VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.5% → 99.6%

Time: 6.9s

Alternatives: 22

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := F \cdot \sin B\\ t_1 := \frac{1}{\sin B}\\ t_2 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -1.1 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, -1, t\_2\right)\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+117}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F, t\_2\right)\\ \mathbf{else}:\\ \;\;\;\;F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{t\_0}, \frac{1}{t\_0}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* F (sin B))) (t_1 (/ 1.0 (sin B))) (t_2 (/ (- x) (tan B))))
   (if (<= F -1.1e+70)
     (fma t_1 -1.0 t_2)
     (if (<= F 2e+117)
       (fma t_1 (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F) t_2)
       (* F (fma -1.0 (/ (* x (cos B)) t_0) (/ 1.0 t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = F * sin(B);
	double t_1 = 1.0 / sin(B);
	double t_2 = -x / tan(B);
	double tmp;
	if (F <= -1.1e+70) {
		tmp = fma(t_1, -1.0, t_2);
	} else if (F <= 2e+117) {
		tmp = fma(t_1, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F), t_2);
	} else {
		tmp = F * fma(-1.0, ((x * cos(B)) / t_0), (1.0 / t_0));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(F * sin(B))
	t_1 = Float64(1.0 / sin(B))
	t_2 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -1.1e+70)
		tmp = fma(t_1, -1.0, t_2);
	elseif (F <= 2e+117)
		tmp = fma(t_1, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F), t_2);
	else
		tmp = Float64(F * fma(-1.0, Float64(Float64(x * cos(B)) / t_0), Float64(1.0 / t_0)));
	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[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.1e+70], N[(t$95$1 * -1.0 + t$95$2), $MachinePrecision], If[LessEqual[F, 2e+117], N[(t$95$1 * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] + t$95$2), $MachinePrecision], N[(F * N[(-1.0 * N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] + N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.1e70

   1. Initial program 77.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.7%

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

   if -1.1e70 < F < 2.0000000000000001e117

   1. Initial program 77.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   if 2.0000000000000001e117 < F

   1. Initial program 77.5%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.9%

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

   4. Applied rewrites 47.9%

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

Alternative 2: 92.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -1.1 \cdot 10^{+70}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, t\_1\right)\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{+236}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ (- x) (tan B))))
   (if (<= F -1.1e+70)
     (fma t_0 -1.0 t_1)
     (if (<= F 1.5e+236)
       (fma t_0 (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F) t_1)
       t_0))))

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 <= -1.1e+70) {
		tmp = fma(t_0, -1.0, t_1);
	} else if (F <= 1.5e+236) {
		tmp = fma(t_0, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F), t_1);
	} else {
		tmp = t_0;
	}
	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 <= -1.1e+70)
		tmp = fma(t_0, -1.0, t_1);
	elseif (F <= 1.5e+236)
		tmp = fma(t_0, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F), t_1);
	else
		tmp = t_0;
	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, -1.1e+70], N[(t$95$0 * -1.0 + t$95$1), $MachinePrecision], If[LessEqual[F, 1.5e+236], N[(t$95$0 * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] + t$95$1), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.1e70

   1. Initial program 77.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.7%

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

   if -1.1e70 < F < 1.4999999999999999e236

   1. Initial program 77.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   if 1.4999999999999999e236 < F

   1. Initial program 77.5%

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

   2. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 16.3%

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

   4. Applied rewrites 16.3%

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

Alternative 3: 92.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -2.2 \cdot 10^{+20}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, t\_1\right)\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{+236}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ (- x) (tan B))))
   (if (<= F -2.2e+20)
     (fma t_0 -1.0 t_1)
     (if (<= F 1.5e+236)
       (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B)) t_1)
       t_0))))

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 <= -2.2e+20) {
		tmp = fma(t_0, -1.0, t_1);
	} else if (F <= 1.5e+236) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), t_1);
	} else {
		tmp = t_0;
	}
	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 <= -2.2e+20)
		tmp = fma(t_0, -1.0, t_1);
	elseif (F <= 1.5e+236)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), t_1);
	else
		tmp = t_0;
	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, -2.2e+20], N[(t$95$0 * -1.0 + t$95$1), $MachinePrecision], If[LessEqual[F, 1.5e+236], 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$1), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.2e20

   1. Initial program 77.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.7%

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

   if -2.2e20 < F < 1.4999999999999999e236

   1. Initial program 77.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 85.7%

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

   if 1.4999999999999999e236 < F

   1. Initial program 77.5%

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

   2. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 16.3%

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

   4. Applied rewrites 16.3%

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

Alternative 4: 90.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -3.2 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{+149}:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{+236}:\\ \;\;\;\;\frac{-x}{\sin B} \cdot \cos B\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))))
   (if (<= F -3.2e+101)
     (fma t_0 -1.0 (/ (- x) (tan B)))
     (if (<= F 2.9e+149)
       (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F (sin B))) (/ x (tan B)))
       (if (<= F 1.5e+236) (* (/ (- x) (sin B)) (cos B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double tmp;
	if (F <= -3.2e+101) {
		tmp = fma(t_0, -1.0, (-x / tan(B)));
	} else if (F <= 2.9e+149) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / sin(B))) - (x / tan(B));
	} else if (F <= 1.5e+236) {
		tmp = (-x / sin(B)) * cos(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -3.2e+101)
		tmp = fma(t_0, -1.0, Float64(Float64(-x) / tan(B)));
	elseif (F <= 2.9e+149)
		tmp = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / sin(B))) - Float64(x / tan(B)));
	elseif (F <= 1.5e+236)
		tmp = Float64(Float64(Float64(-x) / sin(B)) * cos(B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -3.2e+101], N[(t$95$0 * -1.0 + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.9e+149], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.5e+236], N[(N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.20000000000000005e101

