, 2.0) ^ -0.5) * Float64(F / B)) - Float64(x / tan(B)));
	elseif (F <= 3.4e+140)
		tmp = t_1;
	elseif (F <= 1.75e+147)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / B) * (2.0 ^ -0.5)));
	else
		tmp = fma(F, Float64(Float64(1.0 / F) / sin(B)), t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, Block[{t$95$1 = 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]}, If[LessEqual[F, -1.75e+165], (-N[(N[(F * N[(N[(-1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x / B), $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, -1e-168], t$95$1, If[LessEqual[F, 2.12e-165], N[(N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.4e+140], t$95$1, If[LessEqual[F, 1.75e+147], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / B), $MachinePrecision] * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.74999999999999998e165

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   6. Applied rewrites 49.7%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 28.5

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

   9. Applied rewrites 28.5%

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

   if -1.74999999999999998e165 < F < -1e-168 or 2.11999999999999998e-165 < F < 3.4e140

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   6. Applied rewrites 58.3%

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

   if -1e-168 < F < 2.11999999999999998e-165

   1. Initial program 76.3%

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

   2. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval 54.1

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

   4. Applied rewrites 54.1%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 48.1

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

   7. Applied rewrites 48.1%

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

      1. metadata-eval 48.1

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

      2. metadata-eval 48.1

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. sub-flip-reverse N/A

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

   9. Applied rewrites 48.2%

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

   if 3.4e140 < F < 1.74999999999999987e147

   1. Initial program 76.3%

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

   2. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval 54.1

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

   4. Applied rewrites 54.1%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 48.1

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

   7. Applied rewrites 48.1%

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

   8. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval 53.4

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

   10. Applied rewrites 53.4%

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

   if 1.74999999999999987e147 < F

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 28.0

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

   7. Applied rewrites 28.0%

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

      1. metadata-eval 28.0

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

      2. metadata-eval 28.0

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

   9. Applied rewrites 32.4%

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

Alternative 8: 76.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x \cdot \cos B}{\sin B}\\ \mathbf{if}\;x \leq -8:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-26}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, -\frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ (* x (cos B)) (sin B)))))
   (if (<= x -8.0)
     t_0
     (if (<= x 6.5e-26)
       (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B)) (- (/ x B)))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -((x * cos(B)) / sin(B));
	double tmp;
	if (x <= -8.0) {
		tmp = t_0;
	} else if (x <= 6.5e-26) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), -(x / B));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(Float64(x * cos(B)) / sin(B)))
	tmp = 0.0
	if (x <= -8.0)
		tmp = t_0;
	elseif (x <= 6.5e-26)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), Float64(-Float64(x / B)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision])}, If[LessEqual[x, -8.0], t$95$0, If[LessEqual[x, 6.5e-26], 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[(x / B), $MachinePrecision])), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -8 or 6.5e-26 < x

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   6. Applied rewrites 49.7%

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

   7. Taylor expanded in F around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-sin.f64 54.9

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

   9. Applied rewrites 54.9%

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

   if -8 < x < 6.5e-26

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   6. Applied rewrites 58.3%

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

Alternative 9: 75.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B}\\ t_1 := -\frac{x}{B}\\ t_2 := {\left(2 + 2 \cdot x\right)}^{-0.5}\\ \mathbf{if}\;F \leq -1.5 \cdot 10^{+203}:\\ \;\;\;\;-\mathsf{fma}\left(F, \frac{\frac{-1}{F \cdot \sin B}}{t\_1}, 1\right) \cdot \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.42:\\ \;\;\;\;t\_1 + t\_0 \cdot \frac{-1}{F}\\ \mathbf{elif}\;F \leq -1 \cdot 10^{-168}:\\ \;\;\;\;t\_1 + t\_0 \cdot t\_2\\ \mathbf{elif}\;F \leq 2.12 \cdot 10^{-165}:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;t\_1 + \frac{F \cdot t\_2}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{1}{F}}{\sin B}, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ F (sin B)))
        (t_1 (- (/ x B)))
        (t_2 (pow (+ 2.0 (* 2.0 x)) -0.5)))
   (if (<= F -1.5e+203)
     (- (* (fma F (/ (/ -1.0 (* F (sin B))) t_1) 1.0) (/ x B)))
     (if (<= F -1.42)
       (+ t_1 (* t_0 (/ -1.0 F)))
       (if (<= F -1e-168)
         (+ t_1 (* t_0 t_2))
         (if (<= F 2.12e-165)
           (- (* (pow (fma 2.0 x 2.0) -0.5) (/ F B)) (/ x (tan B)))
           (if (<= F 1.4)
             (+ t_1 (/ (* F t_2) (sin B)))
             (fma F (/ (/ 1.0 F) (sin B)) t_1))))))))

