VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.9% → 99.6%

Time: 7.1s

Alternatives: 22

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -1.16 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\tan B}, x, -1 \cdot t\_0\right)\\ \mathbf{elif}\;F \leq 42000000:\\ \;\;\;\;\left(-\frac{x}{\sin B} \cdot \cos B\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))))
   (if (<= F -1.16e+24)
     (fma (/ -1.0 (tan B)) x (* -1.0 t_0))
     (if (<= F 42000000.0)
       (+
        (- (* (/ x (sin B)) (cos B)))
        (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
       (- t_0 (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double tmp;
	if (F <= -1.16e+24) {
		tmp = fma((-1.0 / tan(B)), x, (-1.0 * t_0));
	} else if (F <= 42000000.0) {
		tmp = -((x / sin(B)) * cos(B)) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = t_0 - (x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -1.16e+24)
		tmp = fma(Float64(-1.0 / tan(B)), x, Float64(-1.0 * t_0));
	elseif (F <= 42000000.0)
		tmp = Float64(Float64(-Float64(Float64(x / sin(B)) * cos(B))) + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))));
	else
		tmp = Float64(t_0 - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.16e+24], N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x + N[(-1.0 * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 42000000.0], N[((-N[(N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $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], N[(t$95$0 - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.16000000000000005e24

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 76.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.9

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

   3. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 55.9%

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

   if -1.16000000000000005e24 < F < 4.2e7

   1. Initial program 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip-rev N/A

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

      4. lift-tan.f64 N/A

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

      5. tan-quot N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-/r/ N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 76.9

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

   3. Applied rewrites 76.9%

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

   if 4.2e7 < F

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 76.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.9

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

   3. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 56.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mul-1-neg N/A

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

      7. distribute-frac-neg N/A

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

      8. sub-flip-reverse N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mult-flip-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-tan.f64 56.3

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

   10. Applied rewrites 56.3%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -1.16 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\tan B}, x, -1 \cdot t\_1\right)\\ \mathbf{elif}\;F \leq 42000000:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1 - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))) (t_1 (/ 1.0 (sin B))))
   (if (<= F -1.16e+24)
     (fma (/ -1.0 (tan B)) x (* -1.0 t_1))
     (if (<= F 42000000.0)
       (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F (sin B))) t_0)
       (- t_1 t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double t_1 = 1.0 / sin(B);
	double tmp;
	if (F <= -1.16e+24) {
		tmp = fma((-1.0 / tan(B)), x, (-1.0 * t_1));
	} else if (F <= 42000000.0) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / sin(B))) - t_0;
	} else {
		tmp = t_1 - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	t_1 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -1.16e+24)
		tmp = fma(Float64(-1.0 / tan(B)), x, Float64(-1.0 * t_1));
	elseif (F <= 42000000.0)
		tmp = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / sin(B))) - t_0);
	else
		tmp = Float64(t_1 - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.16e+24], N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x + N[(-1.0 * t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 42000000.0], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(t$95$1 - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.16000000000000005e24

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 76.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.9

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

   3. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 55.9%

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

   if -1.16000000000000005e24 < F < 4.2e7

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-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 76.9

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

   3. Applied rewrites 77.0%

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

   if 4.2e7 < F

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 76.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.9

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

   3. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 56.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mul-1-neg N/A

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

      7. distribute-frac-neg N/A

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

      8. sub-flip-reverse N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mult-flip-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-tan.f64 56.3

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

   10. Applied rewrites 56.3%

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

Alternative 3: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -370000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\tan B}, x, -1 \cdot t\_0\right)\\ \mathbf{elif}\;F \leq 0.00031:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))))
   (if (<= F -370000000.0)
     (fma (/ -1.0 (tan B)) x (* -1.0 t_0))
     (if (<= F 0.00031)
       (fma F (/ (pow (fma 2.0 x 2.0) -0.5) (sin B)) (/ (- x) (tan B)))
       (- t_0 (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double tmp;
	if (F <= -370000000.0) {
		tmp = fma((-1.0 / tan(B)), x, (-1.0 * t_0));
	} else if (F <= 0.00031) {
		tmp = fma(F, (pow(fma(2.0, x, 2.0), -0.5) / sin(B)), (-x / tan(B)));
	} else {
		tmp = t_0 - (x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -370000000.0)
		tmp = fma(Float64(-1.0 / tan(B)), x, Float64(-1.0 * t_0));
	elseif (F <= 0.00031)
		tmp = fma(F, Float64((fma(2.0, x, 2.0) ^ -0.5) / sin(B)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(t_0 - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -370000000.0], N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x + N[(-1.0 * t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.00031], N[(F * N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.7e8

