VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.2% → 99.7%

Time: 6.9s

Alternatives: 21

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{1}{\sin B}, t\_0\right)\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -2e+19)
     (fma -1.0 (/ 1.0 (sin B)) t_0)
     (if (<= F 3.8e+23)
       (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B)) t_0)
       (/ (- 1.0 (* (cos B) x)) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -2e+19) {
		tmp = fma(-1.0, (1.0 / sin(B)), t_0);
	} else if (F <= 3.8e+23) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), t_0);
	} else {
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -2e+19)
		tmp = fma(-1.0, Float64(1.0 / sin(B)), t_0);
	elseif (F <= 3.8e+23)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), t_0);
	else
		tmp = Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2e+19], N[(-1.0 * N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, 3.8e+23], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2e19

   1. Initial program 76.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. mult-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. mult-flip N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 56.6%

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

   if -2e19 < F < 3.79999999999999975e23

   1. Initial program 76.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

   if 3.79999999999999975e23 < F

   1. Initial program 76.2%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.8

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

   4. Applied rewrites 47.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.8

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

   6. Applied rewrites 50.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. associate-*l/ N/A

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

   8. Applied rewrites 56.3%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -200000000:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{1}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 3.8 \cdot 10^{+23}:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -200000000.0)
   (fma -1.0 (/ 1.0 (sin B)) (/ (- x) (tan B)))
   (if (<= F 3.8e+23)
     (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F (sin B))) (/ x (tan B)))
     (/ (- 1.0 (* (cos B) x)) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -200000000.0) {
		tmp = fma(-1.0, (1.0 / sin(B)), (-x / tan(B)));
	} else if (F <= 3.8e+23) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / sin(B))) - (x / tan(B));
	} else {
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -200000000.0)
		tmp = fma(-1.0, Float64(1.0 / sin(B)), Float64(Float64(-x) / tan(B)));
	elseif (F <= 3.8e+23)
		tmp = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / sin(B))) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -200000000.0], N[(-1.0 * N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.8e+23], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2e8

   1. Initial program 76.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. mult-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. mult-flip N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 56.6%

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

   if -2e8 < F < 3.79999999999999975e23

   1. Initial program 76.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.2

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

   3. Applied rewrites 76.3%

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

   if 3.79999999999999975e23 < F

   1. Initial program 76.2%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.8

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

   4. Applied rewrites 47.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.8

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

   6. Applied rewrites 50.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. associate-*l/ N/A

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

   8. Applied rewrites 56.3%

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

Alternative 3: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -1.52:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{1}{\sin B}, t\_0\right)\\ \mathbf{elif}\;F \leq 1.65:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5}}{\sin B}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -1.52)
     (fma -1.0 (/ 1.0 (sin B)) t_0)
     (if (<= F 1.65)
       (fma F (/ (pow (fma 2.0 x 2.0) -0.5) (sin B)) t_0)
       (/ (- 1.0 (* (cos B) x)) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -1.52) {
		tmp = fma(-1.0, (1.0 / sin(B)), t_0);
	} else if (F <= 1.65) {
		tmp = fma(F, (pow(fma(2.0, x, 2.0), -0.5) / sin(B)), t_0);
	} else {
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -1.52)
		tmp = fma(-1.0, Float64(1.0 / sin(B)), t_0);
	elseif (F <= 1.65)
		tmp = fma(F, Float64((fma(2.0, x, 2.0) ^ -0.5) / sin(B)), t_0);
	else
		tmp = Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.52], N[(-1.0 * N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, 1.65], N[(F * N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.52

   1. Initial program 76.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. mult-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. mult-flip N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 56.6%

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

   if -1.52 < F < 1.6499999999999999

   1. Initial program 76.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

   4. Taylor expanded in F around 0

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

   6. Applied rewrites 55.5%

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

   if 1.6499999999999999 < F

   1. Initial program 76.2%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.8

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

   4. Applied rewrites 47.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.8

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

   6. Applied rewrites 50.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. associate-*l/ N/A

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

   8. Applied rewrites 56.3%

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

Alternative 4: 92.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.182:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{1}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 1700:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.182)
   (fma -1.0 (/ 1.0 (sin B)) (/ (- x) (tan B)))
   (if (<= F 1700.0)
     (+
      (- (* x (/ 1.0 (tan B))))
      (* (/ F B) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
     (/ (- 1.0 (* (cos B) x)) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.182) {
		tmp = fma(-1.0, (1.0 / sin(B)), (-x / tan(B)));
	} else if (F <= 1700.0) {
		tmp = -(x * (1.0 / tan(B))) + ((F / B) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.182)
		tmp = fma(-1.0, Float64(1.0 / sin(B)), Float64(Float64(-x) / tan(B)));
	elseif (F <= 1700.0)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / B) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))));
	else
		tmp = Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.182], N[(-1.0 * N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1700.0], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / B), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.182

   1. Initial program 76.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. mult-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. mult-flip N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 56.6%

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

   if -0.182 < F < 1700

   1. Initial program 76.2%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 61.4

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

   4. Applied rewrites 61.4%

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

   if 1700 < F

   1. Initial program 76.2%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.8

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

   4. Applied rewrites 47.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.8

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

   6. Applied rewrites 50.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. associate-*l/ N/A

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

   8. Applied rewrites 56.3%

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

Alternative 5: 89.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{B}\right)\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -1.55 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{1}{\sin B}, t\_1\right)\\ \mathbf{elif}\;F \leq -3.7 \cdot 10^{-227}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-139}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 500000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B)) (/ (- x) B)))
        (t_1 (/ (- x) (tan B))))
   (if (<= F -1.55e+19)
     (fma -1.0 (/ 1.0 (sin B)) t_1)
     (if (<= F -3.7e-227)
       t_0
       (if (<= F 5e-139)
         t_1
         (if (<= F 500000.0) t_0 (/ (- 1.0 (* (cos B) x)) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), (-x / B));
	double t_1 = -x / tan(B);
	double tmp;
	if (F <= -1.55e+19) {
		tmp = fma(-1.0, (1.0 / sin(B)), t_1);
	} else if (F <= -3.7e-227) {
		tmp = t_0;
	} else if (F <= 5e-139) {
		tmp = t_1;
	} else if (F <= 500000.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), Float64(Float64(-x) / B))
	t_1 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -1.55e+19)
		tmp = fma(-1.0, Float64(1.0 / sin(B)), t_1);
	elseif (F <= -3.7e-227)
		tmp = t_0;
	elseif (F <= 5e-139)
		tmp = t_1;
	elseif (F <= 500000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.55e+19], N[(-1.0 * N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[F, -3.7e-227], t$95$0, If[LessEqual[F, 5e-139], t$95$1, If[LessEqual[F, 500000.0], t$95$0, N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.55e19