C

double code(double F, double B, double x) {
	double t_0 = F / sin(B);
	double t_1 = -(x / B);
	double t_2 = pow((2.0 + (2.0 * x)), -0.5);
	double tmp;
	if (F <= -1.5e+203) {
		tmp = -(fma(F, ((-1.0 / (F * sin(B))) / t_1), 1.0) * (x / B));
	} else if (F <= -1.42) {
		tmp = t_1 + (t_0 * (-1.0 / F));
	} else if (F <= -1e-168) {
		tmp = t_1 + (t_0 * t_2);
	} else if (F <= 2.12e-165) {
		tmp = (pow(fma(2.0, x, 2.0), -0.5) * (F / B)) - (x / tan(B));
	} else if (F <= 1.4) {
		tmp = t_1 + ((F * t_2) / sin(B));
	} else {
		tmp = fma(F, ((1.0 / F) / sin(B)), t_1);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(F / sin(B))
	t_1 = Float64(-Float64(x / B))
	t_2 = Float64(2.0 + Float64(2.0 * x)) ^ -0.5
	tmp = 0.0
	if (F <= -1.5e+203)
		tmp = Float64(-Float64(fma(F, Float64(Float64(-1.0 / Float64(F * sin(B))) / t_1), 1.0) * Float64(x / B)));
	elseif (F <= -1.42)
		tmp = Float64(t_1 + Float64(t_0 * Float64(-1.0 / F)));
	elseif (F <= -1e-168)
		tmp = Float64(t_1 + Float64(t_0 * t_2));
	elseif (F <= 2.12e-165)
		tmp = Float64(Float64((fma(2.0, x, 2.0) ^ -0.5) * Float64(F / B)) - Float64(x / tan(B)));
	elseif (F <= 1.4)
		tmp = Float64(t_1 + Float64(Float64(F * t_2) / sin(B)));
	else
		tmp = fma(F, Float64(Float64(1.0 / F) / sin(B)), t_1);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(x / B), $MachinePrecision])}, Block[{t$95$2 = N[Power[N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[F, -1.5e+203], (-N[(N[(F * N[(N[(-1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x / B), $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, -1.42], N[(t$95$1 + N[(t$95$0 * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1e-168], N[(t$95$1 + N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.12e-165], N[(N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4], N[(t$95$1 + N[(N[(F * t$95$2), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -1.5e203

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   6. Applied rewrites 49.7%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 28.5