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 76.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.9

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

   3. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 55.9%

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

   if -3.7e8 < F < 3.1e-4

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. 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 85.2%

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

   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 56.3%

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

   if 3.1e-4 < F

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 76.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.9

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

   3. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 56.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mul-1-neg N/A

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

      7. distribute-frac-neg N/A

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

      8. sub-flip-reverse N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mult-flip-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-tan.f64 56.3

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

   10. Applied rewrites 56.3%

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

Alternative 4: 92.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}\\ t_2 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -370000:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\tan B}, x, -1 \cdot t\_2\right)\\ \mathbf{elif}\;F \leq -3.2 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{B}, x, \frac{t\_1}{\frac{\sin B}{F}}\right)\\ \mathbf{elif}\;F \leq 920000:\\ \;\;\;\;\frac{F}{B} \cdot t\_1 - t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2 - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B)))
        (t_1 (pow (fma x 2.0 (fma F F 2.0)) -0.5))
        (t_2 (/ 1.0 (sin B))))
   (if (<= F -370000.0)
     (fma (/ -1.0 (tan B)) x (* -1.0 t_2))
     (if (<= F -3.2e-58)
       (fma (/ -1.0 B) x (/ t_1 (/ (sin B) F)))
       (if (<= F 920000.0) (- (* (/ F B) t_1) t_0) (- t_2 t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double t_1 = pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5);
	double t_2 = 1.0 / sin(B);
	double tmp;
	if (F <= -370000.0) {
		tmp = fma((-1.0 / tan(B)), x, (-1.0 * t_2));
	} else if (F <= -3.2e-58) {
		tmp = fma((-1.0 / B), x, (t_1 / (sin(B) / F)));
	} else if (F <= 920000.0) {
		tmp = ((F / B) * t_1) - t_0;
	} else {
		tmp = t_2 - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	t_1 = fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5
	t_2 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -370000.0)
		tmp = fma(Float64(-1.0 / tan(B)), x, Float64(-1.0 * t_2));
	elseif (F <= -3.2e-58)
		tmp = fma(Float64(-1.0 / B), x, Float64(t_1 / Float64(sin(B) / F)));
	elseif (F <= 920000.0)
		tmp = Float64(Float64(Float64(F / B) * t_1) - t_0);
	else
		tmp = Float64(t_2 - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$2 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -370000.0], N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x + N[(-1.0 * t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.2e-58], N[(N[(-1.0 / B), $MachinePrecision] * x + N[(t$95$1 / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 920000.0], N[(N[(N[(F / B), $MachinePrecision] * t$95$1), $MachinePrecision] - t$95$0), $MachinePrecision], N[(t$95$2 - t$95$0), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.7e5

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 76.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.9

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

   3. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 55.9%

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

   if -3.7e5 < F < -3.2000000000000001e-58

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 76.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.9

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

   3. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-flip N/A

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

      4. mult-flip-rev N/A

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

      5. lower-/.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 77.7

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

   5. Applied rewrites 77.7%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 50.5%

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

   if -3.2000000000000001e-58 < F < 9.2e5

   1. Initial program 76.9%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.0

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

   4. Applied rewrites 62.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. distribute-frac-neg N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower--.f64 N/A

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

   6. Applied rewrites 62.1%

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

   if 9.2e5 < F

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 76.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.9

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

   3. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 56.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mul-1-neg N/A

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

      7. distribute-frac-neg N/A

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

      8. sub-flip-reverse N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mult-flip-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-tan.f64 56.3

