VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.9% → 99.3%

Time: 7.1s

Alternatives: 22

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% 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: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -2.7 \cdot 10^{+155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B \cdot F} \cdot \left(-F\right)\\ \mathbf{elif}\;F \leq 0.0009:\\ \;\;\;\;\mathsf{fma}\left(F, {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot t\_0, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \frac{x}{\tan B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))))
   (if (<= F -2.7e+155)
     (* (/ (fma (cos B) x 1.0) (* (sin B) F)) (- F))
     (if (<= F 0.0009)
       (fma F (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) t_0) (/ (- x) (tan B)))
       (- t_0 (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double tmp;
	if (F <= -2.7e+155) {
		tmp = (fma(cos(B), x, 1.0) / (sin(B) * F)) * -F;
	} else if (F <= 0.0009) {
		tmp = fma(F, (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * t_0), (-x / tan(B)));
	} else {
		tmp = t_0 - (x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -2.7e+155)
		tmp = Float64(Float64(fma(cos(B), x, 1.0) / Float64(sin(B) * F)) * Float64(-F));
	elseif (F <= 0.0009)
		tmp = fma(F, Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * t_0), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(t_0 - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.7e+155], N[(N[(N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * (-F)), $MachinePrecision], If[LessEqual[F, 0.0009], N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * t$95$0), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.6999999999999999e155

   1. Initial program 76.9%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.8%

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

   4. Applied rewrites 49.8%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 49.9%

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

   if -2.6999999999999999e155 < F < 8.9999999999999998e-4

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 85.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lower-*.f64 85.0%

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

      5. lift-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 85.0%

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

   5. Applied rewrites 85.0%

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

   if 8.9999999999999998e-4 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 85.0%

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

   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 55.2%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-lft-identity N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 55.2%

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

   8. Applied rewrites 55.2%

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

Alternative 2: 99.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.7 \cdot 10^{+155}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B \cdot F} \cdot \left(-F\right)\\ \mathbf{elif}\;F \leq 0.0009:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2.7e+155)
     (* (/ (fma (cos B) x 1.0) (* (sin B) F)) (- F))
     (if (<= F 0.0009)
       (- (/ F (* (sin B) (sqrt (fma 2.0 x (fma F F 2.0))))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2.7e+155) {
		tmp = (fma(cos(B), x, 1.0) / (sin(B) * F)) * -F;
	} else if (F <= 0.0009) {
		tmp = (F / (sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.7e+155)
		tmp = Float64(Float64(fma(cos(B), x, 1.0) / Float64(sin(B) * F)) * Float64(-F));
	elseif (F <= 0.0009)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.7e+155], N[(N[(N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * (-F)), $MachinePrecision], If[LessEqual[F, 0.0009], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.6999999999999999e155

   1. Initial program 76.9%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.8%

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

   4. Applied rewrites 49.8%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 49.9%

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

   if -2.6999999999999999e155 < F < 8.9999999999999998e-4

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 85.0%

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

      1. lift-/.f64 N/A

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

      2. div-flip N/A

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

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

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

      4. lower-unsound-/.f64 85.0%

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

      5. lift-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 85.0%

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

   5. Applied rewrites 85.0%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

   7. Applied rewrites 85.0%

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

   if 8.9999999999999998e-4 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 85.0%

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

   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 55.2%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-lft-identity N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 55.2%

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

   8. Applied rewrites 55.2%

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

Alternative 3: 92.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}\\ t_2 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -44000:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B \cdot F} \cdot \left(-F\right)\\ \mathbf{elif}\;F \leq -2.7 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(F, t\_1 \cdot t\_2, -1 \cdot \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{-79}:\\ \;\;\;\;\frac{F}{B} \cdot t\_1 - t\_0\\ \mathbf{elif}\;F \leq 0.0009:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2 - t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B)))
        (t_1 (pow (fma x 2.0 (fma F F 2.0)) -0.5))
        (t_2 (/ 1.0 (sin B))))
   (if (<= F -44000.0)
     (* (/ (fma (cos B) x 1.0) (* (sin B) F)) (- F))
     (if (<= F -2.7e-46)
       (fma F (* t_1 t_2) (* -1.0 (/ x B)))
       (if (<= F 3.9e-79)
         (- (* (/ F B) t_1) t_0)
         (if (<= F 0.0009)
           (+
            (- (/ x B))
            (*
             (/ F (sin B))
             (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
           (- t_2 t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double t_1 = pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5);
	double t_2 = 1.0 / sin(B);
	double tmp;
	if (F <= -44000.0) {
		tmp = (fma(cos(B), x, 1.0) / (sin(B) * F)) * -F;
	} else if (F <= -2.7e-46) {
		tmp = fma(F, (t_1 * t_2), (-1.0 * (x / B)));
	} else if (F <= 3.9e-79) {
		tmp = ((F / B) * t_1) - t_0;
	} else if (F <= 0.0009) {
		tmp = -(x / B) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = t_2 - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	t_1 = fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5
	t_2 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -44000.0)
		tmp = Float64(Float64(fma(cos(B), x, 1.0) / Float64(sin(B) * F)) * Float64(-F));
	elseif (F <= -2.7e-46)
		tmp = fma(F, Float64(t_1 * t_2), Float64(-1.0 * Float64(x / B)));
	elseif (F <= 3.9e-79)
		tmp = Float64(Float64(Float64(F / B) * t_1) - t_0);
	elseif (F <= 0.0009)
		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)))));
	else
		tmp = Float64(t_2 - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -44000.0], N[(N[(N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * (-F)), $MachinePrecision], If[LessEqual[F, -2.7e-46], N[(F * N[(t$95$1 * t$95$2), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.9e-79], N[(N[(N[(F / B), $MachinePrecision] * t$95$1), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.0009], 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], N[(t$95$2 - t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -44000

   1. Initial program 76.9%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.8%

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

   4. Applied rewrites 49.8%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-*.f64 N/A

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

   6. Applied rewrites 49.9%

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

   if -44000 < F < -2.7e-46

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 85.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lower-*.f64 85.0%

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

      5. lift-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 85.0%

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 57.6%

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

   8. Applied rewrites 57.6%

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

   if -2.7e-46 < F < 3.9000000000000001e-79

   1. Initial program 76.9%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.3%

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

   4. Applied rewrites 62.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. lift-/.f64 N/A

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

      11. lower--.f64 N/A

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

   6. Applied rewrites 62.4%

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

   if 3.9000000000000001e-79 < F < 8.9999999999999998e-4

   1. Initial program 76.9%

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

   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 49.7%

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

   4. Applied rewrites 49.7%

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

   if 8.9999999999999998e-4 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 85.0%

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

   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 55.2%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-lft-identity N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 55.2%