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

   9. Applied rewrites 28.5%

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

   if -1.5e203 < F < -1.4199999999999999

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 28.5

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

   7. Applied rewrites 28.5%

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

   if -1.4199999999999999 < F < -1e-168

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval 35.8

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

   7. Applied rewrites 35.8%

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

   if -1e-168 < F < 2.11999999999999998e-165

   1. Initial program 76.3%

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

   2. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval 54.1

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

   4. Applied rewrites 54.1%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 48.1

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

   7. Applied rewrites 48.1%

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

      1. metadata-eval 48.1

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

      2. metadata-eval 48.1

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. sub-flip-reverse N/A

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

   9. Applied rewrites 48.2%

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

   if 2.11999999999999998e-165 < F < 1.3999999999999999

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-sin.f64 36.4

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

   7. Applied rewrites 36.4%

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

   if 1.3999999999999999 < F

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 28.0

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

   7. Applied rewrites 28.0%

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

      1. metadata-eval 28.0

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

      2. metadata-eval 28.0

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

   9. Applied rewrites 32.4%

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

Alternative 10: 75.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{B}\\ t_1 := \frac{F}{\sin B}\\ t_2 := t\_0 + t\_1 \cdot {\left(2 + 2 \cdot x\right)}^{-0.5}\\ \mathbf{if}\;F \leq -1.5 \cdot 10^{+203}:\\ \;\;\;\;-\mathsf{fma}\left(F, \frac{\frac{-1}{F \cdot \sin B}}{t\_0}, 1\right) \cdot \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.42:\\ \;\;\;\;t\_0 + t\_1 \cdot \frac{-1}{F}\\ \mathbf{elif}\;F \leq -1 \cdot 10^{-168}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-169}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot {2}^{-0.5}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{1}{F}}{\sin B}, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x B)))
        (t_1 (/ F (sin B)))
        (t_2 (+ t_0 (* t_1 (pow (+ 2.0 (* 2.0 x)) -0.5)))))
   (if (<= F -1.5e+203)
     (- (* (fma F (/ (/ -1.0 (* F (sin B))) t_0) 1.0) (/ x B)))
     (if (<= F -1.42)
       (+ t_0 (* t_1 (/ -1.0 F)))
       (if (<= F -1e-168)
         t_2
         (if (<= F 1.65e-169)
           (+ (- (* x (/ 1.0 (tan B)))) (* (/ F B) (pow 2.0 -0.5)))
           (if (<= F 1.4) t_2 (fma F (/ (/ 1.0 F) (sin B)) t_0))))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double t_1 = F / sin(B);
	double t_2 = t_0 + (t_1 * pow((2.0 + (2.0 * x)), -0.5));
	double tmp;
	if (F <= -1.5e+203) {
		tmp = -(fma(F, ((-1.0 / (F * sin(B))) / t_0), 1.0) * (x / B));
	} else if (F <= -1.42) {
		tmp = t_0 + (t_1 * (-1.0 / F));
	} else if (F <= -1e-168) {
		tmp = t_2;
	} else if (F <= 1.65e-169) {
		tmp = -(x * (1.0 / tan(B))) + ((F / B) * pow(2.0, -0.5));
	} else if (F <= 1.4) {
		tmp = t_2;
	} else {
		tmp = fma(F, ((1.0 / F) / sin(B)), t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	t_1 = Float64(F / sin(B))
	t_2 = Float64(t_0 + Float64(t_1 * (Float64(2.0 + Float64(2.0 * x)) ^ -0.5)))
	tmp = 0.0
	if (F <= -1.5e+203)
		tmp = Float64(-Float64(fma(F, Float64(Float64(-1.0 / Float64(F * sin(B))) / t_0), 1.0) * Float64(x / B)));
	elseif (F <= -1.42)
		tmp = Float64(t_0 + Float64(t_1 * Float64(-1.0 / F)));
	elseif (F <= -1e-168)
		tmp = t_2;
	elseif (F <= 1.65e-169)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / B) * (2.0 ^ -0.5)));
	elseif (F <= 1.4)
		tmp = t_2;
	else
		tmp = fma(F, Float64(Float64(1.0 / F) / sin(B)), t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, Block[{t$95$1 = N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 + N[(t$95$1 * N[Power[N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.5e+203], (-N[(N[(F * N[(N[(-1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x / B), $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, -1.42], N[(t$95$0 + N[(t$95$1 * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1e-168], t$95$2, If[LessEqual[F, 1.65e-169], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / B), $MachinePrecision] * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4], t$95$2, N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.5e203

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   6. Applied rewrites 49.7%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 28.5

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

   9. Applied rewrites 28.5%

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

   if -1.5e203 < F < -1.4199999999999999

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 28.5

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

   7. Applied rewrites 28.5%

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

   if -1.4199999999999999 < F < -1e-168 or 1.65000000000000013e-169 < F < 1.3999999999999999

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval 35.8

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

   7. Applied rewrites 35.8%

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

   if -1e-168 < F < 1.65000000000000013e-169

   1. Initial program 76.3%

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

   2. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval 54.1

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

   4. Applied rewrites 54.1%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 48.1

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

   7. Applied rewrites 48.1%

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

   8. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval 53.4

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

   10. Applied rewrites 53.4%

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

   if 1.3999999999999999 < F

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 28.0

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

   7. Applied rewrites 28.0%

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

      1. metadata-eval 28.0

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

      2. metadata-eval 28.0

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

   9. Applied rewrites 32.4%

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

Alternative 11: 75.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{B}\\ t_1 := {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{B}\\ \mathbf{if}\;F \leq -1.75 \cdot 10^{+165}:\\ \;\;\;\;-\mathsf{fma}\left(F, \frac{\frac{-1}{F \cdot \sin B}}{t\_0}, 1\right) \cdot \frac{x}{B}\\ \mathbf{elif}\;F \leq -1 \cdot 10^{-168}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{-169}:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{+38}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{1}{F}}{\sin B}, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x B)))
        (t_1
         (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F (sin B))) (/ x B))))
   (if (<= F -1.75e+165)
     (- (* (fma F (/ (/ -1.0 (* F (sin B))) t_0) 1.0) (/ x B)))
     (if (<= F -1e-168)
       t_1
       (if (<= F 1.65e-169)
         (- (* (pow (fma 2.0 x 2.0) -0.5) (/ F B)) (/ x (tan B)))
         (if (<= F 4.5e+38) t_1 (fma F (/ (/ 1.0 F) (sin B)) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double t_1 = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / sin(B))) - (x / B);
	double tmp;
	if (F <= -1.75e+165) {
		tmp = -(fma(F, ((-1.0 / (F * sin(B))) / t_0), 1.0) * (x / B));
	} else if (F <= -1e-168) {
		tmp = t_1;
	} else if (F <= 1.65e-169) {
		tmp = (pow(fma(2.0, x, 2.0), -0.5) * (F / B)) - (x / tan(B));
	} else if (F <= 4.5e+38) {
		tmp = t_1;
	} else {
		tmp = fma(F, ((1.0 / F) / sin(B)), t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	t_1 = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / sin(B))) - Float64(x / B))
	tmp = 0.0
	if (F <= -1.75e+165)
		tmp = Float64(-Float64(fma(F, Float64(Float64(-1.0 / Float64(F * sin(B))) / t_0), 1.0) * Float64(x / B)));
	elseif (F <= -1e-168)
		tmp = t_1;
	elseif (F <= 1.65e-169)
		tmp = Float64(Float64((fma(2.0, x, 2.0) ^ -0.5) * Float64(F / B)) - Float64(x / tan(B)));
	elseif (F <= 4.5e+38)
		tmp = t_1;
	else
		tmp = fma(F, Float64(Float64(1.0 / F) / sin(B)), t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, Block[{t$95$1 = 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 / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.75e+165], (-N[(N[(F * N[(N[(-1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x / B), $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, -1e-168], t$95$1, If[LessEqual[F, 1.65e-169], N[(N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.5e+38], t$95$1, N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.74999999999999998e165