   1. Initial program 77.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.7%

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

   if -3.20000000000000005e101 < F < 2.9000000000000002e149

   1. Initial program 77.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 77.5%

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

   3. Applied rewrites 77.7%

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

   if 2.9000000000000002e149 < F < 1.4999999999999999e236

   1. Initial program 77.5%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 57.0%

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

   4. Applied rewrites 57.0%

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

   6. Applied rewrites 57.0%

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

      1. lift-tan.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. tan-quot N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 57.0%

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

   8. Applied rewrites 57.0%

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

   if 1.4999999999999999e236 < F

   1. Initial program 77.5%

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

   2. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 16.3%

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

   4. Applied rewrites 16.3%

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

Alternative 5: 87.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -2.05 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, t\_1\right)\\ \mathbf{elif}\;F \leq 3 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5}}{\sin B}, t\_1\right)\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{+112}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, F, -x\right)}} \cdot t\_0\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{+236}:\\ \;\;\;\;\frac{-x}{\sin B} \cdot \cos B\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ (- x) (tan B))))
   (if (<= F -2.05e-22)
     (fma t_0 -1.0 t_1)
     (if (<= F 3e-10)
       (fma F (/ (pow (fma x 2.0 2.0) -0.5) (sin B)) t_1)
       (if (<= F 1.8e+112)
         (*
          (/ 1.0 (/ 1.0 (fma (pow (fma x 2.0 (fma F F 2.0)) -0.5) F (- x))))
          t_0)
         (if (<= F 1.5e+236) (* (/ (- x) (sin B)) (cos B)) t_0))))))

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 <= -2.05e-22) {
		tmp = fma(t_0, -1.0, t_1);
	} else if (F <= 3e-10) {
		tmp = fma(F, (pow(fma(x, 2.0, 2.0), -0.5) / sin(B)), t_1);
	} else if (F <= 1.8e+112) {
		tmp = (1.0 / (1.0 / fma(pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5), F, -x))) * t_0;
	} else if (F <= 1.5e+236) {
		tmp = (-x / sin(B)) * cos(B);
	} else {
		tmp = t_0;
	}
	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 <= -2.05e-22)
		tmp = fma(t_0, -1.0, t_1);
	elseif (F <= 3e-10)
		tmp = fma(F, Float64((fma(x, 2.0, 2.0) ^ -0.5) / sin(B)), t_1);
	elseif (F <= 1.8e+112)
		tmp = Float64(Float64(1.0 / Float64(1.0 / fma((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5), F, Float64(-x)))) * t_0);
	elseif (F <= 1.5e+236)
		tmp = Float64(Float64(Float64(-x) / sin(B)) * cos(B));
	else
		tmp = t_0;
	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, -2.05e-22], N[(t$95$0 * -1.0 + t$95$1), $MachinePrecision], If[LessEqual[F, 3e-10], N[(F * N[(N[Power[N[(x * 2.0 + 2.0), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[F, 1.8e+112], N[(N[(1.0 / N[(1.0 / N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[F, 1.5e+236], N[(N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -2.05e-22

   1. Initial program 77.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.7%

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

   if -2.05e-22 < F < 3e-10

   1. Initial program 77.5%

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

   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 55.5%

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

   4. Applied rewrites 55.5%

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

   6. Applied rewrites 56.9%

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

   if 3e-10 < F < 1.8e112

   1. Initial program 77.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. add-to-fraction N/A

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

      6. div-flip N/A

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

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

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

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

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 60.0%

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

   6. Applied rewrites 60.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. mult-flip N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

   8. Applied rewrites 60.0%

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

   if 1.8e112 < F < 1.4999999999999999e236

   1. Initial program 77.5%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 57.0%

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

   4. Applied rewrites 57.0%

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

   6. Applied rewrites 57.0%

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

      1. lift-tan.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. tan-quot N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 57.0%

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

   8. Applied rewrites 57.0%

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

   if 1.4999999999999999e236 < F

   1. Initial program 77.5%

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

   2. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 16.3%

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

   4. Applied rewrites 16.3%

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

Alternative 6: 87.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -2.05 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 3 \cdot 10^{-10}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5} \cdot F - \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{+112}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, F, -x\right)}} \cdot t\_0\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{+236}:\\ \;\;\;\;\frac{-x}{\sin B} \cdot \cos B\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))))
   (if (<= F -2.05e-22)
     (fma t_0 -1.0 (/ (- x) (tan B)))
     (if (<= F 3e-10)
       (/ (- (* (pow (fma x 2.0 2.0) -0.5) F) (* (cos B) x)) (sin B))
       (if (<= F 1.8e+112)
         (*
          (/ 1.0 (/ 1.0 (fma (pow (fma x 2.0 (fma F F 2.0)) -0.5) F (- x))))
          t_0)
         (if (<= F 1.5e+236) (* (/ (- x) (sin B)) (cos B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double tmp;
	if (F <= -2.05e-22) {
		tmp = fma(t_0, -1.0, (-x / tan(B)));
	} else if (F <= 3e-10) {
		tmp = ((pow(fma(x, 2.0, 2.0), -0.5) * F) - (cos(B) * x)) / sin(B);
	} else if (F <= 1.8e+112) {
		tmp = (1.0 / (1.0 / fma(pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5), F, -x))) * t_0;
	} else if (F <= 1.5e+236) {
		tmp = (-x / sin(B)) * cos(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -2.05e-22)
		tmp = fma(t_0, -1.0, Float64(Float64(-x) / tan(B)));
	elseif (F <= 3e-10)
		tmp = Float64(Float64(Float64((fma(x, 2.0, 2.0) ^ -0.5) * F) - Float64(cos(B) * x)) / sin(B));
	elseif (F <= 1.8e+112)
		tmp = Float64(Float64(1.0 / Float64(1.0 / fma((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5), F, Float64(-x)))) * t_0);
	elseif (F <= 1.5e+236)
		tmp = Float64(Float64(Float64(-x) / sin(B)) * cos(B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.05e-22], N[(t$95$0 * -1.0 + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3e-10], N[(N[(N[(N[Power[N[(x * 2.0 + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.8e+112], N[(N[(1.0 / N[(1.0 / N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[F, 1.5e+236], N[(N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -2.05e-22