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

   10. Applied rewrites 56.3%

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

Alternative 5: 83.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.05 \cdot 10^{+161}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, -1 \cdot \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 920000:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.05e+161)
   (fma F (/ -1.0 (* F (sin B))) (* -1.0 (/ x B)))
   (if (<= F 920000.0)
     (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) B) (/ (- x) (tan B)))
     (- (/ 1.0 (sin B)) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.05e+161) {
		tmp = fma(F, (-1.0 / (F * sin(B))), (-1.0 * (x / B)));
	} else if (F <= 920000.0) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / B), (-x / tan(B)));
	} else {
		tmp = (1.0 / sin(B)) - (x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.05e+161)
		tmp = fma(F, Float64(-1.0 / Float64(F * sin(B))), Float64(-1.0 * Float64(x / B)));
	elseif (F <= 920000.0)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / B), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.05e+161], N[(F * N[(-1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 920000.0], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / B), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.0500000000000001e161

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. 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 85.2%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.0

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

   6. Applied rewrites 58.0%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 34.7

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

   9. Applied rewrites 34.7%

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

   if -3.0500000000000001e161 < F < 9.2e5

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. 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 85.2%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   if 9.2e5 < F

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 76.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.9

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

   3. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 56.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mul-1-neg N/A

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

      7. distribute-frac-neg N/A

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

      8. sub-flip-reverse N/A

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

      9. lower--.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. mult-flip-rev N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-tan.f64 56.3

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

   10. Applied rewrites 56.3%

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

Alternative 6: 79.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ t_1 := -1 \cdot \frac{x}{B}\\ t_2 := {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}\\ t_3 := \mathsf{fma}\left(F, \frac{t\_2}{B}, \frac{-x}{\tan B}\right)\\ \mathbf{if}\;t\_0 \leq -1000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{t\_2}{\sin B}, t\_1\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (+
          (- (* x (/ 1.0 (tan B))))
          (*
           (/ F (sin B))
           (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))
        (t_1 (* -1.0 (/ x B)))
        (t_2 (pow (fma 2.0 x (fma F F 2.0)) -0.5))
        (t_3 (fma F (/ t_2 B) (/ (- x) (tan B)))))
   (if (<= t_0 -1000.0)
     t_3
     (if (<= t_0 2.0)
       (fma F (/ t_2 (sin B)) t_1)
       (if (<= t_0 INFINITY) t_3 (fma F (/ -1.0 (* F (sin B))) t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	double t_1 = -1.0 * (x / B);
	double t_2 = pow(fma(2.0, x, fma(F, F, 2.0)), -0.5);
	double t_3 = fma(F, (t_2 / B), (-x / tan(B)));
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = t_3;
	} else if (t_0 <= 2.0) {
		tmp = fma(F, (t_2 / sin(B)), t_1);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = fma(F, (-1.0 / (F * sin(B))), t_1);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = 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)))))
	t_1 = Float64(-1.0 * Float64(x / B))
	t_2 = fma(2.0, x, fma(F, F, 2.0)) ^ -0.5
	t_3 = fma(F, Float64(t_2 / B), Float64(Float64(-x) / tan(B)))
	tmp = 0.0
	if (t_0 <= -1000.0)
		tmp = t_3;
	elseif (t_0 <= 2.0)
		tmp = fma(F, Float64(t_2 / sin(B)), t_1);
	elseif (t_0 <= Inf)
		tmp = t_3;
	else
		tmp = fma(F, Float64(-1.0 / Float64(F * sin(B))), t_1);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$3 = N[(F * N[(t$95$2 / B), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1000.0], t$95$3, If[LessEqual[t$95$0, 2.0], N[(F * N[(t$95$2 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$3, N[(F * N[(-1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < -1e3 or 2 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < +inf.0

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. 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 85.2%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 70.3%

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

   if -1e3 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 2

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. 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 85.2%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.0

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

   6. Applied rewrites 58.0%

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

   if +inf.0 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))))))

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. 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 85.2%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.0