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   6. Applied rewrites 49.7%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 28.5

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

   9. Applied rewrites 28.5%

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

   if -1.74999999999999998e165 < F < -1e-168 or 1.65000000000000013e-169 < F < 4.4999999999999998e38

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 50.1

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

   6. Applied rewrites 50.1%

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

   if -1e-168 < F < 1.65000000000000013e-169

   1. Initial program 76.3%

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

   2. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval 54.1

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

   4. Applied rewrites 54.1%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 48.1

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

   7. Applied rewrites 48.1%

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

      1. metadata-eval 48.1

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

      2. metadata-eval 48.1

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. sub-flip-reverse N/A

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

   9. Applied rewrites 48.2%

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

   if 4.4999999999999998e38 < F

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 28.0

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

   7. Applied rewrites 28.0%

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

      1. metadata-eval 28.0

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

      2. metadata-eval 28.0

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

   9. Applied rewrites 32.4%

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

Alternative 12: 68.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{B}\\ t_1 := \frac{F}{\sin B}\\ \mathbf{if}\;F \leq -1.5 \cdot 10^{+203}:\\ \;\;\;\;-\mathsf{fma}\left(F, \frac{\frac{-1}{F \cdot \sin B}}{t\_0}, 1\right) \cdot \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.42:\\ \;\;\;\;t\_0 + t\_1 \cdot \frac{-1}{F}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;t\_0 + t\_1 \cdot {\left(2 + 2 \cdot x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{1}{F}}{\sin B}, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x B))) (t_1 (/ F (sin B))))
   (if (<= F -1.5e+203)
     (- (* (fma F (/ (/ -1.0 (* F (sin B))) t_0) 1.0) (/ x B)))
     (if (<= F -1.42)
       (+ t_0 (* t_1 (/ -1.0 F)))
       (if (<= F 1.4)
         (+ t_0 (* t_1 (pow (+ 2.0 (* 2.0 x)) -0.5)))
         (fma F (/ (/ 1.0 F) (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double t_1 = F / sin(B);
	double tmp;
	if (F <= -1.5e+203) {
		tmp = -(fma(F, ((-1.0 / (F * sin(B))) / t_0), 1.0) * (x / B));
	} else if (F <= -1.42) {
		tmp = t_0 + (t_1 * (-1.0 / F));
	} else if (F <= 1.4) {
		tmp = t_0 + (t_1 * pow((2.0 + (2.0 * x)), -0.5));
	} else {
		tmp = fma(F, ((1.0 / F) / sin(B)), t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	t_1 = Float64(F / sin(B))
	tmp = 0.0
	if (F <= -1.5e+203)
		tmp = Float64(-Float64(fma(F, Float64(Float64(-1.0 / Float64(F * sin(B))) / t_0), 1.0) * Float64(x / B)));
	elseif (F <= -1.42)
		tmp = Float64(t_0 + Float64(t_1 * Float64(-1.0 / F)));
	elseif (F <= 1.4)
		tmp = Float64(t_0 + Float64(t_1 * (Float64(2.0 + Float64(2.0 * x)) ^ -0.5)));
	else
		tmp = fma(F, Float64(Float64(1.0 / F) / sin(B)), t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, Block[{t$95$1 = N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.5e+203], (-N[(N[(F * N[(N[(-1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x / B), $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, -1.42], N[(t$95$0 + N[(t$95$1 * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4], N[(t$95$0 + N[(t$95$1 * N[Power[N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.5e203

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   6. Applied rewrites 49.7%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 28.5