   1. Initial program 76.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. mult-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. mult-flip N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 56.6%

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

   if -1.55e19 < F < -3.69999999999999978e-227 or 5.00000000000000034e-139 < F < 5e5

   1. Initial program 76.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. mult-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. mult-flip N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 57.3

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

   8. Applied rewrites 57.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 57.3

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 57.3

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

      12. lift-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lift-/.f64 N/A

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

      15. distribute-neg-frac N/A

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

      16. lift-neg.f64 N/A

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

      17. lower-/.f64 57.3

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

   10. Applied rewrites 57.3%

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

   if -3.69999999999999978e-227 < F < 5.00000000000000034e-139

   1. Initial program 76.2%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

      1. metadata-eval 55.4

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

      2. metadata-eval 55.4

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

      3. metadata-eval 55.4

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

      4. metadata-eval 55.4

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

   6. Applied rewrites 55.5%

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

   if 5e5 < F

   1. Initial program 76.2%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.8

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

   4. Applied rewrites 47.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.8

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

   6. Applied rewrites 50.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l* N/A

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

      9. lift-/.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-/r* N/A

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

      12. associate-*l/ N/A

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

   8. Applied rewrites 56.3%

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

Alternative 6: 76.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F, \frac{1}{\sin B}, -1 \cdot \frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= x -2.1e+16)
     t_0
     (if (<= x 1.8e-34)
       (fma
        (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F)
        (/ 1.0 (sin B))
        (* -1.0 (/ x B)))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (x <= -2.1e+16) {
		tmp = t_0;
	} else if (x <= 1.8e-34) {
		tmp = fma((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F), (1.0 / sin(B)), (-1.0 * (x / B)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (x <= -2.1e+16)
		tmp = t_0;
	elseif (x <= 1.8e-34)
		tmp = fma(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F), Float64(1.0 / sin(B)), Float64(-1.0 * Float64(x / B)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.1e+16], t$95$0, If[LessEqual[x, 1.8e-34], N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] * N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1e16 or 1.80000000000000004e-34 < x

   1. Initial program 76.2%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

      1. metadata-eval 55.4

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

      2. metadata-eval 55.4

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

      3. metadata-eval 55.4

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

      4. metadata-eval 55.4

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

   6. Applied rewrites 55.5%

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

   if -2.1e16 < x < 1.80000000000000004e-34

   1. Initial program 76.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. mult-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. mult-flip N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 57.3

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

   8. Applied rewrites 57.3%

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

Alternative 7: 76.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{+16}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= x -2.1e+16)
     t_0
     (if (<= x 1.8e-34)
       (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B)) (/ (- x) B))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (x <= -2.1e+16) {
		tmp = t_0;
	} else if (x <= 1.8e-34) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), (-x / B));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (x <= -2.1e+16)
		tmp = t_0;
	elseif (x <= 1.8e-34)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), Float64(Float64(-x) / B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.1e+16], t$95$0, If[LessEqual[x, 1.8e-34], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.1e16 or 1.80000000000000004e-34 < x

   1. Initial program 76.2%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

      1. metadata-eval 55.4

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

      2. metadata-eval 55.4

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

      3. metadata-eval 55.4

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

      4. metadata-eval 55.4

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

   6. Applied rewrites 55.5%

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

   if -2.1e16 < x < 1.80000000000000004e-34

   1. Initial program 76.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. mult-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. mult-flip N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 57.3

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

   8. Applied rewrites 57.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lift-*.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 57.3

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 57.3

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

      12. lift-*.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lift-/.f64 N/A

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

      15. distribute-neg-frac N/A

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

      16. lift-neg.f64 N/A

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

      17. lower-/.f64 57.3

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

   10. Applied rewrites 57.3%

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

Alternative 8: 56.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 6.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 6.5e-6)
   (/ (- (* F (pow (+ 2.0 (fma 2.0 x (pow F 2.0))) -0.5)) x) B)
   (/ (- x) (tan B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 6.5e-6) {
		tmp = ((F * pow((2.0 + fma(2.0, x, pow(F, 2.0))), -0.5)) - x) / B;
	} else {
		tmp = -x / tan(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 6.5e-6)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + fma(2.0, x, (F ^ 2.0))) ^ -0.5)) - x) / B);
	else
		tmp = Float64(Float64(-x) / tan(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 6.5e-6], N[(N[(N[(F * N[Power[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 6.4999999999999996e-6

   1. Initial program 76.2%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 43.3%

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

   if 6.4999999999999996e-6 < B

   1. Initial program 76.2%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

      1. metadata-eval 55.4

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

      2. metadata-eval 55.4

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

      3. metadata-eval 55.4

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

      4. metadata-eval 55.4

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

   6. Applied rewrites 55.5%

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

Alternative 9: 56.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 6.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F, \frac{1}{B}, -1 \cdot \frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 6.5e-6)
   (fma (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F) (/ 1.0 B) (* -1.0 (/ x B)))
   (/ (- x) (tan B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 6.5e-6) {
		tmp = fma((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F), (1.0 / B), (-1.0 * (x / B)));
	} else {
		tmp = -x / tan(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 6.5e-6)
		tmp = fma(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F), Float64(1.0 / B), Float64(-1.0 * Float64(x / B)));
	else
		tmp = Float64(Float64(-x) / tan(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 6.5e-6], N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] * N[(1.0 / B), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 6.4999999999999996e-6