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

   6. Applied rewrites 58.0%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 34.7

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

   9. Applied rewrites 34.7%

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

Alternative 7: 77.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ t_1 := -1 \cdot \frac{x}{B}\\ t_2 := \frac{F}{B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{\tan B}\\ \mathbf{if}\;t\_0 \leq -1000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, t\_1\right)\\ \mathbf{elif}\;t\_0 \leq \infty:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (+
          (- (* x (/ 1.0 (tan B))))
          (*
           (/ F (sin B))
           (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))
        (t_1 (* -1.0 (/ x B)))
        (t_2
         (- (* (/ F B) (pow (fma x 2.0 (fma F F 2.0)) -0.5)) (/ x (tan B)))))
   (if (<= t_0 -1000.0)
     t_2
     (if (<= t_0 2.0)
       (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B)) t_1)
       (if (<= t_0 INFINITY) t_2 (fma F (/ -1.0 (* F (sin B))) t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	double t_1 = -1.0 * (x / B);
	double t_2 = ((F / B) * pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5)) - (x / tan(B));
	double tmp;
	if (t_0 <= -1000.0) {
		tmp = t_2;
	} else if (t_0 <= 2.0) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), t_1);
	} else if (t_0 <= ((double) INFINITY)) {
		tmp = t_2;
	} else {
		tmp = fma(F, (-1.0 / (F * sin(B))), t_1);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = 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)))))
	t_1 = Float64(-1.0 * Float64(x / B))
	t_2 = Float64(Float64(Float64(F / B) * (fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5)) - Float64(x / tan(B)))
	tmp = 0.0
	if (t_0 <= -1000.0)
		tmp = t_2;
	elseif (t_0 <= 2.0)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), t_1);
	elseif (t_0 <= Inf)
		tmp = t_2;
	else
		tmp = fma(F, Float64(-1.0 / Float64(F * sin(B))), t_1);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = 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]}, Block[{t$95$1 = N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(F / B), $MachinePrecision] * N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1000.0], t$95$2, If[LessEqual[t$95$0, 2.0], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[t$95$0, Infinity], t$95$2, N[(F * N[(-1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < -1e3 or 2 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < +inf.0

   1. Initial program 76.9%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.0

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

   4. Applied rewrites 62.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. distribute-frac-neg N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower--.f64 N/A

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

   6. Applied rewrites 62.1%

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

   if -1e3 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 2

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. 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 85.2%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.0

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

   6. Applied rewrites 58.0%

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

   if +inf.0 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))))))

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. 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 85.2%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.0

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

   6. Applied rewrites 58.0%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 34.7

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

   9. Applied rewrites 34.7%

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

Alternative 8: 77.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.95 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\tan B}, x, 1 \cdot \frac{1}{B}\right)\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, -1 \cdot \frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -1.95e-10)
   (fma (/ -1.0 (tan B)) x (* 1.0 (/ 1.0 B)))
   (if (<= x 7.2e-9)
     (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B)) (* -1.0 (/ x B)))
     (* -1.0 (/ (* x (cos B)) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -1.95e-10) {
		tmp = fma((-1.0 / tan(B)), x, (1.0 * (1.0 / B)));
	} else if (x <= 7.2e-9) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), (-1.0 * (x / B)));
	} else {
		tmp = -1.0 * ((x * cos(B)) / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -1.95e-10)
		tmp = fma(Float64(-1.0 / tan(B)), x, Float64(1.0 * Float64(1.0 / B)));
	elseif (x <= 7.2e-9)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), Float64(-1.0 * Float64(x / B)));
	else
		tmp = Float64(-1.0 * Float64(Float64(x * cos(B)) / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -1.95e-10], N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x + N[(1.0 * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.2e-9], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.95e-10

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 76.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.9

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

   3. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 56.3%

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

   9. Taylor expanded in B around 0

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

       1. lower-/.f64 54.0

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

   11. Applied rewrites 54.0%

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

   if -1.95e-10 < x < 7.2e-9

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. 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 85.2%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.0