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

   9. Applied rewrites 28.5%

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

   if -1.5e203 < F < -1.4199999999999999

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 28.5

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

   7. Applied rewrites 28.5%

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

   if -1.4199999999999999 < F < 1.3999999999999999

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval 35.8

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

   7. Applied rewrites 35.8%

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

   if 1.3999999999999999 < F

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 28.0

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

   7. Applied rewrites 28.0%

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

      1. metadata-eval 28.0

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

      2. metadata-eval 28.0

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

   9. Applied rewrites 32.4%

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

Alternative 13: 68.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;F \leq -1.5 \cdot 10^{+203}:\\ \;\;\;\;-\mathsf{fma}\left(F, \frac{\frac{-1}{F \cdot \sin B}}{t\_0}, 1\right) \cdot \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.42:\\ \;\;\;\;t\_0 + \frac{F}{\sin B} \cdot \frac{-1}{F}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;t\_0 + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{-0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{1}{F}}{\sin B}, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x B))))
   (if (<= F -1.5e+203)
     (- (* (fma F (/ (/ -1.0 (* F (sin B))) t_0) 1.0) (/ x B)))
     (if (<= F -1.42)
       (+ t_0 (* (/ F (sin B)) (/ -1.0 F)))
       (if (<= F 1.4)
         (+ t_0 (/ (* F (pow (+ 2.0 (* 2.0 x)) -0.5)) (sin B)))
         (fma F (/ (/ 1.0 F) (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (F <= -1.5e+203) {
		tmp = -(fma(F, ((-1.0 / (F * sin(B))) / t_0), 1.0) * (x / B));
	} else if (F <= -1.42) {
		tmp = t_0 + ((F / sin(B)) * (-1.0 / F));
	} else if (F <= 1.4) {
		tmp = t_0 + ((F * pow((2.0 + (2.0 * x)), -0.5)) / sin(B));
	} else {
		tmp = fma(F, ((1.0 / F) / sin(B)), t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	tmp = 0.0
	if (F <= -1.5e+203)
		tmp = Float64(-Float64(fma(F, Float64(Float64(-1.0 / Float64(F * sin(B))) / t_0), 1.0) * Float64(x / B)));
	elseif (F <= -1.42)
		tmp = Float64(t_0 + Float64(Float64(F / sin(B)) * Float64(-1.0 / F)));
	elseif (F <= 1.4)
		tmp = Float64(t_0 + Float64(Float64(F * (Float64(2.0 + Float64(2.0 * x)) ^ -0.5)) / sin(B)));
	else
		tmp = fma(F, Float64(Float64(1.0 / F) / sin(B)), t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[F, -1.5e+203], (-N[(N[(F * N[(N[(-1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x / B), $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, -1.42], N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4], N[(t$95$0 + N[(N[(F * N[Power[N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.5e203

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   6. Applied rewrites 49.7%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 28.5

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

   9. Applied rewrites 28.5%

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

   if -1.5e203 < F < -1.4199999999999999

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 28.5

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

   7. Applied rewrites 28.5%

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

   if -1.4199999999999999 < F < 1.3999999999999999

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-sin.f64 36.4

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

   7. Applied rewrites 36.4%

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

   if 1.3999999999999999 < F

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 28.0

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

   7. Applied rewrites 28.0%

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

      1. metadata-eval 28.0

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

      2. metadata-eval 28.0

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

   9. Applied rewrites 32.4%

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

Alternative 14: 61.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;F \leq -1.5 \cdot 10^{+203}:\\ \;\;\;\;-\mathsf{fma}\left(F, \frac{\frac{-1}{F \cdot \sin B}}{t\_0}, 1\right) \cdot \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.4 \cdot 10^{+16}:\\ \;\;\;\;t\_0 + \frac{F}{\sin B} \cdot \frac{-1}{F}\\ \mathbf{elif}\;F \leq 440:\\ \;\;\;\;\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{1}{F}}{\sin B}, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x B))))
   (if (<= F -1.5e+203)
     (- (* (fma F (/ (/ -1.0 (* F (sin B))) t_0) 1.0) (/ x B)))
     (if (<= F -1.4e+16)
       (+ t_0 (* (/ F (sin B)) (/ -1.0 F)))
       (if (<= F 440.0)
         (/ (- (* F (pow (+ 2.0 (fma 2.0 x (pow F 2.0))) -0.5)) x) B)
         (fma F (/ (/ 1.0 F) (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (F <= -1.5e+203) {
		tmp = -(fma(F, ((-1.0 / (F * sin(B))) / t_0), 1.0) * (x / B));
	} else if (F <= -1.4e+16) {
		tmp = t_0 + ((F / sin(B)) * (-1.0 / F));
	} else if (F <= 440.0) {
		tmp = ((F * pow((2.0 + fma(2.0, x, pow(F, 2.0))), -0.5)) - x) / B;
	} else {
		tmp = fma(F, ((1.0 / F) / sin(B)), t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	tmp = 0.0
	if (F <= -1.5e+203)
		tmp = Float64(-Float64(fma(F, Float64(Float64(-1.0 / Float64(F * sin(B))) / t_0), 1.0) * Float64(x / B)));
	elseif (F <= -1.4e+16)
		tmp = Float64(t_0 + Float64(Float64(F / sin(B)) * Float64(-1.0 / F)));
	elseif (F <= 440.0)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + fma(2.0, x, (F ^ 2.0))) ^ -0.5)) - x) / B);
	else
		tmp = fma(F, Float64(Float64(1.0 / F) / sin(B)), t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[F, -1.5e+203], (-N[(N[(F * N[(N[(-1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x / B), $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, -1.4e+16], N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 440.0], 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[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.5e203

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   6. Applied rewrites 49.7%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 28.5