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

   8. Applied rewrites 55.2%

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

Alternative 4: 86.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}\\ t_2 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -44000:\\ \;\;\;\;\left(\frac{x}{B \cdot F} + \frac{t\_2}{F}\right) \cdot \left(-F\right)\\ \mathbf{elif}\;F \leq -2.7 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(F, t\_1 \cdot t\_2, -1 \cdot \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{-79}:\\ \;\;\;\;\frac{F}{B} \cdot t\_1 - t\_0\\ \mathbf{elif}\;F \leq 0.0009:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_2 - t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B)))
        (t_1 (pow (fma x 2.0 (fma F F 2.0)) -0.5))
        (t_2 (/ 1.0 (sin B))))
   (if (<= F -44000.0)
     (* (+ (/ x (* B F)) (/ t_2 F)) (- F))
     (if (<= F -2.7e-46)
       (fma F (* t_1 t_2) (* -1.0 (/ x B)))
       (if (<= F 3.9e-79)
         (- (* (/ F B) t_1) t_0)
         (if (<= F 0.0009)
           (+
            (- (/ x B))
            (*
             (/ F (sin B))
             (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
           (- t_2 t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double t_1 = pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5);
	double t_2 = 1.0 / sin(B);
	double tmp;
	if (F <= -44000.0) {
		tmp = ((x / (B * F)) + (t_2 / F)) * -F;
	} else if (F <= -2.7e-46) {
		tmp = fma(F, (t_1 * t_2), (-1.0 * (x / B)));
	} else if (F <= 3.9e-79) {
		tmp = ((F / B) * t_1) - t_0;
	} else if (F <= 0.0009) {
		tmp = -(x / B) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = t_2 - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	t_1 = fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5
	t_2 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -44000.0)
		tmp = Float64(Float64(Float64(x / Float64(B * F)) + Float64(t_2 / F)) * Float64(-F));
	elseif (F <= -2.7e-46)
		tmp = fma(F, Float64(t_1 * t_2), Float64(-1.0 * Float64(x / B)));
	elseif (F <= 3.9e-79)
		tmp = Float64(Float64(Float64(F / B) * t_1) - t_0);
	elseif (F <= 0.0009)
		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)))));
	else
		tmp = Float64(t_2 - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -44000.0], N[(N[(N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 / F), $MachinePrecision]), $MachinePrecision] * (-F)), $MachinePrecision], If[LessEqual[F, -2.7e-46], N[(F * N[(t$95$1 * t$95$2), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.9e-79], N[(N[(N[(F / B), $MachinePrecision] * t$95$1), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.0009], 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], N[(t$95$2 - t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -44000

   1. Initial program 76.9%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.8%

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 33.7%

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

   7. Applied rewrites 33.7%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-*.f64 N/A

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

   9. Applied rewrites 33.7%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-/.f64 N/A

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

       5. lower-/.f64 33.8%

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

   11. Applied rewrites 33.8%

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

   if -44000 < F < -2.7e-46

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 85.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lower-*.f64 85.0%

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

      5. lift-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 85.0%

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 57.6%

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

   8. Applied rewrites 57.6%

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

   if -2.7e-46 < F < 3.9000000000000001e-79

   1. Initial program 76.9%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.3%

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

   4. Applied rewrites 62.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. lift-/.f64 N/A

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

      11. lower--.f64 N/A

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

   6. Applied rewrites 62.4%

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

   if 3.9000000000000001e-79 < F < 8.9999999999999998e-4

   1. Initial program 76.9%

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

   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 49.7%

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

   4. Applied rewrites 49.7%

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

   if 8.9999999999999998e-4 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 85.0%

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

   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 55.2%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-lft-identity N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 55.2%

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

   8. Applied rewrites 55.2%

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

Alternative 5: 85.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}\\ t_2 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -44000:\\ \;\;\;\;\left(\frac{x}{B \cdot F} + \frac{t\_2}{F}\right) \cdot \left(-F\right)\\ \mathbf{elif}\;F \leq -2.7 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(F, t\_1 \cdot t\_2, -1 \cdot \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{F}{B} \cdot t\_1 - t\_0\\ \mathbf{elif}\;F \leq 0.0009:\\ \;\;\;\;\frac{F \cdot {\left(2 + {F}^{2}\right)}^{-0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_2 - t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B)))
        (t_1 (pow (fma x 2.0 (fma F F 2.0)) -0.5))
        (t_2 (/ 1.0 (sin B))))
   (if (<= F -44000.0)
     (* (+ (/ x (* B F)) (/ t_2 F)) (- F))
     (if (<= F -2.7e-46)
       (fma F (* t_1 t_2) (* -1.0 (/ x B)))
       (if (<= F 6.8e-78)
         (- (* (/ F B) t_1) t_0)
         (if (<= F 0.0009)
           (/ (* F (pow (+ 2.0 (pow F 2.0)) -0.5)) (sin B))
           (- t_2 t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double t_1 = pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5);
	double t_2 = 1.0 / sin(B);
	double tmp;
	if (F <= -44000.0) {
		tmp = ((x / (B * F)) + (t_2 / F)) * -F;
	} else if (F <= -2.7e-46) {
		tmp = fma(F, (t_1 * t_2), (-1.0 * (x / B)));
	} else if (F <= 6.8e-78) {
		tmp = ((F / B) * t_1) - t_0;
	} else if (F <= 0.0009) {
		tmp = (F * pow((2.0 + pow(F, 2.0)), -0.5)) / sin(B);
	} else {
		tmp = t_2 - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	t_1 = fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5
	t_2 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -44000.0)
		tmp = Float64(Float64(Float64(x / Float64(B * F)) + Float64(t_2 / F)) * Float64(-F));
	elseif (F <= -2.7e-46)
		tmp = fma(F, Float64(t_1 * t_2), Float64(-1.0 * Float64(x / B)));
	elseif (F <= 6.8e-78)
		tmp = Float64(Float64(Float64(F / B) * t_1) - t_0);
	elseif (F <= 0.0009)
		tmp = Float64(Float64(F * (Float64(2.0 + (F ^ 2.0)) ^ -0.5)) / sin(B));
	else
		tmp = Float64(t_2 - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -44000.0], N[(N[(N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 / F), $MachinePrecision]), $MachinePrecision] * (-F)), $MachinePrecision], If[LessEqual[F, -2.7e-46], N[(F * N[(t$95$1 * t$95$2), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.8e-78], N[(N[(N[(F / B), $MachinePrecision] * t$95$1), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.0009], N[(N[(F * N[Power[N[(2.0 + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(t$95$2 - t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -44000

   1. Initial program 76.9%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.8%

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 33.7%

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

   7. Applied rewrites 33.7%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-*.f64 N/A

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

   9. Applied rewrites 33.7%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-/.f64 N/A

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

       5. lower-/.f64 33.8%

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

   11. Applied rewrites 33.8%

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

   if -44000 < F < -2.7e-46

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 85.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lower-*.f64 85.0%