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

   6. Applied rewrites 58.0%

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

   if 7.2e-9 < x

   1. Initial program 76.9%

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

   2. Taylor expanded in F around 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 55.8

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

   4. Applied rewrites 55.8%

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

Alternative 9: 77.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{-1}{\tan B}, x, 1 \cdot \frac{1}{B}\right)\\ \mathbf{if}\;x \leq -1.95 \cdot 10^{-10}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.22:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, -1 \cdot \frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (fma (/ -1.0 (tan B)) x (* 1.0 (/ 1.0 B)))))
   (if (<= x -1.95e-10)
     t_0
     (if (<= x 1.22)
       (fma
        F
        (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B))
        (* -1.0 (/ x B)))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = fma((-1.0 / tan(B)), x, (1.0 * (1.0 / B)));
	double tmp;
	if (x <= -1.95e-10) {
		tmp = t_0;
	} else if (x <= 1.22) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), (-1.0 * (x / B)));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(Float64(-1.0 / tan(B)), x, Float64(1.0 * Float64(1.0 / B)))
	tmp = 0.0
	if (x <= -1.95e-10)
		tmp = t_0;
	elseif (x <= 1.22)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), Float64(-1.0 * Float64(x / B)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x + N[(1.0 * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.95e-10], t$95$0, If[LessEqual[x, 1.22], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.95e-10 or 1.21999999999999997 < x

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 76.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.9

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

   3. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 56.3%

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

   9. Taylor expanded in B around 0

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

       1. lower-/.f64 54.0

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

   11. Applied rewrites 54.0%

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

   if -1.95e-10 < x < 1.21999999999999997

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. 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 85.2%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.0

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

   6. Applied rewrites 58.0%

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

Alternative 10: 75.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{-1}{\tan B}, x, 1 \cdot \frac{1}{B}\right)\\ \mathbf{if}\;x \leq -1.4 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.0057:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{F}{\sin B}, \frac{-x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (fma (/ -1.0 (tan B)) x (* 1.0 (/ 1.0 B)))))
   (if (<= x -1.4e-11)
     t_0
     (if (<= x 0.0057)
       (fma (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F (sin B)) (/ (- x) B))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = fma((-1.0 / tan(B)), x, (1.0 * (1.0 / B)));
	double tmp;
	if (x <= -1.4e-11) {
		tmp = t_0;
	} else if (x <= 0.0057) {
		tmp = fma(pow(fma(2.0, x, fma(F, F, 2.0)), -0.5), (F / sin(B)), (-x / B));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(Float64(-1.0 / tan(B)), x, Float64(1.0 * Float64(1.0 / B)))
	tmp = 0.0
	if (x <= -1.4e-11)
		tmp = t_0;
	elseif (x <= 0.0057)
		tmp = fma((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5), Float64(F / sin(B)), Float64(Float64(-x) / B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x + N[(1.0 * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.4e-11], t$95$0, If[LessEqual[x, 0.0057], N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4e-11 or 0.0057000000000000002 < x

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 76.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.9

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

   3. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 56.3%

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

   9. Taylor expanded in B around 0

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

       1. lower-/.f64 54.0

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

   11. Applied rewrites 54.0%

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

   if -1.4e-11 < x < 0.0057000000000000002

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. 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 85.2%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.0

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

   6. Applied rewrites 58.0%

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

      1. lift-fma.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. lift-sin.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lower-/.f64 49.8

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

      10. lift-*.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. associate-*r/ N/A

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

      13. lift-*.f64 N/A

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

      14. lower-/.f64 49.8

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

      15. lift-*.f64 N/A

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

      16. mul-1-neg N/A

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

      17. lower-neg.f64 49.8

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

   8. Applied rewrites 49.8%

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

Alternative 11: 71.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -1 \cdot \frac{x}{B}\\ \mathbf{if}\;F \leq -370000000:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, t\_0\right)\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5}}{\sin B}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\tan B}, x, 1 \cdot \frac{1}{B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* -1.0 (/ x B))))
   (if (<= F -370000000.0)
     (fma F (/ -1.0 (* F (sin B))) t_0)
     (if (<= F 9e-5)
       (fma F (/ (pow (fma 2.0 x 2.0) -0.5) (sin B)) t_0)
       (fma (/ -1.0 (tan B)) x (* 1.0 (/ 1.0 B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 * (x / B);
	double tmp;
	if (F <= -370000000.0) {
		tmp = fma(F, (-1.0 / (F * sin(B))), t_0);
	} else if (F <= 9e-5) {
		tmp = fma(F, (pow(fma(2.0, x, 2.0), -0.5) / sin(B)), t_0);
	} else {
		tmp = fma((-1.0 / tan(B)), x, (1.0 * (1.0 / B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 * Float64(x / B))
	tmp = 0.0
	if (F <= -370000000.0)
		tmp = fma(F, Float64(-1.0 / Float64(F * sin(B))), t_0);
	elseif (F <= 9e-5)
		tmp = fma(F, Float64((fma(2.0, x, 2.0) ^ -0.5) / sin(B)), t_0);
	else
		tmp = fma(Float64(-1.0 / tan(B)), x, Float64(1.0 * Float64(1.0 / B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -370000000.0], N[(F * N[(-1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, 9e-5], 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[Tan[B], $MachinePrecision]), $MachinePrecision] * x + N[(1.0 * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.7e8