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

   9. Applied rewrites 28.5%

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

   if -1.5e203 < F < -1.4e16

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 28.5

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

   7. Applied rewrites 28.5%

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

   if -1.4e16 < F < 440

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.2%

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

   if 440 < F

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 28.0

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

   7. Applied rewrites 28.0%

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

      1. metadata-eval 28.0

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

      2. metadata-eval 28.0

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

   9. Applied rewrites 32.4%

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

Alternative 15: 59.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;F \leq -1.4 \cdot 10^{+16}:\\ \;\;\;\;t\_0 + \frac{F}{\sin B} \cdot \frac{-1}{F}\\ \mathbf{elif}\;F \leq 440:\\ \;\;\;\;\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{1}{F}}{\sin B}, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x B))))
   (if (<= F -1.4e+16)
     (+ t_0 (* (/ F (sin B)) (/ -1.0 F)))
     (if (<= F 440.0)
       (/ (- (* F (pow (+ 2.0 (fma 2.0 x (pow F 2.0))) -0.5)) x) B)
       (fma F (/ (/ 1.0 F) (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (F <= -1.4e+16) {
		tmp = t_0 + ((F / sin(B)) * (-1.0 / F));
	} else if (F <= 440.0) {
		tmp = ((F * pow((2.0 + fma(2.0, x, pow(F, 2.0))), -0.5)) - x) / B;
	} else {
		tmp = fma(F, ((1.0 / F) / sin(B)), t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	tmp = 0.0
	if (F <= -1.4e+16)
		tmp = Float64(t_0 + Float64(Float64(F / sin(B)) * Float64(-1.0 / F)));
	elseif (F <= 440.0)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + fma(2.0, x, (F ^ 2.0))) ^ -0.5)) - x) / B);
	else
		tmp = fma(F, Float64(Float64(1.0 / F) / sin(B)), t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[F, -1.4e+16], N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 440.0], 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[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4e16

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 28.5

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

   7. Applied rewrites 28.5%

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

   if -1.4e16 < F < 440

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.2%

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

   if 440 < F

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 28.0

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

   7. Applied rewrites 28.0%

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

      1. metadata-eval 28.0

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

      2. metadata-eval 28.0

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

   9. Applied rewrites 32.4%

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

Alternative 16: 59.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.42 \cdot 10^{+65}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 440:\\ \;\;\;\;\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{1}{F}}{\sin B}, -\frac{x}{B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.42e+65)
   (/ -1.0 (sin B))
   (if (<= F 440.0)
     (/ (- (* F (pow (+ 2.0 (fma 2.0 x (pow F 2.0))) -0.5)) x) B)
     (fma F (/ (/ 1.0 F) (sin B)) (- (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.42e+65) {
		tmp = -1.0 / sin(B);
	} else if (F <= 440.0) {
		tmp = ((F * pow((2.0 + fma(2.0, x, pow(F, 2.0))), -0.5)) - x) / B;
	} else {
		tmp = fma(F, ((1.0 / F) / sin(B)), -(x / B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.42e+65)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 440.0)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + fma(2.0, x, (F ^ 2.0))) ^ -0.5)) - x) / B);
	else
		tmp = fma(F, Float64(Float64(1.0 / F) / sin(B)), Float64(-Float64(x / B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.42e+65], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 440.0], 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[(F * N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + (-N[(x / B), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.42000000000000012e65

   1. Initial program 76.3%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 17.8

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

   4. Applied rewrites 17.8%

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

   if -1.42000000000000012e65 < F < 440

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.2%

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

   if 440 < F

   1. Initial program 76.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 28.0

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

   7. Applied rewrites 28.0%

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

      1. metadata-eval 28.0

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

      2. metadata-eval 28.0

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. associate-*l/ N/A

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

      8. associate-/l* N/A

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

   9. Applied rewrites 32.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{\frac{1}{F}}{\sin B}, -\frac{x}{B}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 58.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;F \leq -1.42 \cdot 10^{+65}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 440:\\ \;\;\;\;t\_0 + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{1}{F}}{\sin B}, t\_0\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x B))))
   (if (<= F -1.42e+65)
     (/ -1.0 (sin B))
     (if (<= F 440.0)
       (+ t_0 (* (/ F B) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
       (fma F (/ (/ 1.0 F) (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (F <= -1.42e+65) {
		tmp = -1.0 / sin(B);
	} else if (F <= 440.0) {
		tmp = t_0 + ((F / B) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = fma(F, ((1.0 / F) / sin(B)), t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	tmp = 0.0
	if (F <= -1.42e+65)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 440.0)
		tmp = Float64(t_0 + 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)), t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[F, -1.42e+65], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 440.0], N[(t$95$0 + 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] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.42000000000000012e65

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.8

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.8%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -1.42000000000000012e65 < F < 440

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. lower-/.f64 35.9

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   7. Applied rewrites 35.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if 440 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 50.1

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 50.1%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in F around inf

      \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{F}} \]
   6. Step-by-step derivation

      1. lower-/.f64 28.0

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\color{blue}{F}} \]

   7. Applied rewrites 28.0%

      \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{F}} \]
   8. Step-by-step derivation