   1. Initial program 76.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. distribute-frac-neg N/A

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

      5. mult-flip-rev N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*r/ N/A

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

      10. mult-flip N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 57.3

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

   8. Applied rewrites 57.3%

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

   9. Taylor expanded in B around 0

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

   11. Applied rewrites 43.3%

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

   if 6.4999999999999996e-6 < B

   1. Initial program 76.2%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

      1. metadata-eval 55.4

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

      2. metadata-eval 55.4

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

      3. metadata-eval 55.4

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

      4. metadata-eval 55.4

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

   6. Applied rewrites 55.5%

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

Alternative 10: 55.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;x \leq -2.3 \cdot 10^{+15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -2.3 \cdot 10^{-194}:\\ \;\;\;\;F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \left|\frac{1}{B \cdot F}\right|\right)\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-225}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= x -2.3e+15)
     t_0
     (if (<= x -2.3e-194)
       (* F (fma -1.0 (/ x (* B F)) (fabs (/ 1.0 (* B F)))))
       (if (<= x 1.4e-225) (/ -1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (x <= -2.3e+15) {
		tmp = t_0;
	} else if (x <= -2.3e-194) {
		tmp = F * fma(-1.0, (x / (B * F)), fabs((1.0 / (B * F))));
	} else if (x <= 1.4e-225) {
		tmp = -1.0 / sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (x <= -2.3e+15)
		tmp = t_0;
	elseif (x <= -2.3e-194)
		tmp = Float64(F * fma(-1.0, Float64(x / Float64(B * F)), abs(Float64(1.0 / Float64(B * F)))));
	elseif (x <= 1.4e-225)
		tmp = Float64(-1.0 / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.3e+15], t$95$0, If[LessEqual[x, -2.3e-194], N[(F * N[(-1.0 * N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[Abs[N[(1.0 / N[(B * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.4e-225], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.3e15 or 1.4e-225 < x

   1. Initial program 76.2%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

      1. metadata-eval 55.4

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

      2. metadata-eval 55.4

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

      3. metadata-eval 55.4

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

      4. metadata-eval 55.4

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

      5. lift-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

   6. Applied rewrites 55.5%

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

   if -2.3e15 < x < -2.30000000000000003e-194

   1. Initial program 76.2%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.8

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

   4. Applied rewrites 47.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 31.5

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

   7. Applied rewrites 31.5%

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

   8. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 24.4

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

   10. Applied rewrites 24.4%

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

       1. lift-/.f64 N/A

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

       2. inv-pow N/A

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

       3. pow-to-exp N/A

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

       4. exp-fabs N/A

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

       5. pow-to-exp N/A

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

       6. inv-pow N/A

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

       7. lift-/.f64 N/A

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

       8. lower-fabs.f64 26.0

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

   12. Applied rewrites 26.0%

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

   if -2.30000000000000003e-194 < x < 1.4e-225

   1. Initial program 76.2%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 17.7

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

   4. Applied rewrites 17.7%

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

Alternative 11: 52.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{-37}:\\ \;\;\;\;-1 \cdot \frac{x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.5e-12)
   (/ -1.0 (sin B))
   (if (<= F 1.5e-37) (* -1.0 (/ x (sin B))) (/ (- 1.0 x) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.5e-12) {
		tmp = -1.0 / sin(B);
	} else if (F <= 1.5e-37) {
		tmp = -1.0 * (x / sin(B));
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.5d-12)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 1.5d-37) then
        tmp = (-1.0d0) * (x / sin(b))
    else
        tmp = (1.0d0 - x) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.5e-12) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 1.5e-37) {
		tmp = -1.0 * (x / Math.sin(B));
	} else {
		tmp = (1.0 - x) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.5e-12:
		tmp = -1.0 / math.sin(B)
	elif F <= 1.5e-37:
		tmp = -1.0 * (x / math.sin(B))
	else:
		tmp = (1.0 - x) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.5e-12)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 1.5e-37)
		tmp = Float64(-1.0 * Float64(x / sin(B)));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.5e-12)
		tmp = -1.0 / sin(B);
	elseif (F <= 1.5e-37)
		tmp = -1.0 * (x / sin(B));
	else
		tmp = (1.0 - x) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.5e-12], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.5e-37], N[(-1.0 * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5000000000000001e-12

   1. Initial program 76.2%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 17.7

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

   4. Applied rewrites 17.7%

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

   if -1.5000000000000001e-12 < F < 1.5e-37

   1. Initial program 76.2%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 31.1%

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

   if 1.5e-37 < F

   1. Initial program 76.2%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.8

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

   4. Applied rewrites 47.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.8

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

   6. Applied rewrites 50.5%

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

   7. Taylor expanded in B around 0

      \[\leadsto \frac{1 - x}{\sin B \cdot F} \cdot F \]
   8. Step-by-step derivation

      1. lower--.f64 35.4

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

   9. Applied rewrites 35.4%

      \[\leadsto \frac{1 - x}{\sin B \cdot F} \cdot F \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

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

       4. associate-/l* N/A

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

       5. mult-flip-rev N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-/r* N/A

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

       9. lift-/.f64 N/A

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

       10. associate-/l* N/A

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

       11. lift-/.f64 N/A

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

       12. rgt-mult-inverse N/A

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

       13. mult-flip N/A

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

   11. Applied rewrites 38.3%

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

Alternative 12: 51.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 4.4 \cdot 10^{-69}:\\ \;\;\;\;-1 \cdot \frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot x\right) \cdot B, B, x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.5e-12)
   (/ -1.0 (sin B))
   (if (<= F 4.4e-69)
     (* -1.0 (/ (fma (* (* -0.3333333333333333 x) B) B x) B))
     (/ (- 1.0 x) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.5e-12) {
		tmp = -1.0 / sin(B);
	} else if (F <= 4.4e-69) {
		tmp = -1.0 * (fma(((-0.3333333333333333 * x) * B), B, x) / B);
	} else {
		tmp = (1.0 - x) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.5e-12)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 4.4e-69)
		tmp = Float64(-1.0 * Float64(fma(Float64(Float64(-0.3333333333333333 * x) * B), B, x) / B));
	else
		tmp = Float64(Float64(1.0 - x) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.5e-12], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.4e-69], N[(-1.0 * N[(N[(N[(N[(-0.3333333333333333 * x), $MachinePrecision] * B), $MachinePrecision] * B + x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5000000000000001e-12