   1. Initial program 77.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.7%

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

   if -2.05e-22 < F < 3e-10

   1. Initial program 77.5%

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

   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 55.5%

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

   4. Applied rewrites 55.5%

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

   6. Applied rewrites 56.5%

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

   if 3e-10 < F < 1.8e112

   1. Initial program 77.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. add-to-fraction N/A

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

      6. div-flip N/A

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

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

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

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

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 60.0%

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

   6. Applied rewrites 60.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. mult-flip N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

   8. Applied rewrites 60.0%

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

   if 1.8e112 < F < 1.4999999999999999e236

   1. Initial program 77.5%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 57.0%

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

   4. Applied rewrites 57.0%

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

   6. Applied rewrites 57.0%

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

      1. lift-tan.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. tan-quot N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 57.0%

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

   8. Applied rewrites 57.0%

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

   if 1.4999999999999999e236 < F

   1. Initial program 77.5%

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

   2. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 16.3%

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

   4. Applied rewrites 16.3%

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

Alternative 7: 80.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -7.2 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq -3.4 \cdot 10^{-100}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-192}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot {\left(2 + 2 \cdot x\right)}^{-0.5}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{+112}:\\ \;\;\;\;\frac{1}{\frac{\sin B}{\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, F, -1 \cdot x\right)}}\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{+236}:\\ \;\;\;\;\frac{-x}{\sin B} \cdot \cos B\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))))
   (if (<= F -7.2e+19)
     (fma t_0 -1.0 (/ (- x) (tan B)))
     (if (<= F -3.4e-100)
       (+
        (- (/ x B))
        (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
       (if (<= F 1.3e-192)
         (+ (- (* x (/ 1.0 (tan B)))) (* (/ F B) (pow (+ 2.0 (* 2.0 x)) -0.5)))
         (if (<= F 1.8e+112)
           (/
            1.0
            (/
             (sin B)
             (fma (pow (fma 2.0 x (fma F F 2.0)) -0.5) F (* -1.0 x))))
           (if (<= F 1.5e+236) (* (/ (- x) (sin B)) (cos B)) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double tmp;
	if (F <= -7.2e+19) {
		tmp = fma(t_0, -1.0, (-x / tan(B)));
	} else if (F <= -3.4e-100) {
		tmp = -(x / B) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else if (F <= 1.3e-192) {
		tmp = -(x * (1.0 / tan(B))) + ((F / B) * pow((2.0 + (2.0 * x)), -0.5));
	} else if (F <= 1.8e+112) {
		tmp = 1.0 / (sin(B) / fma(pow(fma(2.0, x, fma(F, F, 2.0)), -0.5), F, (-1.0 * x)));
	} else if (F <= 1.5e+236) {
		tmp = (-x / sin(B)) * cos(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -7.2e+19)
		tmp = fma(t_0, -1.0, Float64(Float64(-x) / tan(B)));
	elseif (F <= -3.4e-100)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))));
	elseif (F <= 1.3e-192)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / B) * (Float64(2.0 + Float64(2.0 * x)) ^ -0.5)));
	elseif (F <= 1.8e+112)
		tmp = Float64(1.0 / Float64(sin(B) / fma((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5), F, Float64(-1.0 * x))));
	elseif (F <= 1.5e+236)
		tmp = Float64(Float64(Float64(-x) / sin(B)) * cos(B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -7.2e+19], N[(t$95$0 * -1.0 + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.4e-100], N[((-N[(x / B), $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], If[LessEqual[F, 1.3e-192], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / B), $MachinePrecision] * N[Power[N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.8e+112], N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F + N[(-1.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.5e+236], N[(N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -7.2e19

   1. Initial program 77.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.7%

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

   if -7.2e19 < F < -3.39999999999999976e-100

   1. Initial program 77.5%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 49.9%

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

   4. Applied rewrites 49.9%

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

   if -3.39999999999999976e-100 < F < 1.3000000000000001e-192

   1. Initial program 77.5%

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

   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 55.5%

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

   4. Applied rewrites 55.5%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 50.2%

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

   if 1.3000000000000001e-192 < F < 1.8e112

   1. Initial program 77.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. add-to-fraction N/A

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

      6. div-flip N/A

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

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

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

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

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 60.0%

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

   6. Applied rewrites 60.0%

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

   if 1.8e112 < F < 1.4999999999999999e236

   1. Initial program 77.5%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 57.0%

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

   4. Applied rewrites 57.0%

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

   6. Applied rewrites 57.0%

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

      1. lift-tan.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. tan-quot N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 57.0%

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

   8. Applied rewrites 57.0%

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

   if 1.4999999999999999e236 < F

   1. Initial program 77.5%

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

   2. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 16.3%

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

   4. Applied rewrites 16.3%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 6 regimes into one program.
4. Add Preprocessing