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

      5. lift-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 85.0%

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 57.6%

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

   8. Applied rewrites 57.6%

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

   if -2.7e-46 < F < 6.8000000000000002e-78

   1. Initial program 76.9%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.3%

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

   4. Applied rewrites 62.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. lift-/.f64 N/A

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

      11. lower--.f64 N/A

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

   6. Applied rewrites 62.4%

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

   if 6.8000000000000002e-78 < F < 8.9999999999999998e-4

   1. Initial program 76.9%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{\sin B}} \]
   3. 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 31.0%

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

   4. Applied rewrites 31.0%

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

   if 8.9999999999999998e-4 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 85.0%

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

   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 55.2%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-lft-identity N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 55.2%

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

   8. Applied rewrites 55.2%

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

Alternative 6: 79.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}\\ t_1 := -1 \cdot \frac{x}{B}\\ t_2 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -44000:\\ \;\;\;\;\left(\frac{x}{B \cdot F} + \frac{t\_2}{F}\right) \cdot \left(-F\right)\\ \mathbf{elif}\;F \leq -2.7 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(F, t\_0 \cdot t\_2, t\_1\right)\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{F}{B} \cdot t\_0 - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.0009:\\ \;\;\;\;\frac{F \cdot {\left(2 + {F}^{2}\right)}^{-0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, 1, t\_1\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (fma x 2.0 (fma F F 2.0)) -0.5))
        (t_1 (* -1.0 (/ x B)))
        (t_2 (/ 1.0 (sin B))))
   (if (<= F -44000.0)
     (* (+ (/ x (* B F)) (/ t_2 F)) (- F))
     (if (<= F -2.7e-46)
       (fma F (* t_0 t_2) t_1)
       (if (<= F 6.8e-78)
         (- (* (/ F B) t_0) (/ x (tan B)))
         (if (<= F 0.0009)
           (/ (* F (pow (+ 2.0 (pow F 2.0)) -0.5)) (sin B))
           (fma t_2 1.0 t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5);
	double t_1 = -1.0 * (x / B);
	double t_2 = 1.0 / sin(B);
	double tmp;
	if (F <= -44000.0) {
		tmp = ((x / (B * F)) + (t_2 / F)) * -F;
	} else if (F <= -2.7e-46) {
		tmp = fma(F, (t_0 * t_2), t_1);
	} else if (F <= 6.8e-78) {
		tmp = ((F / B) * t_0) - (x / tan(B));
	} else if (F <= 0.0009) {
		tmp = (F * pow((2.0 + pow(F, 2.0)), -0.5)) / sin(B);
	} else {
		tmp = fma(t_2, 1.0, t_1);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5
	t_1 = Float64(-1.0 * Float64(x / B))
	t_2 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -44000.0)
		tmp = Float64(Float64(Float64(x / Float64(B * F)) + Float64(t_2 / F)) * Float64(-F));
	elseif (F <= -2.7e-46)
		tmp = fma(F, Float64(t_0 * t_2), t_1);
	elseif (F <= 6.8e-78)
		tmp = Float64(Float64(Float64(F / B) * t_0) - Float64(x / tan(B)));
	elseif (F <= 0.0009)
		tmp = Float64(Float64(F * (Float64(2.0 + (F ^ 2.0)) ^ -0.5)) / sin(B));
	else
		tmp = fma(t_2, 1.0, t_1);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -44000.0], N[(N[(N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 / F), $MachinePrecision]), $MachinePrecision] * (-F)), $MachinePrecision], If[LessEqual[F, -2.7e-46], N[(F * N[(t$95$0 * t$95$2), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[F, 6.8e-78], N[(N[(N[(F / B), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0009], N[(N[(F * N[Power[N[(2.0 + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(t$95$2 * 1.0 + t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -44000

   1. Initial program 76.9%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.8%

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 33.7%

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

   7. Applied rewrites 33.7%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-*.f64 N/A

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

   9. Applied rewrites 33.7%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-/.f64 N/A

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

       5. lower-/.f64 33.8%

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

   11. Applied rewrites 33.8%

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

   if -44000 < F < -2.7e-46

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 85.0%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-/.f64 N/A

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

      4. lower-*.f64 85.0%

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

      5. lift-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 85.0%

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 57.6%

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

   8. Applied rewrites 57.6%

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

   if -2.7e-46 < F < 6.8000000000000002e-78

   1. Initial program 76.9%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.3%

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

   4. Applied rewrites 62.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. lift-/.f64 N/A

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

      11. lower--.f64 N/A

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

   6. Applied rewrites 62.4%

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

   if 6.8000000000000002e-78 < F < 8.9999999999999998e-4

   1. Initial program 76.9%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{\sin B}} \]
   3. 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 31.0%

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

   4. Applied rewrites 31.0%

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

   if 8.9999999999999998e-4 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 85.0%

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

   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 55.2%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 35.8%

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

   9. Applied rewrites 35.8%

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

Alternative 7: 78.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := -1 \cdot \frac{x}{B}\\ t_1 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -44000:\\ \;\;\;\;\left(\frac{x}{B \cdot F} + \frac{t\_1}{F}\right) \cdot \left(-F\right)\\ \mathbf{elif}\;F \leq -2.7 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, t\_0\right)\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{F}{B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.0009:\\ \;\;\;\;\frac{F \cdot {\left(2 + {F}^{2}\right)}^{-0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, 1, t\_0\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* -1.0 (/ x B))) (t_1 (/ 1.0 (sin B))))
   (if (<= F -44000.0)
     (* (+ (/ x (* B F)) (/ t_1 F)) (- F))
     (if (<= F -2.7e-46)
       (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B)) t_0)
       (if (<= F 6.8e-78)
         (- (* (/ F B) (pow (fma x 2.0 (fma F F 2.0)) -0.5)) (/ x (tan B)))
         (if (<= F 0.0009)
           (/ (* F (pow (+ 2.0 (pow F 2.0)) -0.5)) (sin B))
           (fma t_1 1.0 t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 * (x / B);
	double t_1 = 1.0 / sin(B);
	double tmp;
	if (F <= -44000.0) {
		tmp = ((x / (B * F)) + (t_1 / F)) * -F;
	} else if (F <= -2.7e-46) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), t_0);
	} else if (F <= 6.8e-78) {
		tmp = ((F / B) * pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5)) - (x / tan(B));
	} else if (F <= 0.0009) {
		tmp = (F * pow((2.0 + pow(F, 2.0)), -0.5)) / sin(B);
	} else {
		tmp = fma(t_1, 1.0, t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 * Float64(x / B))
	t_1 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -44000.0)
		tmp = Float64(Float64(Float64(x / Float64(B * F)) + Float64(t_1 / F)) * Float64(-F));
	elseif (F <= -2.7e-46)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), t_0);
	elseif (F <= 6.8e-78)
		tmp = Float64(Float64(Float64(F / B) * (fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5)) - Float64(x / tan(B)));
	elseif (F <= 0.0009)
		tmp = Float64(Float64(F * (Float64(2.0 + (F ^ 2.0)) ^ -0.5)) / sin(B));
	else
		tmp = fma(t_1, 1.0, t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -44000.0], N[(N[(N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 / F), $MachinePrecision]), $MachinePrecision] * (-F)), $MachinePrecision], If[LessEqual[F, -2.7e-46], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, 6.8e-78], N[(N[(N[(F / B), $MachinePrecision] * N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0009], N[(N[(F * N[Power[N[(2.0 + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * 1.0 + t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -44000