   1. Initial program 76.2%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 17.7

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

   4. Applied rewrites 17.7%

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

   if -1.5000000000000001e-12 < F < 4.4e-69

   1. Initial program 76.2%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

   5. Taylor expanded in B around 0

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

      1. metadata-eval N/A

         \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]

      2. metadata-eval N/A

         \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]

   7. Applied rewrites 28.7%

      \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) + x}{B} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) + x}{B} \]

      4. *-commutative N/A

         \[\leadsto -1 \cdot \frac{\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot {B}^{2} + x}{B} \]

      5. lift-pow.f64 N/A

         \[\leadsto -1 \cdot \frac{\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot {B}^{2} + x}{B} \]

      6. unpow2 N/A

         \[\leadsto -1 \cdot \frac{\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot \left(B \cdot B\right) + x}{B} \]

      7. associate-*r* N/A

         \[\leadsto -1 \cdot \frac{\left(\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot B\right) \cdot B + x}{B} \]

      8. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot B, B, x\right)}{B} \]

      9. lower-*.f64 28.9

         \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B, B, x\right)}{B} \]

      10. lift--.f64 N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot B, B, x\right)}{B} \]

      11. lift-*.f64 N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot B, B, x\right)}{B} \]

      12. lift-*.f64 N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot B, B, x\right)}{B} \]

      13. distribute-rgt-out-- N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(x \cdot \left(\frac{-1}{2} - \frac{-1}{6}\right)\right) \cdot B, B, x\right)}{B} \]

      14. *-commutative N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\left(\frac{-1}{2} - \frac{-1}{6}\right) \cdot x\right) \cdot B, B, x\right)}{B} \]

      15. lower-*.f64 N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\left(\frac{-1}{2} - \frac{-1}{6}\right) \cdot x\right) \cdot B, B, x\right)}{B} \]

      16. metadata-eval 28.9

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot x\right) \cdot B, B, x\right)}{B} \]

   9. Applied rewrites 28.9%

      \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot x\right) \cdot B, B, x\right)}{B} \]

   if 4.4e-69 < F

   1. Initial program 76.2%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.8

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

   4. Applied rewrites 47.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.8

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

   6. Applied rewrites 50.5%

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

   7. Taylor expanded in B around 0

      \[\leadsto \frac{1 - x}{\sin B \cdot F} \cdot F \]
   8. Step-by-step derivation

      1. lower--.f64 35.4

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

   9. Applied rewrites 35.4%

      \[\leadsto \frac{1 - x}{\sin B \cdot F} \cdot F \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

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

       4. associate-/l* N/A

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

       5. mult-flip-rev N/A

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

       6. lift-*.f64 N/A

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

       7. *-commutative N/A

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

       8. associate-/r* N/A

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

       9. lift-/.f64 N/A

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

       10. associate-/l* N/A

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

       11. lift-/.f64 N/A

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

       12. rgt-mult-inverse N/A

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

       13. mult-flip N/A

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

   11. Applied rewrites 38.3%

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

Alternative 13: 44.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.32 \cdot 10^{-21}:\\ \;\;\;\;-1 \cdot \frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot x\right) \cdot B, B, x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.5e-12)
   (/ -1.0 (sin B))
   (if (<= F 1.32e-21)
     (* -1.0 (/ (fma (* (* -0.3333333333333333 x) B) B x) B))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.5e-12) {
		tmp = -1.0 / sin(B);
	} else if (F <= 1.32e-21) {
		tmp = -1.0 * (fma(((-0.3333333333333333 * x) * B), B, x) / B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.5e-12)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 1.32e-21)
		tmp = Float64(-1.0 * Float64(fma(Float64(Float64(-0.3333333333333333 * x) * B), B, x) / B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.5e-12], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.32e-21], N[(-1.0 * N[(N[(N[(N[(-0.3333333333333333 * x), $MachinePrecision] * B), $MachinePrecision] * B + x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5000000000000001e-12

   1. Initial program 76.2%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 17.7

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

   4. Applied rewrites 17.7%

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

   if -1.5000000000000001e-12 < F < 1.32e-21

   1. Initial program 76.2%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

   5. Taylor expanded in B around 0

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

      1. metadata-eval N/A

         \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]

      2. metadata-eval N/A

         \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]

   7. Applied rewrites 28.7%

      \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) + x}{B} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) + x}{B} \]

      4. *-commutative N/A

         \[\leadsto -1 \cdot \frac{\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot {B}^{2} + x}{B} \]

      5. lift-pow.f64 N/A

         \[\leadsto -1 \cdot \frac{\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot {B}^{2} + x}{B} \]

      6. unpow2 N/A

         \[\leadsto -1 \cdot \frac{\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot \left(B \cdot B\right) + x}{B} \]

      7. associate-*r* N/A

         \[\leadsto -1 \cdot \frac{\left(\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot B\right) \cdot B + x}{B} \]

      8. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot B, B, x\right)}{B} \]

      9. lower-*.f64 28.9

         \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B, B, x\right)}{B} \]

      10. lift--.f64 N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot B, B, x\right)}{B} \]

      11. lift-*.f64 N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot B, B, x\right)}{B} \]

      12. lift-*.f64 N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot B, B, x\right)}{B} \]

      13. distribute-rgt-out-- N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(x \cdot \left(\frac{-1}{2} - \frac{-1}{6}\right)\right) \cdot B, B, x\right)}{B} \]

      14. *-commutative N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\left(\frac{-1}{2} - \frac{-1}{6}\right) \cdot x\right) \cdot B, B, x\right)}{B} \]

      15. lower-*.f64 N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\left(\frac{-1}{2} - \frac{-1}{6}\right) \cdot x\right) \cdot B, B, x\right)}{B} \]