Alternative 8: 80.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{if}\;F \leq -7.2 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq -3.4 \cdot 10^{-100}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot t\_1\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-192}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot t\_1\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{+112}:\\ \;\;\;\;\frac{1}{\frac{\sin B}{\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, F, -1 \cdot x\right)}}\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{+236}:\\ \;\;\;\;\frac{-x}{\sin B} \cdot \cos B\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B)))
        (t_1 (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
   (if (<= F -7.2e+19)
     (fma t_0 -1.0 (/ (- x) (tan B)))
     (if (<= F -3.4e-100)
       (+ (- (/ x B)) (* (/ F (sin B)) t_1))
       (if (<= F 1.3e-192)
         (+ (- (* x (/ 1.0 (tan B)))) (* (/ F B) t_1))
         (if (<= F 1.8e+112)
           (/
            1.0
            (/
             (sin B)
             (fma (pow (fma 2.0 x (fma F F 2.0)) -0.5) F (* -1.0 x))))
           (if (<= F 1.5e+236) (* (/ (- x) (sin B)) (cos B)) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0));
	double tmp;
	if (F <= -7.2e+19) {
		tmp = fma(t_0, -1.0, (-x / tan(B)));
	} else if (F <= -3.4e-100) {
		tmp = -(x / B) + ((F / sin(B)) * t_1);
	} else if (F <= 1.3e-192) {
		tmp = -(x * (1.0 / tan(B))) + ((F / B) * t_1);
	} else if (F <= 1.8e+112) {
		tmp = 1.0 / (sin(B) / fma(pow(fma(2.0, x, fma(F, F, 2.0)), -0.5), F, (-1.0 * x)));
	} else if (F <= 1.5e+236) {
		tmp = (-x / sin(B)) * cos(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0))
	tmp = 0.0
	if (F <= -7.2e+19)
		tmp = fma(t_0, -1.0, Float64(Float64(-x) / tan(B)));
	elseif (F <= -3.4e-100)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F / sin(B)) * t_1));
	elseif (F <= 1.3e-192)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / B) * t_1));
	elseif (F <= 1.8e+112)
		tmp = Float64(1.0 / Float64(sin(B) / fma((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5), F, Float64(-1.0 * x))));
	elseif (F <= 1.5e+236)
		tmp = Float64(Float64(Float64(-x) / sin(B)) * cos(B));
	else
		tmp = t_0;
	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[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]}, If[LessEqual[F, -7.2e+19], N[(t$95$0 * -1.0 + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.4e-100], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.3e-192], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / B), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.8e+112], N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F + N[(-1.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.5e+236], N[(N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -7.2e19

   1. Initial program 77.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.7%

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

   if -7.2e19 < F < -3.39999999999999976e-100

   1. Initial program 77.5%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 49.9%

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

   4. Applied rewrites 49.9%

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

   if -3.39999999999999976e-100 < F < 1.3000000000000001e-192

   1. Initial program 77.5%

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

   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 63.4%

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

   4. Applied rewrites 63.4%

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

   if 1.3000000000000001e-192 < F < 1.8e112

   1. Initial program 77.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. add-to-fraction N/A

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

      6. div-flip N/A

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

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

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

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

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 60.0%

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

   6. Applied rewrites 60.0%

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

   if 1.8e112 < F < 1.4999999999999999e236

   1. Initial program 77.5%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 57.0%

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

   4. Applied rewrites 57.0%

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

   6. Applied rewrites 57.0%

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

      1. lift-tan.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. tan-quot N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 57.0%

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

   8. Applied rewrites 57.0%

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

   if 1.4999999999999999e236 < F

   1. Initial program 77.5%

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

   2. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 16.3%

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

   4. Applied rewrites 16.3%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 6 regimes into one program.
4. Add Preprocessing

Alternative 9: 77.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-61}:\\ \;\;\;\;\frac{-x}{\sin B} \cdot \cos B\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\frac{1}{\frac{1}{\mathsf{fma}\left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, F, -x\right)}} \cdot \frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -1.7e-61)
   (* (/ (- x) (sin B)) (cos B))
   (if (<= x 5e-78)
     (*
      (/ 1.0 (/ 1.0 (fma (pow (fma x 2.0 (fma F F 2.0)) -0.5) F (- x))))
      (/ 1.0 (sin B)))
     (/ (- x) (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -1.7e-61) {
		tmp = (-x / sin(B)) * cos(B);
	} else if (x <= 5e-78) {
		tmp = (1.0 / (1.0 / fma(pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5), F, -x))) * (1.0 / sin(B));
	} else {
		tmp = -x / tan(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -1.7e-61)
		tmp = Float64(Float64(Float64(-x) / sin(B)) * cos(B));
	elseif (x <= 5e-78)
		tmp = Float64(Float64(1.0 / Float64(1.0 / fma((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5), F, Float64(-x)))) * Float64(1.0 / sin(B)));
	else
		tmp = Float64(Float64(-x) / tan(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -1.7e-61], N[(N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e-78], N[(N[(1.0 / N[(1.0 / N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F + (-x)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6999999999999999e-61

   1. Initial program 77.5%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 57.0%

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

   4. Applied rewrites 57.0%

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

   6. Applied rewrites 57.0%

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

      1. lift-tan.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. tan-quot N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 57.0%