      1. metadata-eval 28.0

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \frac{1}{F} \]

      2. metadata-eval 28.0

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \frac{1}{F} \]

      3. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot \frac{1}{F}} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \frac{1}{F} + \left(-\frac{x}{B}\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \frac{1}{F}} + \left(-\frac{x}{B}\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot \frac{1}{F} + \left(-\frac{x}{B}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot \frac{1}{F}}{\sin B}} + \left(-\frac{x}{B}\right) \]

      8. associate-/l* N/A

         \[\leadsto \color{blue}{F \cdot \frac{\frac{1}{F}}{\sin B}} + \left(-\frac{x}{B}\right) \]

   9. Applied rewrites 32.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{\frac{1}{F}}{\sin B}, -\frac{x}{B}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 52.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.42 \cdot 10^{+65}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 45000:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.42e+65)
   (/ -1.0 (sin B))
   (if (<= F 45000.0)
     (+
      (- (/ x B))
      (* (/ F B) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.42e+65) {
		tmp = -1.0 / sin(B);
	} else if (F <= 45000.0) {
		tmp = -(x / B) + ((F / B) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.42d+65)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 45000.0d0) then
        tmp = -(x / b) + ((f / b) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.42e+65) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 45000.0) {
		tmp = -(x / B) + ((F / B) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.42e+65:
		tmp = -1.0 / math.sin(B)
	elif F <= 45000.0:
		tmp = -(x / B) + ((F / B) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.42e+65)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 45000.0)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F / B) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.42e+65)
		tmp = -1.0 / sin(B);
	elseif (F <= 45000.0)
		tmp = -(x / B) + ((F / B) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.42e+65], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 45000.0], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F / B), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.42000000000000012e65

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.8

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.8%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -1.42000000000000012e65 < F < 45000

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. lower-/.f64 35.9

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   7. Applied rewrites 35.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if 45000 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.2

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 52.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2300:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-27}:\\ \;\;\;\;\left(-x \cdot \frac{1}{B}\right) + \frac{F}{B} \cdot {\left(2 + 2 \cdot x\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2300.0)
   (/ -1.0 (sin B))
   (if (<= F 1.2e-27)
     (+ (- (* x (/ 1.0 B))) (* (/ F B) (pow (+ 2.0 (* 2.0 x)) -0.5)))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2300.0) {
		tmp = -1.0 / sin(B);
	} else if (F <= 1.2e-27) {
		tmp = -(x * (1.0 / B)) + ((F / B) * pow((2.0 + (2.0 * x)), -0.5));
	} 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 <= (-2300.0d0)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 1.2d-27) then
        tmp = -(x * (1.0d0 / b)) + ((f / b) * ((2.0d0 + (2.0d0 * x)) ** (-0.5d0)))
    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 <= -2300.0) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 1.2e-27) {
		tmp = -(x * (1.0 / B)) + ((F / B) * Math.pow((2.0 + (2.0 * x)), -0.5));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2300.0:
		tmp = -1.0 / math.sin(B)
	elif F <= 1.2e-27:
		tmp = -(x * (1.0 / B)) + ((F / B) * math.pow((2.0 + (2.0 * x)), -0.5))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2300.0)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 1.2e-27)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / B))) + Float64(Float64(F / B) * (Float64(2.0 + Float64(2.0 * x)) ^ -0.5)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2300.0)
		tmp = -1.0 / sin(B);
	elseif (F <= 1.2e-27)
		tmp = -(x * (1.0 / B)) + ((F / B) * ((2.0 + (2.0 * x)) ^ -0.5));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2300.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.2e-27], N[((-N[(x * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / B), $MachinePrecision] * N[Power[N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2300

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.8

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.8%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -2300 < F < 1.20000000000000001e-27

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{{\left(2 + 2 \cdot x\right)}^{\frac{-1}{2}}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(2 + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(2 + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      3. lower-pow.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(2 + 2 \cdot x\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      4. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(2 + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(2 + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{\color{blue}{2}}\right)\right)} \]

      6. metadata-eval N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(2 + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      7. metadata-eval 54.1

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(2 + 2 \cdot x\right)}^{-0.5} \]

   4. Applied rewrites 54.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{{\left(2 + 2 \cdot x\right)}^{-0.5}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(2 + 2 \cdot x\right)}^{\frac{-1}{2}} \]
   6. Step-by-step derivation

      1. lower-/.f64 48.1

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\color{blue}{B}} \cdot {\left(2 + 2 \cdot x\right)}^{-0.5} \]

   7. Applied rewrites 48.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(2 + 2 \cdot x\right)}^{-0.5} \]

   8. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{B}}\right) + \frac{F}{B} \cdot {\left(2 + 2 \cdot x\right)}^{\frac{-1}{2}} \]
   9. Step-by-step derivation

      1. lower-/.f64 28.7

         \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{F}{B} \cdot {\left(2 + 2 \cdot x\right)}^{-0.5} \]