   1. Initial program 76.9%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.8%

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 33.7%

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

   7. Applied rewrites 33.7%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-*.f64 N/A

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

   9. Applied rewrites 33.7%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-/.f64 N/A

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

       5. lower-/.f64 33.8%

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

   11. Applied rewrites 33.8%

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

   if -44000 < F < -2.7e-46

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 85.0%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 57.6%

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

   6. Applied rewrites 57.6%

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

   if -2.7e-46 < F < 6.8000000000000002e-78

   1. Initial program 76.9%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.3%

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

   4. Applied rewrites 62.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

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

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip N/A

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

      10. lift-/.f64 N/A

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

      11. lower--.f64 N/A

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

   6. Applied rewrites 62.4%

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

   if 6.8000000000000002e-78 < F < 8.9999999999999998e-4

   1. Initial program 76.9%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{\sin B}} \]
   3. 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 31.0%

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

   4. Applied rewrites 31.0%

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

   if 8.9999999999999998e-4 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 85.0%

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

   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 55.2%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 35.8%

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

   9. Applied rewrites 35.8%

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

Alternative 8: 78.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := -1 \cdot \frac{x}{B}\\ t_1 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -44000:\\ \;\;\;\;\left(\frac{x}{B \cdot F} + \frac{t\_1}{F}\right) \cdot \left(-F\right)\\ \mathbf{elif}\;F \leq -1.72 \cdot 10^{-46}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, t\_0\right)\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{-78}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\tan B}, x, \frac{F}{B} \cdot {\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5}\right)\\ \mathbf{elif}\;F \leq 0.0009:\\ \;\;\;\;\frac{F \cdot {\left(2 + {F}^{2}\right)}^{-0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, 1, t\_0\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* -1.0 (/ x B))) (t_1 (/ 1.0 (sin B))))
   (if (<= F -44000.0)
     (* (+ (/ x (* B F)) (/ t_1 F)) (- F))
     (if (<= F -1.72e-46)
       (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B)) t_0)
       (if (<= F 6.8e-78)
         (fma (/ -1.0 (tan B)) x (* (/ F B) (pow (fma x 2.0 2.0) -0.5)))
         (if (<= F 0.0009)
           (/ (* F (pow (+ 2.0 (pow F 2.0)) -0.5)) (sin B))
           (fma t_1 1.0 t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 * (x / B);
	double t_1 = 1.0 / sin(B);
	double tmp;
	if (F <= -44000.0) {
		tmp = ((x / (B * F)) + (t_1 / F)) * -F;
	} else if (F <= -1.72e-46) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), t_0);
	} else if (F <= 6.8e-78) {
		tmp = fma((-1.0 / tan(B)), x, ((F / B) * pow(fma(x, 2.0, 2.0), -0.5)));
	} else if (F <= 0.0009) {
		tmp = (F * pow((2.0 + pow(F, 2.0)), -0.5)) / sin(B);
	} else {
		tmp = fma(t_1, 1.0, t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 * Float64(x / B))
	t_1 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -44000.0)
		tmp = Float64(Float64(Float64(x / Float64(B * F)) + Float64(t_1 / F)) * Float64(-F));
	elseif (F <= -1.72e-46)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), t_0);
	elseif (F <= 6.8e-78)
		tmp = fma(Float64(-1.0 / tan(B)), x, Float64(Float64(F / B) * (fma(x, 2.0, 2.0) ^ -0.5)));
	elseif (F <= 0.0009)
		tmp = Float64(Float64(F * (Float64(2.0 + (F ^ 2.0)) ^ -0.5)) / sin(B));
	else
		tmp = fma(t_1, 1.0, t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -44000.0], N[(N[(N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 / F), $MachinePrecision]), $MachinePrecision] * (-F)), $MachinePrecision], If[LessEqual[F, -1.72e-46], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, 6.8e-78], N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x + N[(N[(F / B), $MachinePrecision] * N[Power[N[(x * 2.0 + 2.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0009], N[(N[(F * N[Power[N[(2.0 + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * 1.0 + t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -44000

   1. Initial program 76.9%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.8%

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 33.7%

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

   7. Applied rewrites 33.7%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-*.f64 N/A

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

   9. Applied rewrites 33.7%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-/.f64 N/A

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

       5. lower-/.f64 33.8%

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

   11. Applied rewrites 33.8%

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

   if -44000 < F < -1.7199999999999999e-46

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 85.0%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 57.6%

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

   6. Applied rewrites 57.6%

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

   if -1.7199999999999999e-46 < F < 6.8000000000000002e-78

   1. Initial program 76.9%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.3%

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

   4. Applied rewrites 62.3%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 62.3%

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

      11. lift-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval 62.3%

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

   6. Applied rewrites 62.3%

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

   7. Taylor expanded in F around 0

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

   9. Applied rewrites 48.9%

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

   if 6.8000000000000002e-78 < F < 8.9999999999999998e-4

   1. Initial program 76.9%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{\sin B}} \]
   3. 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 31.0%

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

   4. Applied rewrites 31.0%

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

   if 8.9999999999999998e-4 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 85.0%

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

   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 55.2%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 35.8%