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

   8. Applied rewrites 57.0%

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

   if -1.6999999999999999e-61 < x < 4.9999999999999996e-78

   1. Initial program 77.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. add-to-fraction N/A

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

      6. div-flip N/A

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

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

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

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

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 60.0%

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

   6. Applied rewrites 60.0%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. metadata-eval N/A

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

      4. mult-flip N/A

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

      5. *-commutative N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

   8. Applied rewrites 60.0%

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

   if 4.9999999999999996e-78 < x

   1. Initial program 77.5%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 57.0%

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

   4. Applied rewrites 57.0%

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

   6. Applied rewrites 57.0%

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

Alternative 10: 77.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-61}:\\ \;\;\;\;\frac{-x}{\sin B} \cdot \cos B\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\frac{1}{\frac{\sin B}{\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, F, -1 \cdot x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -1.7e-61)
   (* (/ (- x) (sin B)) (cos B))
   (if (<= x 5e-78)
     (/
      1.0
      (/ (sin B) (fma (pow (fma 2.0 x (fma F F 2.0)) -0.5) F (* -1.0 x))))
     (/ (- x) (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -1.7e-61) {
		tmp = (-x / sin(B)) * cos(B);
	} else if (x <= 5e-78) {
		tmp = 1.0 / (sin(B) / fma(pow(fma(2.0, x, fma(F, F, 2.0)), -0.5), F, (-1.0 * x)));
	} else {
		tmp = -x / tan(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -1.7e-61)
		tmp = Float64(Float64(Float64(-x) / sin(B)) * cos(B));
	elseif (x <= 5e-78)
		tmp = Float64(1.0 / Float64(sin(B) / fma((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5), F, Float64(-1.0 * x))));
	else
		tmp = Float64(Float64(-x) / tan(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -1.7e-61], N[(N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e-78], N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F + N[(-1.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6999999999999999e-61

   1. Initial program 77.5%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 57.0%

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

   4. Applied rewrites 57.0%

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

   6. Applied rewrites 57.0%

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

      1. lift-tan.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. tan-quot N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 57.0%

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

   8. Applied rewrites 57.0%

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

   if -1.6999999999999999e-61 < x < 4.9999999999999996e-78

   1. Initial program 77.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. add-to-fraction N/A

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

      6. div-flip N/A

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

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

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

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

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 60.0%

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

   6. Applied rewrites 60.0%

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

   if 4.9999999999999996e-78 < x

   1. Initial program 77.5%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 57.0%

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

   4. Applied rewrites 57.0%

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

   6. Applied rewrites 57.0%

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

Alternative 11: 72.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -1.7 \cdot 10^{-61}:\\ \;\;\;\;\frac{-x}{\sin B} \cdot \cos B\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\frac{1}{\sin B} \cdot \left({\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5} \cdot F\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -1.7e-61)
   (* (/ (- x) (sin B)) (cos B))
   (if (<= x 5e-78)
     (* (/ 1.0 (sin B)) (* (pow (fma F F 2.0) -0.5) F))
     (/ (- x) (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -1.7e-61) {
		tmp = (-x / sin(B)) * cos(B);
	} else if (x <= 5e-78) {
		tmp = (1.0 / sin(B)) * (pow(fma(F, F, 2.0), -0.5) * F);
	} else {
		tmp = -x / tan(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -1.7e-61)
		tmp = Float64(Float64(Float64(-x) / sin(B)) * cos(B));
	elseif (x <= 5e-78)
		tmp = Float64(Float64(1.0 / sin(B)) * Float64((fma(F, F, 2.0) ^ -0.5) * F));
	else
		tmp = Float64(Float64(-x) / tan(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -1.7e-61], N[(N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e-78], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(F * F + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.6999999999999999e-61

   1. Initial program 77.5%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 57.0%

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

   4. Applied rewrites 57.0%

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

   6. Applied rewrites 57.0%

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

      1. lift-tan.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. tan-quot N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-cos.f64 N/A

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

      6. associate-/r/ N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 57.0%

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

   8. Applied rewrites 57.0%

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

   if -1.6999999999999999e-61 < x < 4.9999999999999996e-78

   1. Initial program 77.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-sin.f64 30.3%

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

   6. Applied rewrites 30.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-/.f64 30.3%

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 30.3%

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-fma.f64 30.3%

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

   8. Applied rewrites 30.3%

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

   if 4.9999999999999996e-78 < x

   1. Initial program 77.5%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 57.0%

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

   4. Applied rewrites 57.0%

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

   6. Applied rewrites 57.0%

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

Alternative 12: 72.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\frac{1}{\sin B} \cdot \left({\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5} \cdot F\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= x -1.7e-61)
     t_0
     (if (<= x 5e-78)
       (* (/ 1.0 (sin B)) (* (pow (fma F F 2.0) -0.5) F))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (x <= -1.7e-61) {
		tmp = t_0;
	} else if (x <= 5e-78) {
		tmp = (1.0 / sin(B)) * (pow(fma(F, F, 2.0), -0.5) * F);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (x <= -1.7e-61)
		tmp = t_0;
	elseif (x <= 5e-78)
		tmp = Float64(Float64(1.0 / sin(B)) * Float64((fma(F, F, 2.0) ^ -0.5) * F));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.7e-61], t$95$0, If[LessEqual[x, 5e-78], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[(F * F + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6999999999999999e-61 or 4.9999999999999996e-78 < x

   1. Initial program 77.5%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 57.0%

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

   4. Applied rewrites 57.0%

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

   6. Applied rewrites 57.0%

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

   if -1.6999999999999999e-61 < x < 4.9999999999999996e-78

   1. Initial program 77.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-sin.f64 30.3%

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

   6. Applied rewrites 30.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-sin.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-/.f64 30.3%