      16. metadata-eval 28.9

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot x\right) \cdot B, B, x\right)}{B} \]

   9. Applied rewrites 28.9%

      \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot x\right) \cdot B, B, x\right)}{B} \]

   if 1.32e-21 < F

   1. Initial program 76.2%

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

   2. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 16.9

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

   4. Applied rewrites 16.9%

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

Alternative 14: 43.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{-69}:\\ \;\;\;\;-1 \cdot \frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot x\right) \cdot B, B, x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.5e-12)
   (/ -1.0 (sin B))
   (if (<= F 9.2e-69)
     (* -1.0 (/ (fma (* (* -0.3333333333333333 x) B) B x) B))
     (- (/ 1.0 B) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.5e-12) {
		tmp = -1.0 / sin(B);
	} else if (F <= 9.2e-69) {
		tmp = -1.0 * (fma(((-0.3333333333333333 * x) * B), B, x) / B);
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.5e-12)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 9.2e-69)
		tmp = Float64(-1.0 * Float64(fma(Float64(Float64(-0.3333333333333333 * x) * B), B, x) / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.5e-12], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.2e-69], N[(-1.0 * N[(N[(N[(N[(-0.3333333333333333 * x), $MachinePrecision] * B), $MachinePrecision] * B + x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5000000000000001e-12

   1. Initial program 76.2%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 17.7

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

   4. Applied rewrites 17.7%

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

   if -1.5000000000000001e-12 < F < 9.2000000000000003e-69

   1. Initial program 76.2%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

   5. Taylor expanded in B around 0

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

      1. metadata-eval N/A

         \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]

      2. metadata-eval N/A

         \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]

   7. Applied rewrites 28.7%

      \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) + x}{B} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) + x}{B} \]

      4. *-commutative N/A

         \[\leadsto -1 \cdot \frac{\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot {B}^{2} + x}{B} \]

      5. lift-pow.f64 N/A

         \[\leadsto -1 \cdot \frac{\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot {B}^{2} + x}{B} \]

      6. unpow2 N/A

         \[\leadsto -1 \cdot \frac{\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot \left(B \cdot B\right) + x}{B} \]

      7. associate-*r* N/A

         \[\leadsto -1 \cdot \frac{\left(\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot B\right) \cdot B + x}{B} \]

      8. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot B, B, x\right)}{B} \]

      9. lower-*.f64 28.9

         \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B, B, x\right)}{B} \]

      10. lift--.f64 N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot B, B, x\right)}{B} \]

      11. lift-*.f64 N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot B, B, x\right)}{B} \]

      12. lift-*.f64 N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot B, B, x\right)}{B} \]

      13. distribute-rgt-out-- N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(x \cdot \left(\frac{-1}{2} - \frac{-1}{6}\right)\right) \cdot B, B, x\right)}{B} \]

      14. *-commutative N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\left(\frac{-1}{2} - \frac{-1}{6}\right) \cdot x\right) \cdot B, B, x\right)}{B} \]

      15. lower-*.f64 N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\left(\frac{-1}{2} - \frac{-1}{6}\right) \cdot x\right) \cdot B, B, x\right)}{B} \]

      16. metadata-eval 28.9

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot x\right) \cdot B, B, x\right)}{B} \]

   9. Applied rewrites 28.9%

      \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot x\right) \cdot B, B, x\right)}{B} \]

   if 9.2000000000000003e-69 < F

   1. Initial program 76.2%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.8

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

   4. Applied rewrites 47.8%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 47.8

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

   6. Applied rewrites 50.5%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 29.7

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

   9. Applied rewrites 29.7%

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

       1. lift-/.f64 N/A

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

       2. lift--.f64 N/A

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

       3. div-sub N/A

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

       4. lift-/.f64 N/A

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

       5. lower--.f64 N/A

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

       6. lower-/.f64 29.7

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

   11. Applied rewrites 29.7%

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

Alternative 15: 39.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.9 \cdot 10^{-61}:\\ \;\;\;\;F \cdot \mathsf{fma}\left(-1, \frac{x}{B \cdot F}, \left|\frac{1}{B \cdot F}\right|\right)\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{-69}:\\ \;\;\;\;-1 \cdot \frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot x\right) \cdot B, B, x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.9e-61)
   (* F (fma -1.0 (/ x (* B F)) (fabs (/ 1.0 (* B F)))))
   (if (<= F 9.2e-69)
     (* -1.0 (/ (fma (* (* -0.3333333333333333 x) B) B x) B))
     (- (/ 1.0 B) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.9e-61) {
		tmp = F * fma(-1.0, (x / (B * F)), fabs((1.0 / (B * F))));
	} else if (F <= 9.2e-69) {
		tmp = -1.0 * (fma(((-0.3333333333333333 * x) * B), B, x) / B);
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.9e-61)
		tmp = Float64(F * fma(-1.0, Float64(x / Float64(B * F)), abs(Float64(1.0 / Float64(B * F)))));
	elseif (F <= 9.2e-69)
		tmp = Float64(-1.0 * Float64(fma(Float64(Float64(-0.3333333333333333 * x) * B), B, x) / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.9e-61], N[(F * N[(-1.0 * N[(x / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[Abs[N[(1.0 / N[(B * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.2e-69], N[(-1.0 * N[(N[(N[(N[(-0.3333333333333333 * x), $MachinePrecision] * B), $MachinePrecision] * B + x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.89999999999999972e-61

   1. Initial program 76.2%

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

   2. Taylor expanded in F around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-cos.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-sin.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 47.8

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

   4. Applied rewrites 47.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 31.5

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

   7. Applied rewrites 31.5%

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

   8. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 24.4

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

   10. Applied rewrites 24.4%

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

       1. lift-/.f64 N/A

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

       2. inv-pow N/A

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

       3. pow-to-exp N/A

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

       4. exp-fabs N/A

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

       5. pow-to-exp N/A

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

       6. inv-pow N/A

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

       7. lift-/.f64 N/A

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

       8. lower-fabs.f64 26.0

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

   12. Applied rewrites 26.0%

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

   if -5.89999999999999972e-61 < F < 9.2000000000000003e-69

   1. Initial program 76.2%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.4

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]