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

   9. Applied rewrites 35.8%

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

Alternative 9: 78.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := -1 \cdot \frac{x}{B}\\ t_1 := \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, t\_0\right)\\ t_2 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -44000:\\ \;\;\;\;\left(\frac{x}{B \cdot F} + \frac{t\_2}{F}\right) \cdot \left(-F\right)\\ \mathbf{elif}\;F \leq -1.72 \cdot 10^{-46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{-79}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\tan B}, x, \frac{F}{B} \cdot {\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5}\right)\\ \mathbf{elif}\;F \leq 0.0009:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_2, 1, t\_0\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* -1.0 (/ x B)))
        (t_1 (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B)) t_0))
        (t_2 (/ 1.0 (sin B))))
   (if (<= F -44000.0)
     (* (+ (/ x (* B F)) (/ t_2 F)) (- F))
     (if (<= F -1.72e-46)
       t_1
       (if (<= F 3.9e-79)
         (fma (/ -1.0 (tan B)) x (* (/ F B) (pow (fma x 2.0 2.0) -0.5)))
         (if (<= F 0.0009) t_1 (fma t_2 1.0 t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 * (x / B);
	double t_1 = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), t_0);
	double t_2 = 1.0 / sin(B);
	double tmp;
	if (F <= -44000.0) {
		tmp = ((x / (B * F)) + (t_2 / F)) * -F;
	} else if (F <= -1.72e-46) {
		tmp = t_1;
	} else if (F <= 3.9e-79) {
		tmp = fma((-1.0 / tan(B)), x, ((F / B) * pow(fma(x, 2.0, 2.0), -0.5)));
	} else if (F <= 0.0009) {
		tmp = t_1;
	} else {
		tmp = fma(t_2, 1.0, t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 * Float64(x / B))
	t_1 = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), t_0)
	t_2 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -44000.0)
		tmp = Float64(Float64(Float64(x / Float64(B * F)) + Float64(t_2 / F)) * Float64(-F));
	elseif (F <= -1.72e-46)
		tmp = t_1;
	elseif (F <= 3.9e-79)
		tmp = fma(Float64(-1.0 / tan(B)), x, Float64(Float64(F / B) * (fma(x, 2.0, 2.0) ^ -0.5)));
	elseif (F <= 0.0009)
		tmp = t_1;
	else
		tmp = fma(t_2, 1.0, t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -44000.0], N[(N[(N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 / F), $MachinePrecision]), $MachinePrecision] * (-F)), $MachinePrecision], If[LessEqual[F, -1.72e-46], t$95$1, If[LessEqual[F, 3.9e-79], N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x + N[(N[(F / B), $MachinePrecision] * N[Power[N[(x * 2.0 + 2.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0009], t$95$1, N[(t$95$2 * 1.0 + t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -44000

   1. Initial program 76.9%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.8%

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 33.7%

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

   7. Applied rewrites 33.7%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-*.f64 N/A

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

   9. Applied rewrites 33.7%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-/.f64 N/A

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

       5. lower-/.f64 33.8%

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

   11. Applied rewrites 33.8%

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

   if -44000 < F < -1.7199999999999999e-46 or 3.9000000000000001e-79 < F < 8.9999999999999998e-4

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 85.0%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 57.6%

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

   6. Applied rewrites 57.6%

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

   if -1.7199999999999999e-46 < F < 3.9000000000000001e-79

   1. Initial program 76.9%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.3%

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

   4. Applied rewrites 62.3%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 62.3%

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

      11. lift-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval 62.3%

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

   6. Applied rewrites 62.3%

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

   7. Taylor expanded in F around 0

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

   9. Applied rewrites 48.9%

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

   if 8.9999999999999998e-4 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 85.0%

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

   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 55.2%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 35.8%

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

   9. Applied rewrites 35.8%

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

Alternative 10: 78.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -580000000:\\ \;\;\;\;\left(\frac{x}{B \cdot F} + \frac{t\_0}{F}\right) \cdot \left(-F\right)\\ \mathbf{elif}\;F \leq 0.0076:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\tan B}, x, \frac{F}{B} \cdot {\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 1, -1 \cdot \frac{x}{B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))))
   (if (<= F -580000000.0)
     (* (+ (/ x (* B F)) (/ t_0 F)) (- F))
     (if (<= F 0.0076)
       (fma (/ -1.0 (tan B)) x (* (/ F B) (pow (fma x 2.0 2.0) -0.5)))
       (fma t_0 1.0 (* -1.0 (/ x B)))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double tmp;
	if (F <= -580000000.0) {
		tmp = ((x / (B * F)) + (t_0 / F)) * -F;
	} else if (F <= 0.0076) {
		tmp = fma((-1.0 / tan(B)), x, ((F / B) * pow(fma(x, 2.0, 2.0), -0.5)));
	} else {
		tmp = fma(t_0, 1.0, (-1.0 * (x / B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -580000000.0)
		tmp = Float64(Float64(Float64(x / Float64(B * F)) + Float64(t_0 / F)) * Float64(-F));
	elseif (F <= 0.0076)
		tmp = fma(Float64(-1.0 / tan(B)), x, Float64(Float64(F / B) * (fma(x, 2.0, 2.0) ^ -0.5)));
	else
		tmp = fma(t_0, 1.0, Float64(-1.0 * Float64(x / B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -580000000.0], N[(N[(N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / F), $MachinePrecision]), $MachinePrecision] * (-F)), $MachinePrecision], If[LessEqual[F, 0.0076], N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x + N[(N[(F / B), $MachinePrecision] * N[Power[N[(x * 2.0 + 2.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 1.0 + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.8e8

   1. Initial program 76.9%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.8%

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 33.7%

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

   7. Applied rewrites 33.7%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-*.f64 N/A

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

   9. Applied rewrites 33.7%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-/.f64 N/A

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

       5. lower-/.f64 33.8%

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

   11. Applied rewrites 33.8%

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

   if -5.8e8 < F < 0.0076

   1. Initial program 76.9%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.3%

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

   4. Applied rewrites 62.3%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 62.3%

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

      11. lift-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval 62.3%

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

   6. Applied rewrites 62.3%

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

   7. Taylor expanded in F around 0

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

   9. Applied rewrites 48.9%

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

   if 0.0076 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 85.0%

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

   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 55.2%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 35.8%

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

   9. Applied rewrites 35.8%

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

Alternative 11: 64.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -185000000:\\ \;\;\;\;\left(\frac{x}{B \cdot F} + \frac{t\_0}{F}\right) \cdot \left(-F\right)\\ \mathbf{elif}\;F \leq 0.0009:\\ \;\;\;\;\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 1, -1 \cdot \frac{x}{B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))))
   (if (<= F -185000000.0)
     (* (+ (/ x (* B F)) (/ t_0 F)) (- F))
     (if (<= F 0.0009)
       (/ (- (* F (pow (+ 2.0 (fma 2.0 x (pow F 2.0))) -0.5)) x) B)
       (fma t_0 1.0 (* -1.0 (/ x B)))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double tmp;
	if (F <= -185000000.0) {
		tmp = ((x / (B * F)) + (t_0 / F)) * -F;
	} else if (F <= 0.0009) {
		tmp = ((F * pow((2.0 + fma(2.0, x, pow(F, 2.0))), -0.5)) - x) / B;
	} else {
		tmp = fma(t_0, 1.0, (-1.0 * (x / B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -185000000.0)
		tmp = Float64(Float64(Float64(x / Float64(B * F)) + Float64(t_0 / F)) * Float64(-F));
	elseif (F <= 0.0009)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + fma(2.0, x, (F ^ 2.0))) ^ -0.5)) - x) / B);
	else
		tmp = fma(t_0, 1.0, Float64(-1.0 * Float64(x / B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -185000000.0], N[(N[(N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 / F), $MachinePrecision]), $MachinePrecision] * (-F)), $MachinePrecision], If[LessEqual[F, 0.0009], N[(N[(N[(F * N[Power[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(t$95$0 * 1.0 + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.85e8