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 30.3%

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. lift-pow.f64 N/A

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

      14. pow2 N/A

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

      15. lift-fma.f64 30.3%

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

   8. Applied rewrites 30.3%

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

Alternative 13: 72.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;x \leq -1.7 \cdot 10^{-61}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-78}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}{\sin B} \cdot F\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= x -1.7e-61)
     t_0
     (if (<= x 5e-78) (* (/ (pow (fma F F 2.0) -0.5) (sin B)) F) t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (x <= -1.7e-61) {
		tmp = t_0;
	} else if (x <= 5e-78) {
		tmp = (pow(fma(F, F, 2.0), -0.5) / sin(B)) * F;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (x <= -1.7e-61)
		tmp = t_0;
	elseif (x <= 5e-78)
		tmp = Float64(Float64((fma(F, F, 2.0) ^ -0.5) / sin(B)) * F);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.7e-61], t$95$0, If[LessEqual[x, 5e-78], N[(N[(N[Power[N[(F * F + 2.0), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.6999999999999999e-61 or 4.9999999999999996e-78 < x

   1. Initial program 77.5%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 57.0%

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

   4. Applied rewrites 57.0%

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

   6. Applied rewrites 57.0%

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

   if -1.6999999999999999e-61 < x < 4.9999999999999996e-78

   1. Initial program 77.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 85.7%

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

   4. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-sin.f64 30.3%

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

   6. Applied rewrites 30.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 30.3%

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-fma.f64 30.3%

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

   8. Applied rewrites 30.3%

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

Alternative 14: 71.2% accurate, 1.9× speedup

Math

\[\mathsf{copysign}\left(1, B\right) \cdot \begin{array}{l} \mathbf{if}\;\left|B\right| \leq 1.4 \cdot 10^{-11}:\\ \;\;\;\;\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{\left|B\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan \left(\left|B\right|\right)}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (*
  (copysign 1.0 B)
  (if (<= (fabs B) 1.4e-11)
    (/ (- (* F (pow (+ 2.0 (fma 2.0 x (pow F 2.0))) -0.5)) x) (fabs B))
    (/ (- x) (tan (fabs B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (fabs(B) <= 1.4e-11) {
		tmp = ((F * pow((2.0 + fma(2.0, x, pow(F, 2.0))), -0.5)) - x) / fabs(B);
	} else {
		tmp = -x / tan(fabs(B));
	}
	return copysign(1.0, B) * tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (abs(B) <= 1.4e-11)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + fma(2.0, x, (F ^ 2.0))) ^ -0.5)) - x) / abs(B));
	else
		tmp = Float64(Float64(-x) / tan(abs(B)));
	end
	return Float64(copysign(1.0, B) * tmp)
end

Wolfram

code[F_, B_, x_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[B]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[B], $MachinePrecision], 1.4e-11], 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] / N[Abs[B], $MachinePrecision]), $MachinePrecision], N[((-x) / N[Tan[N[Abs[B], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if B < 1.4e-11

   1. Initial program 77.5%

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

   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 44.7%

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

   if 1.4e-11 < B

   1. Initial program 77.5%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 57.0%

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

   4. Applied rewrites 57.0%

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

   6. Applied rewrites 57.0%

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

Alternative 15: 71.1% accurate, 2.2× speedup

Math

\[\mathsf{copysign}\left(1, B\right) \cdot \begin{array}{l} \mathbf{if}\;\left|B\right| \leq 1.4 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{\frac{\left|B\right|}{\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, F, -1 \cdot x\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan \left(\left|B\right|\right)}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (*
  (copysign 1.0 B)
  (if (<= (fabs B) 1.4e-11)
    (/
     1.0
     (/ (fabs B) (fma (pow (fma 2.0 x (fma F F 2.0)) -0.5) F (* -1.0 x))))
    (/ (- x) (tan (fabs B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (fabs(B) <= 1.4e-11) {
		tmp = 1.0 / (fabs(B) / fma(pow(fma(2.0, x, fma(F, F, 2.0)), -0.5), F, (-1.0 * x)));
	} else {
		tmp = -x / tan(fabs(B));
	}
	return copysign(1.0, B) * tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (abs(B) <= 1.4e-11)
		tmp = Float64(1.0 / Float64(abs(B) / fma((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5), F, Float64(-1.0 * x))));
	else
		tmp = Float64(Float64(-x) / tan(abs(B)));
	end
	return Float64(copysign(1.0, B) * tmp)
end

Wolfram

code[F_, B_, x_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[B]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[B], $MachinePrecision], 1.4e-11], N[(1.0 / N[(N[Abs[B], $MachinePrecision] / N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F + N[(-1.0 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-x) / N[Tan[N[Abs[B], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if B < 1.4e-11