      2. metadata-eval N/A

         \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]

   7. Applied rewrites 28.7%

      \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) + x}{B} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) + x}{B} \]

      4. *-commutative N/A

         \[\leadsto -1 \cdot \frac{\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot {B}^{2} + x}{B} \]

      5. lift-pow.f64 N/A

         \[\leadsto -1 \cdot \frac{\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot {B}^{2} + x}{B} \]

      6. unpow2 N/A

         \[\leadsto -1 \cdot \frac{\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot \left(B \cdot B\right) + x}{B} \]

      7. associate-*r* N/A

         \[\leadsto -1 \cdot \frac{\left(\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot B\right) \cdot B + x}{B} \]

      8. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot B, B, x\right)}{B} \]

      9. lower-*.f64 28.9

         \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B, B, x\right)}{B} \]

      10. lift--.f64 N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot B, B, x\right)}{B} \]

      11. lift-*.f64 N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot B, B, x\right)}{B} \]

      12. lift-*.f64 N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot B, B, x\right)}{B} \]

      13. distribute-rgt-out-- N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(x \cdot \left(\frac{-1}{2} - \frac{-1}{6}\right)\right) \cdot B, B, x\right)}{B} \]

      14. *-commutative N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\left(\frac{-1}{2} - \frac{-1}{6}\right) \cdot x\right) \cdot B, B, x\right)}{B} \]

      15. lower-*.f64 N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\left(\frac{-1}{2} - \frac{-1}{6}\right) \cdot x\right) \cdot B, B, x\right)}{B} \]

      16. metadata-eval 28.9

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot x\right) \cdot B, B, x\right)}{B} \]

   9. Applied rewrites 28.9%

      \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot x\right) \cdot B, B, x\right)}{B} \]

   if 9.2000000000000003e-69 < F

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.f64 47.8

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.8%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \cdot \color{blue}{F} \]

      3. lower-*.f64 47.8

         \[\leadsto \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \cdot \color{blue}{F} \]

   6. Applied rewrites 50.5%

      \[\leadsto \frac{1 - \cos B \cdot x}{\sin B \cdot F} \cdot \color{blue}{F} \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{1 - x}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 - x}{B} \]

      2. lower--.f64 29.7

         \[\leadsto \frac{1 - x}{B} \]

   9. Applied rewrites 29.7%

      \[\leadsto \frac{1 - x}{\color{blue}{B}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{1 - x}{B} \]

       2. lift--.f64 N/A

          \[\leadsto \frac{1 - x}{B} \]

       3. div-sub N/A

          \[\leadsto \frac{1}{B} - \frac{x}{\color{blue}{B}} \]

       4. lift-/.f64 N/A

          \[\leadsto \frac{1}{B} - \frac{x}{B} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{1}{B} - \frac{x}{\color{blue}{B}} \]

       6. lower-/.f64 29.7

          \[\leadsto \frac{1}{B} - \frac{x}{B} \]

   11. Applied rewrites 29.7%

       \[\leadsto \frac{1}{B} - \frac{x}{\color{blue}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 36.9% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{-69}:\\ \;\;\;\;-1 \cdot \frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot x\right) \cdot B, B, x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.5e-12)
   (/ -1.0 B)
   (if (<= F 9.2e-69)
     (* -1.0 (/ (fma (* (* -0.3333333333333333 x) B) B x) B))
     (- (/ 1.0 B) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.5e-12) {
		tmp = -1.0 / B;
	} else if (F <= 9.2e-69) {
		tmp = -1.0 * (fma(((-0.3333333333333333 * x) * B), B, x) / B);
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.5e-12)
		tmp = Float64(-1.0 / B);
	elseif (F <= 9.2e-69)
		tmp = Float64(-1.0 * Float64(fma(Float64(Float64(-0.3333333333333333 * x) * B), B, x) / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.5e-12], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 9.2e-69], N[(-1.0 * N[(N[(N[(N[(-0.3333333333333333 * x), $MachinePrecision] * B), $MachinePrecision] * B + x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5000000000000001e-12

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.7

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-1}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 10.8%

      \[\leadsto \frac{-1}{B} \]

   if -1.5000000000000001e-12 < F < 9.2000000000000003e-69

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.4

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]

      2. metadata-eval N/A

         \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]

   7. Applied rewrites 28.7%

      \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) + x}{B} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{{B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) + x}{B} \]

      4. *-commutative N/A

         \[\leadsto -1 \cdot \frac{\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot {B}^{2} + x}{B} \]

      5. lift-pow.f64 N/A

         \[\leadsto -1 \cdot \frac{\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot {B}^{2} + x}{B} \]

      6. unpow2 N/A

         \[\leadsto -1 \cdot \frac{\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot \left(B \cdot B\right) + x}{B} \]

      7. associate-*r* N/A

         \[\leadsto -1 \cdot \frac{\left(\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot B\right) \cdot B + x}{B} \]

      8. lower-fma.f64 N/A

         \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot B, B, x\right)}{B} \]

      9. lower-*.f64 28.9

         \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B, B, x\right)}{B} \]

      10. lift--.f64 N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot B, B, x\right)}{B} \]

      11. lift-*.f64 N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot B, B, x\right)}{B} \]

      12. lift-*.f64 N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right) \cdot B, B, x\right)}{B} \]

      13. distribute-rgt-out-- N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(x \cdot \left(\frac{-1}{2} - \frac{-1}{6}\right)\right) \cdot B, B, x\right)}{B} \]

      14. *-commutative N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\left(\frac{-1}{2} - \frac{-1}{6}\right) \cdot x\right) \cdot B, B, x\right)}{B} \]

      15. lower-*.f64 N/A

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(\left(\frac{-1}{2} - \frac{-1}{6}\right) \cdot x\right) \cdot B, B, x\right)}{B} \]