   1. Initial program 76.9%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.8%

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 33.7%

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

   7. Applied rewrites 33.7%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-*.f64 N/A

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

   9. Applied rewrites 33.7%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. associate-/r* N/A

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

       4. lift-/.f64 N/A

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

       5. lower-/.f64 33.8%

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

   11. Applied rewrites 33.8%

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

   if -1.85e8 < F < 8.9999999999999998e-4

   1. Initial program 76.9%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 43.7%

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

   if 8.9999999999999998e-4 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 85.0%

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

   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 55.2%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 35.8%

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

   9. Applied rewrites 35.8%

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

Alternative 12: 64.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -185000000:\\ \;\;\;\;\left(\frac{x}{B \cdot F} + \frac{\frac{1}{F}}{\sin B}\right) \cdot \left(-F\right)\\ \mathbf{elif}\;F \leq 0.0009:\\ \;\;\;\;\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, 1, -1 \cdot \frac{x}{B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -185000000.0)
   (* (+ (/ x (* B F)) (/ (/ 1.0 F) (sin B))) (- F))
   (if (<= F 0.0009)
     (/ (- (* F (pow (+ 2.0 (fma 2.0 x (pow F 2.0))) -0.5)) x) B)
     (fma (/ 1.0 (sin B)) 1.0 (* -1.0 (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -185000000.0) {
		tmp = ((x / (B * F)) + ((1.0 / F) / sin(B))) * -F;
	} else if (F <= 0.0009) {
		tmp = ((F * pow((2.0 + fma(2.0, x, pow(F, 2.0))), -0.5)) - x) / B;
	} else {
		tmp = fma((1.0 / sin(B)), 1.0, (-1.0 * (x / B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -185000000.0)
		tmp = Float64(Float64(Float64(x / Float64(B * F)) + Float64(Float64(1.0 / F) / sin(B))) * Float64(-F));
	elseif (F <= 0.0009)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + fma(2.0, x, (F ^ 2.0))) ^ -0.5)) - x) / B);
	else
		tmp = fma(Float64(1.0 / sin(B)), 1.0, Float64(-1.0 * Float64(x / B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -185000000.0], N[(N[(N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-F)), $MachinePrecision], If[LessEqual[F, 0.0009], N[(N[(N[(F * N[Power[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * 1.0 + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.85e8

   1. Initial program 76.9%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.8%

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 33.7%

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

   7. Applied rewrites 33.7%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-*.f64 N/A

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

   9. Applied rewrites 33.7%

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

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. *-commutative N/A

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

       4. associate-/r* N/A

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

       5. lift-/.f64 N/A

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

       6. lower-/.f64 33.7%

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

   11. Applied rewrites 33.7%

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

   if -1.85e8 < F < 8.9999999999999998e-4

   1. Initial program 76.9%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 43.7%

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

   if 8.9999999999999998e-4 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 85.0%

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

   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 55.2%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 35.8%

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

   9. Applied rewrites 35.8%

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

Alternative 13: 64.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -185000000:\\ \;\;\;\;\left(\frac{x}{B \cdot F} - \frac{-1}{\sin B \cdot F}\right) \cdot \left(-F\right)\\ \mathbf{elif}\;F \leq 0.0009:\\ \;\;\;\;\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, 1, -1 \cdot \frac{x}{B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -185000000.0)
   (* (- (/ x (* B F)) (/ -1.0 (* (sin B) F))) (- F))
   (if (<= F 0.0009)
     (/ (- (* F (pow (+ 2.0 (fma 2.0 x (pow F 2.0))) -0.5)) x) B)
     (fma (/ 1.0 (sin B)) 1.0 (* -1.0 (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -185000000.0) {
		tmp = ((x / (B * F)) - (-1.0 / (sin(B) * F))) * -F;
	} else if (F <= 0.0009) {
		tmp = ((F * pow((2.0 + fma(2.0, x, pow(F, 2.0))), -0.5)) - x) / B;
	} else {
		tmp = fma((1.0 / sin(B)), 1.0, (-1.0 * (x / B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -185000000.0)
		tmp = Float64(Float64(Float64(x / Float64(B * F)) - Float64(-1.0 / Float64(sin(B) * F))) * Float64(-F));
	elseif (F <= 0.0009)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + fma(2.0, x, (F ^ 2.0))) ^ -0.5)) - x) / B);
	else
		tmp = fma(Float64(1.0 / sin(B)), 1.0, Float64(-1.0 * Float64(x / B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -185000000.0], N[(N[(N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision] - N[(-1.0 / N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * (-F)), $MachinePrecision], If[LessEqual[F, 0.0009], N[(N[(N[(F * N[Power[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * 1.0 + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.85e8

   1. Initial program 76.9%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.8%

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

   4. Applied rewrites 49.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 33.7%

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

   7. Applied rewrites 33.7%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-*.f64 N/A

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

   9. Applied rewrites 33.7%

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

       1. lift-+.f64 N/A

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

       2. add-flip N/A

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

       3. lower--.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. *-commutative N/A

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

       7. lift-sin.f64 N/A

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

       8. distribute-neg-frac N/A

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

       9. metadata-eval N/A

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

       10. lower-/.f64 N/A

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

       11. lift-sin.f64 N/A

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

       12. *-commutative N/A

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

       13. lift-*.f64 33.7%

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

   11. Applied rewrites 33.7%

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

   if -1.85e8 < F < 8.9999999999999998e-4

   1. Initial program 76.9%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 43.7%

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

   if 8.9999999999999998e-4 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 85.0%

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

   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 55.2%

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, \color{blue}{1}, \frac{-x}{\tan B}\right) \]

   7. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, 1, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, 1, -1 \cdot \color{blue}{\frac{x}{B}}\right) \]

      2. lower-/.f64 35.8%

         \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, 1, -1 \cdot \frac{x}{\color{blue}{B}}\right) \]

   9. Applied rewrites 35.8%

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, 1, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 59.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1360000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 0.0009:\\ \;\;\;\;\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, 1, -1 \cdot \frac{x}{B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1360000.0)
   (/ -1.0 (sin B))
   (if (<= F 0.0009)
     (/ (- (* F (pow (+ 2.0 (fma 2.0 x (pow F 2.0))) -0.5)) x) B)
     (fma (/ 1.0 (sin B)) 1.0 (* -1.0 (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1360000.0) {
		tmp = -1.0 / sin(B);
	} else if (F <= 0.0009) {
		tmp = ((F * pow((2.0 + fma(2.0, x, pow(F, 2.0))), -0.5)) - x) / B;
	} else {
		tmp = fma((1.0 / sin(B)), 1.0, (-1.0 * (x / B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1360000.0)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 0.0009)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + fma(2.0, x, (F ^ 2.0))) ^ -0.5)) - x) / B);
	else
		tmp = fma(Float64(1.0 / sin(B)), 1.0, Float64(-1.0 * Float64(x / B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1360000.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0009], N[(N[(N[(F * N[Power[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * 1.0 + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.36e6