   1. Initial program 77.5%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. add-to-fraction N/A

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

      6. div-flip N/A

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

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

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

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

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

   3. Applied rewrites 85.6%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 60.0%

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

   6. Applied rewrites 60.0%

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

   7. Taylor expanded in B around 0

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

   9. Applied rewrites 44.6%

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

   if 1.4e-11 < B

   1. Initial program 77.5%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 57.0%

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

   4. Applied rewrites 57.0%

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

   6. Applied rewrites 57.0%

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

Alternative 16: 59.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-245}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-227}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= x -1e-245)
     t_0
     (if (<= x 5.2e-227)
       (/ 1.0 (sin B))
       (if (<= x 6.8e-78) (/ -1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (x <= -1e-245) {
		tmp = t_0;
	} else if (x <= 5.2e-227) {
		tmp = 1.0 / sin(B);
	} else if (x <= 6.8e-78) {
		tmp = -1.0 / sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -x / tan(b)
    if (x <= (-1d-245)) then
        tmp = t_0
    else if (x <= 5.2d-227) then
        tmp = 1.0d0 / sin(b)
    else if (x <= 6.8d-78) then
        tmp = (-1.0d0) / sin(b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -x / Math.tan(B);
	double tmp;
	if (x <= -1e-245) {
		tmp = t_0;
	} else if (x <= 5.2e-227) {
		tmp = 1.0 / Math.sin(B);
	} else if (x <= 6.8e-78) {
		tmp = -1.0 / Math.sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -x / math.tan(B)
	tmp = 0
	if x <= -1e-245:
		tmp = t_0
	elif x <= 5.2e-227:
		tmp = 1.0 / math.sin(B)
	elif x <= 6.8e-78:
		tmp = -1.0 / math.sin(B)
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (x <= -1e-245)
		tmp = t_0;
	elseif (x <= 5.2e-227)
		tmp = Float64(1.0 / sin(B));
	elseif (x <= 6.8e-78)
		tmp = Float64(-1.0 / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -x / tan(B);
	tmp = 0.0;
	if (x <= -1e-245)
		tmp = t_0;
	elseif (x <= 5.2e-227)
		tmp = 1.0 / sin(B);
	elseif (x <= 6.8e-78)
		tmp = -1.0 / sin(B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1e-245], t$95$0, If[LessEqual[x, 5.2e-227], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.8e-78], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.9999999999999993e-246 or 6.80000000000000023e-78 < x

   1. Initial program 77.5%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 57.0%

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

   4. Applied rewrites 57.0%

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

   6. Applied rewrites 57.0%

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

   if -9.9999999999999993e-246 < x < 5.20000000000000023e-227

   1. Initial program 77.5%

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

   2. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 16.3%

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

   4. Applied rewrites 16.3%

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

   if 5.20000000000000023e-227 < x < 6.80000000000000023e-78

   1. Initial program 77.5%

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

   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 17.1%

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

   4. Applied rewrites 17.1%

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

Alternative 17: 44.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.75 \cdot 10^{+22}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-43}:\\ \;\;\;\;-1 \cdot \frac{x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.75e+22)
   (/ -1.0 (sin B))
   (if (<= F 3.1e-43) (* -1.0 (/ x (sin B))) (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.75e+22) {
		tmp = -1.0 / sin(B);
	} else if (F <= 3.1e-43) {
		tmp = -1.0 * (x / sin(B));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.75d+22)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 3.1d-43) then
        tmp = (-1.0d0) * (x / sin(b))
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.75e+22) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 3.1e-43) {
		tmp = -1.0 * (x / Math.sin(B));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.75e+22:
		tmp = -1.0 / math.sin(B)
	elif F <= 3.1e-43:
		tmp = -1.0 * (x / math.sin(B))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.75e+22)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 3.1e-43)
		tmp = Float64(-1.0 * Float64(x / sin(B)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.75e+22)
		tmp = -1.0 / sin(B);
	elseif (F <= 3.1e-43)
		tmp = -1.0 * (x / sin(B));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.75e+22], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.1e-43], N[(-1.0 * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.75e22

   1. Initial program 77.5%

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

   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 17.1%

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

   4. Applied rewrites 17.1%

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

   if -1.75e22 < F < 3.0999999999999999e-43

   1. Initial program 77.5%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 57.0%

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

   4. Applied rewrites 57.0%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 32.3%

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

   if 3.0999999999999999e-43 < F

   1. Initial program 77.5%

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

   2. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 16.3%

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

   4. Applied rewrites 16.3%

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

Alternative 18: 43.4% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.75 \cdot 10^{+22}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-43}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.75e+22)
   (/ -1.0 (sin B))
   (if (<= F 3.1e-43) (- (/ x B)) (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.75e+22) {
		tmp = -1.0 / sin(B);
	} else if (F <= 3.1e-43) {
		tmp = -(x / B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.75d+22)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 3.1d-43) then
        tmp = -(x / b)
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.75e+22) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 3.1e-43) {
		tmp = -(x / B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.75e+22:
		tmp = -1.0 / math.sin(B)
	elif F <= 3.1e-43:
		tmp = -(x / B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.75e+22)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 3.1e-43)
		tmp = Float64(-Float64(x / B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.75e+22)
		tmp = -1.0 / sin(B);
	elseif (F <= 3.1e-43)
		tmp = -(x / B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.75e+22], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.1e-43], (-N[(x / B), $MachinePrecision]), N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.75e22

   1. Initial program 77.5%

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

   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 17.1%

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

   4. Applied rewrites 17.1%

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

   if -1.75e22 < F < 3.0999999999999999e-43

   1. Initial program 77.5%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 57.0%

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

   4. Applied rewrites 57.0%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 30.0%

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

   7. Applied rewrites 30.0%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 30.0%

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

   9. Applied rewrites 30.0%

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

   if 3.0999999999999999e-43 < F

   1. Initial program 77.5%

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

   2. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 16.3%

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

   4. Applied rewrites 16.3%

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

Alternative 19: 36.6% accurate, 2.6× speedup

Math

\[\begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;x \leq -3.9 \cdot 10^{-81}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x B))))
   (if (<= x -3.9e-81) t_0 (if (<= x 3.8e-15) (/ -1.0 (sin B)) t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (x <= -3.9e-81) {
		tmp = t_0;
	} else if (x <= 3.8e-15) {
		tmp = -1.0 / sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -(x / b)
    if (x <= (-3.9d-81)) then
        tmp = t_0
    else if (x <= 3.8d-15) then
        tmp = (-1.0d0) / sin(b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (x <= -3.9e-81) {
		tmp = t_0;
	} else if (x <= 3.8e-15) {
		tmp = -1.0 / Math.sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -(x / B)
	tmp = 0
	if x <= -3.9e-81:
		tmp = t_0
	elif x <= 3.8e-15:
		tmp = -1.0 / math.sin(B)
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	tmp = 0.0
	if (x <= -3.9e-81)
		tmp = t_0;
	elseif (x <= 3.8e-15)
		tmp = Float64(-1.0 / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -(x / B);
	tmp = 0.0;
	if (x <= -3.9e-81)
		tmp = t_0;
	elseif (x <= 3.8e-15)
		tmp = -1.0 / sin(B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[x, -3.9e-81], t$95$0, If[LessEqual[x, 3.8e-15], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.89999999999999985e-81 or 3.8000000000000002e-15 < x