   10. Applied rewrites 28.7%

       \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{B}}\right) + \frac{F}{B} \cdot {\left(2 + 2 \cdot x\right)}^{-0.5} \]

   if 1.20000000000000001e-27 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.2

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 45.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9.8 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{-47}:\\ \;\;\;\;-1 \cdot \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9.8e-18)
   (/ -1.0 (sin B))
   (if (<= F 3.9e-47) (- (* 1.0 (/ x B))) (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.8e-18) {
		tmp = -1.0 / sin(B);
	} else if (F <= 3.9e-47) {
		tmp = -(1.0 * (x / B));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-9.8d-18)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 3.9d-47) then
        tmp = -(1.0d0 * (x / b))
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.8e-18) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 3.9e-47) {
		tmp = -(1.0 * (x / B));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -9.8e-18:
		tmp = -1.0 / math.sin(B)
	elif F <= 3.9e-47:
		tmp = -(1.0 * (x / B))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9.8e-18)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 3.9e-47)
		tmp = Float64(-Float64(1.0 * Float64(x / B)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -9.8e-18)
		tmp = -1.0 / sin(B);
	elseif (F <= 3.9e-47)
		tmp = -(1.0 * (x / B));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9.8e-18], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.9e-47], (-N[(1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.8000000000000002e-18

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in 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 17.8

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.8%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -9.8000000000000002e-18 < F < 3.89999999999999978e-47

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 50.1

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 50.1%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 49.7%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(F, \frac{\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}}{-\frac{x}{B}}, 1\right) \cdot \frac{x}{B}} \]

   7. Taylor expanded in F around 0

      \[\leadsto -\color{blue}{1} \cdot \frac{x}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 29.4%

      \[\leadsto -\color{blue}{1} \cdot \frac{x}{B} \]

   if 3.89999999999999978e-47 < F

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.2

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 32.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 4.7 \cdot 10^{-29}:\\ \;\;\;\;-1 \cdot \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 4.7e-29) (- (* 1.0 (/ x B))) (/ -1.0 (sin B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 4.7e-29) {
		tmp = -(1.0 * (x / B));
	} else {
		tmp = -1.0 / sin(B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 4.7d-29) then
        tmp = -(1.0d0 * (x / b))
    else
        tmp = (-1.0d0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 4.7e-29) {
		tmp = -(1.0 * (x / B));
	} else {
		tmp = -1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 4.7e-29:
		tmp = -(1.0 * (x / B))
	else:
		tmp = -1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 4.7e-29)
		tmp = Float64(-Float64(1.0 * Float64(x / B)));
	else
		tmp = Float64(-1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 4.7e-29)
		tmp = -(1.0 * (x / B));
	else
		tmp = -1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 4.7e-29], (-N[(1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 4.6999999999999998e-29

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 50.1

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 50.1%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   6. Applied rewrites 49.7%

      \[\leadsto \color{blue}{-\mathsf{fma}\left(F, \frac{\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}}{-\frac{x}{B}}, 1\right) \cdot \frac{x}{B}} \]

   7. Taylor expanded in F around 0

      \[\leadsto -\color{blue}{1} \cdot \frac{x}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 29.4%

      \[\leadsto -\color{blue}{1} \cdot \frac{x}{B} \]

   if 4.6999999999999998e-29 < B

   1. Initial program 76.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.8

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.8%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 29.4% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ -1 \cdot \frac{x}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (- (* 1.0 (/ x B))))

C

double code(double F, double B, double x) {
	return -(1.0 * (x / B));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -(1.0d0 * (x / b))
end function

Java

public static double code(double F, double B, double x) {
	return -(1.0 * (x / B));
}

Python

def code(F, B, x):
	return -(1.0 * (x / B))

Julia

function code(F, B, x)
	return Float64(-Float64(1.0 * Float64(x / B)))
end

MATLAB

function tmp = code(F, B, x)
	tmp = -(1.0 * (x / B));
end

Wolfram

code[F_, B_, x_] := (-N[(1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 76.3%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in B around 0

   \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
3. Step-by-step derivation

   1. lower-/.f64 50.1

      \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

4. Applied rewrites 50.1%

   \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

6. Applied rewrites 49.7%

   \[\leadsto \color{blue}{-\mathsf{fma}\left(F, \frac{\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}}{-\frac{x}{B}}, 1\right) \cdot \frac{x}{B}} \]

7. Taylor expanded in F around 0

   \[\leadsto -\color{blue}{1} \cdot \frac{x}{B} \]
8. Step-by-step derivation

9. Applied rewrites 29.4%

   \[\leadsto -\color{blue}{1} \cdot \frac{x}{B} \]
10. Add Preprocessing

Alternative 23: 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 76.3%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in F around -inf

   \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

   2. lower-sin.f64 17.8

      \[\leadsto \frac{-1}{\sin B} \]

4. Applied rewrites 17.8%

   \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

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 2025148 
(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))))))