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.5%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.5%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -1.36e6 < F < 8.9999999999999998e-4

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   if 8.9999999999999998e-4 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 85.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F, \frac{-x}{\tan B}\right)} \]

   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 55.2%

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, \color{blue}{1}, \frac{-x}{\tan B}\right) \]

   7. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, 1, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, 1, -1 \cdot \color{blue}{\frac{x}{B}}\right) \]

      2. lower-/.f64 35.8%

         \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, 1, -1 \cdot \frac{x}{\color{blue}{B}}\right) \]

   9. Applied rewrites 35.8%

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, 1, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 59.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1360000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 0.0009:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, 1, -1 \cdot \frac{x}{B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1360000.0)
   (/ -1.0 (sin B))
   (if (<= F 0.0009)
     (+
      (- (/ x B))
      (* (/ F B) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
     (fma (/ 1.0 (sin B)) 1.0 (* -1.0 (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1360000.0) {
		tmp = -1.0 / sin(B);
	} else if (F <= 0.0009) {
		tmp = -(x / B) + ((F / B) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = fma((1.0 / sin(B)), 1.0, (-1.0 * (x / B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1360000.0)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 0.0009)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F / B) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))));
	else
		tmp = fma(Float64(1.0 / sin(B)), 1.0, Float64(-1.0 * Float64(x / B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1360000.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0009], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F / B), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * 1.0 + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.36e6

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.5%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.5%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -1.36e6 < F < 8.9999999999999998e-4

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 62.3%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\color{blue}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 62.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. lower-/.f64 35.7%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   7. Applied rewrites 35.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if 8.9999999999999998e-4 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 85.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sin B}, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F, \frac{-x}{\tan B}\right)} \]

   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 55.2%

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, \color{blue}{1}, \frac{-x}{\tan B}\right) \]

   7. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, 1, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, 1, -1 \cdot \color{blue}{\frac{x}{B}}\right) \]

      2. lower-/.f64 35.8%

         \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, 1, -1 \cdot \frac{x}{\color{blue}{B}}\right) \]

   9. Applied rewrites 35.8%

      \[\leadsto \mathsf{fma}\left(\frac{1}{\sin B}, 1, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 52.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1360000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{+25}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1360000.0)
   (/ -1.0 (sin B))
   (if (<= F 2.4e+25)
     (+
      (- (/ x B))
      (* (/ F B) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1360000.0) {
		tmp = -1.0 / sin(B);
	} else if (F <= 2.4e+25) {
		tmp = -(x / B) + ((F / B) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1360000.0d0)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 2.4d+25) then
        tmp = -(x / b) + ((f / b) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1360000.0) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 2.4e+25) {
		tmp = -(x / B) + ((F / B) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1360000.0:
		tmp = -1.0 / math.sin(B)
	elif F <= 2.4e+25:
		tmp = -(x / B) + ((F / B) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1360000.0)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 2.4e+25)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F / B) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1360000.0)
		tmp = -1.0 / sin(B);
	elseif (F <= 2.4e+25)
		tmp = -(x / B) + ((F / B) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1360000.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.4e+25], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F / B), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.36e6

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.5%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.5%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -1.36e6 < F < 2.4e25

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 62.3%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\color{blue}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 62.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. lower-/.f64 35.7%

         \[\leadsto \left(-\frac{x}{\color{blue}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   7. Applied rewrites 35.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if 2.4e25 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   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 16.7%

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 16.7%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 52.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -920000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}, \frac{F}{B}, \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot 0.3333333333333333 - x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -920000000.0)
   (/ -1.0 (sin B))
   (if (<= F 2.4e+25)
     (fma
      (/ 1.0 (sqrt (fma 2.0 x (fma F F 2.0))))
      (/ F B)
      (/ (- (* (* (* B B) x) 0.3333333333333333) x) B))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -920000000.0) {
		tmp = -1.0 / sin(B);
	} else if (F <= 2.4e+25) {
		tmp = fma((1.0 / sqrt(fma(2.0, x, fma(F, F, 2.0)))), (F / B), (((((B * B) * x) * 0.3333333333333333) - x) / B));
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -920000000.0)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 2.4e+25)
		tmp = fma(Float64(1.0 / sqrt(fma(2.0, x, fma(F, F, 2.0)))), Float64(F / B), Float64(Float64(Float64(Float64(Float64(B * B) * x) * 0.3333333333333333) - x) / B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -920000000.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.4e+25], N[(N[(1.0 / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(F / B), $MachinePrecision] + N[(N[(N[(N[(N[(B * B), $MachinePrecision] * x), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.2e8

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.5%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.5%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -9.2e8 < F < 2.4e25

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 62.3%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\color{blue}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 62.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{\color{blue}{B}} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\frac{1}{3} \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      5. lower-pow.f64 35.6%

         \[\leadsto \frac{0.3333333333333333 \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   7. Applied rewrites 35.6%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   9. Applied rewrites 35.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}, \frac{F}{B}, \frac{\left(\left(B \cdot B\right) \cdot x\right) \cdot 0.3333333333333333 - x}{B}\right)} \]

   if 2.4e25 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   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 16.7%

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 16.7%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 44.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -370:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 9.4 \cdot 10^{+26}:\\ \;\;\;\;-1 \cdot \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -370.0)
   (/ -1.0 (sin B))
   (if (<= F 9.4e+26) (* -1.0 (/ x B)) (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -370.0) {
		tmp = -1.0 / sin(B);
	} else if (F <= 9.4e+26) {
		tmp = -1.0 * (x / B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-370.0d0)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 9.4d+26) then
        tmp = (-1.0d0) * (x / b)
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -370.0) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 9.4e+26) {
		tmp = -1.0 * (x / B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -370.0:
		tmp = -1.0 / math.sin(B)
	elif F <= 9.4e+26:
		tmp = -1.0 * (x / B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -370.0)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 9.4e+26)
		tmp = Float64(-1.0 * Float64(x / B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -370.0)
		tmp = -1.0 / sin(B);
	elseif (F <= 9.4e+26)
		tmp = -1.0 * (x / B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -370.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.4e+26], N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -370

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.5%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.5%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -370 < F < 9.3999999999999995e26

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \color{blue}{\left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{\color{blue}{x \cdot \cos B}}{F \cdot \sin B}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \color{blue}{\cos B}}{F \cdot \sin B}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}\right)\right) \]

      11. lower-sin.f64 49.8%

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

   4. Applied rewrites 49.8%

      \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      5. lower-/.f64 28.4%

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

   7. Applied rewrites 28.4%

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{\color{blue}{B}} \]