      16. metadata-eval 28.9

          \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot x\right) \cdot B, B, x\right)}{B} \]

   9. Applied rewrites 28.9%

      \[\leadsto -1 \cdot \frac{\mathsf{fma}\left(\left(-0.3333333333333333 \cdot x\right) \cdot B, B, x\right)}{B} \]

   if 9.2000000000000003e-69 < F

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.f64 47.8

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.8%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \cdot \color{blue}{F} \]

      3. lower-*.f64 47.8

         \[\leadsto \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \cdot \color{blue}{F} \]

   6. Applied rewrites 50.5%

      \[\leadsto \frac{1 - \cos B \cdot x}{\sin B \cdot F} \cdot \color{blue}{F} \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{1 - x}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 - x}{B} \]

      2. lower--.f64 29.7

         \[\leadsto \frac{1 - x}{B} \]

   9. Applied rewrites 29.7%

      \[\leadsto \frac{1 - x}{\color{blue}{B}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{1 - x}{B} \]

       2. lift--.f64 N/A

          \[\leadsto \frac{1 - x}{B} \]

       3. div-sub N/A

          \[\leadsto \frac{1}{B} - \frac{x}{\color{blue}{B}} \]

       4. lift-/.f64 N/A

          \[\leadsto \frac{1}{B} - \frac{x}{B} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{1}{B} - \frac{x}{\color{blue}{B}} \]

       6. lower-/.f64 29.7

          \[\leadsto \frac{1}{B} - \frac{x}{B} \]

   11. Applied rewrites 29.7%

       \[\leadsto \frac{1}{B} - \frac{x}{\color{blue}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 36.9% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{-69}:\\ \;\;\;\;-\frac{\mathsf{fma}\left(-0.3333333333333333 \cdot x, B \cdot B, x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.5e-12)
   (/ -1.0 B)
   (if (<= F 9.2e-69)
     (- (/ (fma (* -0.3333333333333333 x) (* B B) x) B))
     (- (/ 1.0 B) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.5e-12) {
		tmp = -1.0 / B;
	} else if (F <= 9.2e-69) {
		tmp = -(fma((-0.3333333333333333 * x), (B * B), x) / B);
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.5e-12)
		tmp = Float64(-1.0 / B);
	elseif (F <= 9.2e-69)
		tmp = Float64(-Float64(fma(Float64(-0.3333333333333333 * x), Float64(B * B), x) / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.5e-12], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 9.2e-69], (-N[(N[(N[(-0.3333333333333333 * x), $MachinePrecision] * N[(B * B), $MachinePrecision] + x), $MachinePrecision] / B), $MachinePrecision]), N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5000000000000001e-12

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.7

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-1}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 10.8%

      \[\leadsto \frac{-1}{B} \]

   if -1.5000000000000001e-12 < F < 9.2000000000000003e-69

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.4

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]

      2. metadata-eval N/A

         \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot x - \frac{-1}{6} \cdot x\right)}{B} \]

   7. Applied rewrites 28.7%

      \[\leadsto -1 \cdot \frac{x + {B}^{2} \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x + {B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right)}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x + {B}^{2} \cdot \left(\frac{-1}{2} \cdot x - \frac{-1}{6} \cdot x\right)}{B}\right) \]

      3. lower-neg.f64 28.7

         \[\leadsto -\frac{x + {B}^{2} \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}{B} \]

   9. Applied rewrites 28.7%

      \[\leadsto \color{blue}{-\frac{\mathsf{fma}\left(-0.3333333333333333 \cdot x, B \cdot B, x\right)}{B}} \]

   if 9.2000000000000003e-69 < F

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.f64 47.8

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.8%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \cdot \color{blue}{F} \]

      3. lower-*.f64 47.8

         \[\leadsto \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \cdot \color{blue}{F} \]

   6. Applied rewrites 50.5%

      \[\leadsto \frac{1 - \cos B \cdot x}{\sin B \cdot F} \cdot \color{blue}{F} \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{1 - x}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 - x}{B} \]

      2. lower--.f64 29.7

         \[\leadsto \frac{1 - x}{B} \]

   9. Applied rewrites 29.7%

      \[\leadsto \frac{1 - x}{\color{blue}{B}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{1 - x}{B} \]

       2. lift--.f64 N/A

          \[\leadsto \frac{1 - x}{B} \]

       3. div-sub N/A

          \[\leadsto \frac{1}{B} - \frac{x}{\color{blue}{B}} \]

       4. lift-/.f64 N/A

          \[\leadsto \frac{1}{B} - \frac{x}{B} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{1}{B} - \frac{x}{\color{blue}{B}} \]

       6. lower-/.f64 29.7

          \[\leadsto \frac{1}{B} - \frac{x}{B} \]

   11. Applied rewrites 29.7%

       \[\leadsto \frac{1}{B} - \frac{x}{\color{blue}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 36.8% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 3.7 \cdot 10^{-64}:\\ \;\;\;\;-1 \cdot \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.5e-12)
   (/ -1.0 B)
   (if (<= F 3.7e-64) (* -1.0 (/ x B)) (- (/ 1.0 B) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.5e-12) {
		tmp = -1.0 / B;
	} else if (F <= 3.7e-64) {
		tmp = -1.0 * (x / B);
	} else {
		tmp = (1.0 / B) - (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 <= (-1.5d-12)) then
        tmp = (-1.0d0) / b
    else if (f <= 3.7d-64) then
        tmp = (-1.0d0) * (x / b)
    else
        tmp = (1.0d0 / b) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.5e-12) {
		tmp = -1.0 / B;
	} else if (F <= 3.7e-64) {
		tmp = -1.0 * (x / B);
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.5e-12:
		tmp = -1.0 / B
	elif F <= 3.7e-64:
		tmp = -1.0 * (x / B)
	else:
		tmp = (1.0 / B) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.5e-12)
		tmp = Float64(-1.0 / B);
	elseif (F <= 3.7e-64)
		tmp = Float64(-1.0 * Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.5e-12)
		tmp = -1.0 / B;
	elseif (F <= 3.7e-64)
		tmp = -1.0 * (x / B);
	else
		tmp = (1.0 / B) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.5e-12], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 3.7e-64], N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5000000000000001e-12