   1. Initial program 77.5%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 57.0%

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

   4. Applied rewrites 57.0%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 30.0%

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

   7. Applied rewrites 30.0%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 30.0%

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

   9. Applied rewrites 30.0%

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

   if -3.89999999999999985e-81 < x < 3.8000000000000002e-15

   1. Initial program 77.5%

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

   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 17.1%

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

   4. Applied rewrites 17.1%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 32.1% accurate, 3.2× speedup

Math

\[\begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;x \leq -1.1 \cdot 10^{-142}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-75}:\\ \;\;\;\;\frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x B))))
   (if (<= x -1.1e-142)
     t_0
     (if (<= x 1.4e-75)
       (/ -1.0 (* B (+ 1.0 (* -0.16666666666666666 (pow B 2.0)))))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (x <= -1.1e-142) {
		tmp = t_0;
	} else if (x <= 1.4e-75) {
		tmp = -1.0 / (B * (1.0 + (-0.16666666666666666 * pow(B, 2.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -(x / b)
    if (x <= (-1.1d-142)) then
        tmp = t_0
    else if (x <= 1.4d-75) then
        tmp = (-1.0d0) / (b * (1.0d0 + ((-0.16666666666666666d0) * (b ** 2.0d0))))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (x <= -1.1e-142) {
		tmp = t_0;
	} else if (x <= 1.4e-75) {
		tmp = -1.0 / (B * (1.0 + (-0.16666666666666666 * Math.pow(B, 2.0))));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -(x / B)
	tmp = 0
	if x <= -1.1e-142:
		tmp = t_0
	elif x <= 1.4e-75:
		tmp = -1.0 / (B * (1.0 + (-0.16666666666666666 * math.pow(B, 2.0))))
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	tmp = 0.0
	if (x <= -1.1e-142)
		tmp = t_0;
	elseif (x <= 1.4e-75)
		tmp = Float64(-1.0 / Float64(B * Float64(1.0 + Float64(-0.16666666666666666 * (B ^ 2.0)))));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -(x / B);
	tmp = 0.0;
	if (x <= -1.1e-142)
		tmp = t_0;
	elseif (x <= 1.4e-75)
		tmp = -1.0 / (B * (1.0 + (-0.16666666666666666 * (B ^ 2.0))));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[x, -1.1e-142], t$95$0, If[LessEqual[x, 1.4e-75], N[(-1.0 / N[(B * N[(1.0 + N[(-0.16666666666666666 * N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.10000000000000008e-142 or 1.39999999999999999e-75 < x

   1. Initial program 77.5%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 57.0%

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

   4. Applied rewrites 57.0%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 30.0%

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

   7. Applied rewrites 30.0%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 30.0%

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

   9. Applied rewrites 30.0%

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

   if -1.10000000000000008e-142 < x < 1.39999999999999999e-75

   1. Initial program 77.5%

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

   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 17.1%

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

   4. Applied rewrites 17.1%

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

   5. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 10.6%

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

   7. Applied rewrites 10.6%

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

Alternative 21: 31.9% accurate, 8.9× speedup

Math

\[\begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;x \leq -3.6 \cdot 10^{-81}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x B))))
   (if (<= x -3.6e-81) t_0 (if (<= x 6.8e-78) (/ -1.0 B) t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (x <= -3.6e-81) {
		tmp = t_0;
	} else if (x <= 6.8e-78) {
		tmp = -1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -(x / b)
    if (x <= (-3.6d-81)) then
        tmp = t_0
    else if (x <= 6.8d-78) then
        tmp = (-1.0d0) / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (x <= -3.6e-81) {
		tmp = t_0;
	} else if (x <= 6.8e-78) {
		tmp = -1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -(x / B)
	tmp = 0
	if x <= -3.6e-81:
		tmp = t_0
	elif x <= 6.8e-78:
		tmp = -1.0 / B
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	tmp = 0.0
	if (x <= -3.6e-81)
		tmp = t_0;
	elseif (x <= 6.8e-78)
		tmp = Float64(-1.0 / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -(x / B);
	tmp = 0.0;
	if (x <= -3.6e-81)
		tmp = t_0;
	elseif (x <= 6.8e-78)
		tmp = -1.0 / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[x, -3.6e-81], t$95$0, If[LessEqual[x, 6.8e-78], N[(-1.0 / B), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.5999999999999999e-81 or 6.80000000000000023e-78 < x

   1. Initial program 77.5%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 57.0%

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

   4. Applied rewrites 57.0%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 30.0%

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

   7. Applied rewrites 30.0%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 30.0%

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

   9. Applied rewrites 30.0%

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

   if -3.5999999999999999e-81 < x < 6.80000000000000023e-78

   1. Initial program 77.5%

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

   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 17.1%

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

   4. Applied rewrites 17.1%

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

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-1}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 10.7%

      \[\leadsto \frac{-1}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 10.7% accurate, 26.5× speedup

Math

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

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 77.5%

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

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 17.1%

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

4. Applied rewrites 17.1%

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

5. Taylor expanded in B around 0

   \[\leadsto \frac{-1}{B} \]
6. Step-by-step derivation

7. Applied rewrites 10.7%

   \[\leadsto \frac{-1}{B} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025181 
(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))))))