   8. Taylor expanded in x around inf

      \[\leadsto -1 \cdot \frac{x}{B} \]
   9. Step-by-step derivation

      1. lower-/.f64 28.8%

         \[\leadsto -1 \cdot \frac{x}{B} \]

   10. Applied rewrites 28.8%

       \[\leadsto -1 \cdot \frac{x}{B} \]

   if 9.3999999999999995e26 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   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 16.7%

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 16.7%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 37.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -370:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -370.0) (/ -1.0 (sin B)) (* -1.0 (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -370.0) {
		tmp = -1.0 / sin(B);
	} else {
		tmp = -1.0 * (x / B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-370.0d0)) then
        tmp = (-1.0d0) / sin(b)
    else
        tmp = (-1.0d0) * (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -370.0) {
		tmp = -1.0 / Math.sin(B);
	} else {
		tmp = -1.0 * (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -370.0:
		tmp = -1.0 / math.sin(B)
	else:
		tmp = -1.0 * (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -370.0)
		tmp = Float64(-1.0 / sin(B));
	else
		tmp = Float64(-1.0 * Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -370.0)
		tmp = -1.0 / sin(B);
	else
		tmp = -1.0 * (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -370.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -370

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.5%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.5%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -370 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \color{blue}{\left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{\color{blue}{x \cdot \cos B}}{F \cdot \sin B}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \color{blue}{\cos B}}{F \cdot \sin B}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}\right)\right) \]

      11. lower-sin.f64 49.8%

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

   4. Applied rewrites 49.8%

      \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      5. lower-/.f64 28.4%

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

   7. Applied rewrites 28.4%

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{\color{blue}{B}} \]

   8. Taylor expanded in x around inf

      \[\leadsto -1 \cdot \frac{x}{B} \]
   9. Step-by-step derivation

      1. lower-/.f64 28.8%

         \[\leadsto -1 \cdot \frac{x}{B} \]

   10. Applied rewrites 28.8%

       \[\leadsto -1 \cdot \frac{x}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 36.3% accurate, 9.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{-56}:\\ \;\;\;\;-\frac{x + 1}{B}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.2e-56) (- (/ (+ x 1.0) B)) (* -1.0 (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.2e-56) {
		tmp = -((x + 1.0) / B);
	} else {
		tmp = -1.0 * (x / B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-5.2d-56)) then
        tmp = -((x + 1.0d0) / b)
    else
        tmp = (-1.0d0) * (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.2e-56) {
		tmp = -((x + 1.0) / B);
	} else {
		tmp = -1.0 * (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5.2e-56:
		tmp = -((x + 1.0) / B)
	else:
		tmp = -1.0 * (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.2e-56)
		tmp = Float64(-Float64(Float64(x + 1.0) / B));
	else
		tmp = Float64(-1.0 * Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5.2e-56)
		tmp = -((x + 1.0) / B);
	else
		tmp = -1.0 * (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.2e-56], (-N[(N[(x + 1.0), $MachinePrecision] / B), $MachinePrecision]), N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -5.1999999999999999e-56

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \color{blue}{\left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{\color{blue}{x \cdot \cos B}}{F \cdot \sin B}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \color{blue}{\cos B}}{F \cdot \sin B}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}\right)\right) \]

      11. lower-sin.f64 49.8%

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

   4. Applied rewrites 49.8%

      \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      5. lower-/.f64 28.4%

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

   7. Applied rewrites 28.4%

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B}\right) \]

      3. lower-neg.f64 28.4%

         \[\leadsto -\frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

   9. Applied rewrites 29.5%

      \[\leadsto \color{blue}{-\frac{x + 1}{B}} \]

   if -5.1999999999999999e-56 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \color{blue}{\left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{\color{blue}{x \cdot \cos B}}{F \cdot \sin B}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \color{blue}{\cos B}}{F \cdot \sin B}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}\right)\right) \]

      11. lower-sin.f64 49.8%

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

   4. Applied rewrites 49.8%

      \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      5. lower-/.f64 28.4%

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

   7. Applied rewrites 28.4%

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{\color{blue}{B}} \]

   8. Taylor expanded in x around inf

      \[\leadsto -1 \cdot \frac{x}{B} \]
   9. Step-by-step derivation

      1. lower-/.f64 28.8%

         \[\leadsto -1 \cdot \frac{x}{B} \]

   10. Applied rewrites 28.8%

       \[\leadsto -1 \cdot \frac{x}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 30.2% accurate, 10.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.26 \cdot 10^{+32}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.26e+32) (/ -1.0 B) (* -1.0 (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.26e+32) {
		tmp = -1.0 / B;
	} else {
		tmp = -1.0 * (x / B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.26d+32)) then
        tmp = (-1.0d0) / b
    else
        tmp = (-1.0d0) * (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.26e+32) {
		tmp = -1.0 / B;
	} else {
		tmp = -1.0 * (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.26e+32:
		tmp = -1.0 / B
	else:
		tmp = -1.0 * (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.26e+32)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(-1.0 * Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.26e+32)
		tmp = -1.0 / B;
	else
		tmp = -1.0 * (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.26e+32], N[(-1.0 / B), $MachinePrecision], N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.26e32

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.5%

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.5%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-1}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 10.8%

      \[\leadsto \frac{-1}{B} \]

   if -1.26e32 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \color{blue}{\left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)}\right) \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{\color{blue}{x \cdot \cos B}}{F \cdot \sin B}\right)\right) \]

      5. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \color{blue}{\cos B}}{F \cdot \sin B}\right)\right) \]

      6. lower-sin.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}\right)\right) \]

      9. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}\right)\right) \]

      11. lower-sin.f64 49.8%

          \[\leadsto -1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right) \]

   4. Applied rewrites 49.8%

      \[\leadsto \color{blue}{-1 \cdot \left(F \cdot \left(\frac{1}{F \cdot \sin B} + \frac{x \cdot \cos B}{F \cdot \sin B}\right)\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

      5. lower-/.f64 28.4%

         \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

   7. Applied rewrites 28.4%

      \[\leadsto -1 \cdot \frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{\color{blue}{B}} \]

   8. Taylor expanded in x around inf

      \[\leadsto -1 \cdot \frac{x}{B} \]
   9. Step-by-step derivation

      1. lower-/.f64 28.8%

         \[\leadsto -1 \cdot \frac{x}{B} \]

   10. Applied rewrites 28.8%

       \[\leadsto -1 \cdot \frac{x}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 10.8% accurate, 26.5× speedup

Math

\[\frac{-1}{B} \]

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 76.9%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in F around -inf

   \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

   2. lower-sin.f64 17.5%

      \[\leadsto \frac{-1}{\sin B} \]

4. Applied rewrites 17.5%

   \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

5. Taylor expanded in B around 0

   \[\leadsto \frac{-1}{B} \]
6. Step-by-step derivation

7. Applied rewrites 10.8%

   \[\leadsto \frac{-1}{B} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025193 
(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))))))