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.7

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-1}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 10.8%

      \[\leadsto \frac{-1}{B} \]

   if -1.5000000000000001e-12 < F < 3.69999999999999999e-64

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.4

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 28.8

         \[\leadsto -1 \cdot \frac{x}{B} \]

   7. Applied rewrites 28.8%

      \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]

   if 3.69999999999999999e-64 < F

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.f64 47.8

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.8%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \cdot \color{blue}{F} \]

      3. lower-*.f64 47.8

         \[\leadsto \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \cdot \color{blue}{F} \]

   6. Applied rewrites 50.5%

      \[\leadsto \frac{1 - \cos B \cdot x}{\sin B \cdot F} \cdot \color{blue}{F} \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{1 - x}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 - x}{B} \]

      2. lower--.f64 29.7

         \[\leadsto \frac{1 - x}{B} \]

   9. Applied rewrites 29.7%

      \[\leadsto \frac{1 - x}{\color{blue}{B}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{1 - x}{B} \]

       2. lift--.f64 N/A

          \[\leadsto \frac{1 - x}{B} \]

       3. div-sub N/A

          \[\leadsto \frac{1}{B} - \frac{x}{\color{blue}{B}} \]

       4. lift-/.f64 N/A

          \[\leadsto \frac{1}{B} - \frac{x}{B} \]

       5. lower--.f64 N/A

          \[\leadsto \frac{1}{B} - \frac{x}{\color{blue}{B}} \]

       6. lower-/.f64 29.7

          \[\leadsto \frac{1}{B} - \frac{x}{B} \]

   11. Applied rewrites 29.7%

       \[\leadsto \frac{1}{B} - \frac{x}{\color{blue}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 36.7% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 3.7 \cdot 10^{-64}:\\ \;\;\;\;-1 \cdot \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.5e-12)
   (/ -1.0 B)
   (if (<= F 3.7e-64) (* -1.0 (/ x B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.5e-12) {
		tmp = -1.0 / B;
	} else if (F <= 3.7e-64) {
		tmp = -1.0 * (x / 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 <= (-1.5d-12)) then
        tmp = (-1.0d0) / b
    else if (f <= 3.7d-64) then
        tmp = (-1.0d0) * (x / 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 <= -1.5e-12) {
		tmp = -1.0 / B;
	} else if (F <= 3.7e-64) {
		tmp = -1.0 * (x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.5e-12:
		tmp = -1.0 / B
	elif F <= 3.7e-64:
		tmp = -1.0 * (x / B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.5e-12)
		tmp = Float64(-1.0 / B);
	elseif (F <= 3.7e-64)
		tmp = Float64(-1.0 * Float64(x / B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.5e-12)
		tmp = -1.0 / B;
	elseif (F <= 3.7e-64)
		tmp = -1.0 * (x / B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.5e-12], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 3.7e-64], N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5000000000000001e-12

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.7

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-1}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 10.8%

      \[\leadsto \frac{-1}{B} \]

   if -1.5000000000000001e-12 < F < 3.69999999999999999e-64

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.4

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 28.8

         \[\leadsto -1 \cdot \frac{x}{B} \]

   7. Applied rewrites 28.8%

      \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]

   if 3.69999999999999999e-64 < F

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.f64 47.8

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.8%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \cdot \color{blue}{F} \]

      3. lower-*.f64 47.8

         \[\leadsto \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \cdot \color{blue}{F} \]

   6. Applied rewrites 50.5%

      \[\leadsto \frac{1 - \cos B \cdot x}{\sin B \cdot F} \cdot \color{blue}{F} \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{1 - x}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 - x}{B} \]

      2. lower--.f64 29.7

         \[\leadsto \frac{1 - x}{B} \]

   9. Applied rewrites 29.7%

      \[\leadsto \frac{1 - x}{\color{blue}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 30.9% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -46000000000:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -46000000000.0) (/ -1.0 B) (/ (- 1.0 x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -46000000000.0) {
		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 <= (-46000000000.0d0)) 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 <= -46000000000.0) {
		tmp = -1.0 / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -46000000000.0:
		tmp = -1.0 / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -46000000000.0)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -46000000000.0)
		tmp = -1.0 / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -46000000000.0], N[(-1.0 / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -4.6e10

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.7

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-1}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 10.8%

      \[\leadsto \frac{-1}{B} \]

   if -4.6e10 < F

   1. Initial program 76.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]

      2. lower-fma.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \color{blue}{\frac{x \cdot \cos B}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      3. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F \cdot \sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{\color{blue}{F} \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \color{blue}{\sin B}}, \frac{1}{F \cdot \sin B}\right) \]

      7. lower-sin.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      8. lower-/.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      9. lower-*.f64 N/A

         \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

      10. lower-sin.f64 47.8

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.8%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto F \cdot \color{blue}{\mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

      2. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \cdot \color{blue}{F} \]

      3. lower-*.f64 47.8

         \[\leadsto \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \cdot \color{blue}{F} \]

   6. Applied rewrites 50.5%

      \[\leadsto \frac{1 - \cos B \cdot x}{\sin B \cdot F} \cdot \color{blue}{F} \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{1 - x}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 - x}{B} \]

      2. lower--.f64 29.7

         \[\leadsto \frac{1 - x}{B} \]

   9. Applied rewrites 29.7%

      \[\leadsto \frac{1 - x}{\color{blue}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 10.8% accurate, 26.5× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 76.2%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in F around -inf

   \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

   2. lower-sin.f64 17.7

      \[\leadsto \frac{-1}{\sin B} \]

4. Applied rewrites 17.7%

   \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

5. Taylor expanded in B around 0

   \[\leadsto \frac{-1}{B} \]
6. Step-by-step derivation

7. Applied rewrites 10.8%

   \[\leadsto \frac{-1}{B} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025142 
(FPCore (F B x)
  :name "VandenBroeck and Keller, Equation (23)"
  :precision binary64
  (+ (- (* x (/ 1.0 (tan B)))) (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))