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. 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 85.2%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.0

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

   6. Applied rewrites 58.0%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 34.7

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

   9. Applied rewrites 34.7%

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

   if -3.7e8 < F < 9.00000000000000057e-5

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. 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 85.2%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 58.0

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

   6. Applied rewrites 58.0%

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

   7. Taylor expanded in F around 0

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

   9. Applied rewrites 37.2%

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

   if 9.00000000000000057e-5 < F

   1. Initial program 76.9%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. metadata-eval N/A

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

      10. lower-/.f64 76.9

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 76.9

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

   3. Applied rewrites 76.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*r/ N/A

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

      4. mult-flip N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 85.1

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

   5. Applied rewrites 85.1%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 56.3%

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

   9. Taylor expanded in B around 0

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

       1. lower-/.f64 54.0

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

   11. Applied rewrites 54.0%

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

Alternative 12: 64.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 90000:\\ \;\;\;\;\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{B} \cdot \frac{1}{F} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 90000.0)
   (/ (- (* F (pow (+ 2.0 (fma 2.0 x (pow F 2.0))) -0.5)) x) B)
   (- (* (/ F B) (/ 1.0 F)) (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 90000.0) {
		tmp = ((F * pow((2.0 + fma(2.0, x, pow(F, 2.0))), -0.5)) - x) / B;
	} else {
		tmp = ((F / B) * (1.0 / F)) - (x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 90000.0)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + fma(2.0, x, (F ^ 2.0))) ^ -0.5)) - x) / B);
	else
		tmp = Float64(Float64(Float64(F / B) * Float64(1.0 / F)) - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 90000.0], N[(N[(N[(F * N[Power[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(F / B), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 9e4

   1. Initial program 76.9%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 43.9%

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

   if 9e4 < B

   1. Initial program 76.9%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.0

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

   4. Applied rewrites 62.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. lift-neg.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. distribute-frac-neg N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lower--.f64 N/A

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

   6. Applied rewrites 62.1%

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

   7. Taylor expanded in F around inf

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

      1. lower-/.f64 46.4

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

   9. Applied rewrites 46.4%

      \[\leadsto \frac{F}{B} \cdot \color{blue}{\frac{1}{F}} - \frac{x}{\tan B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 58.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 90000:\\ \;\;\;\;\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{\tan B}, x, 1 \cdot \frac{1}{B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 90000.0)
   (/ (- (* F (pow (+ 2.0 (fma 2.0 x (pow F 2.0))) -0.5)) x) B)
   (fma (/ -1.0 (tan B)) x (* 1.0 (/ 1.0 B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 90000.0) {
		tmp = ((F * pow((2.0 + fma(2.0, x, pow(F, 2.0))), -0.5)) - x) / B;
	} else {
		tmp = fma((-1.0 / tan(B)), x, (1.0 * (1.0 / B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 90000.0)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + fma(2.0, x, (F ^ 2.0))) ^ -0.5)) - x) / B);
	else
		tmp = fma(Float64(-1.0 / tan(B)), x, Float64(1.0 * Float64(1.0 / B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 90000.0], N[(N[(N[(F * N[Power[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x + N[(1.0 * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 9e4

   1. Initial program 76.9%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 43.9%

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

   if 9e4 < B

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. lift-neg.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{1}{\tan B}}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{1}{\tan B} \cdot x}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      5. distribute-lft-neg-in N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{\tan B}\right)\right) \cdot x} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(\frac{1}{\tan B}\right), x, \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(\color{blue}{\frac{1}{\tan B}}\right), x, \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      8. distribute-neg-frac N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(1\right)}{\tan B}}, x, \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{-1}}{\tan B}, x, \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      10. lower-/.f64 76.9

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{-1}{\tan B}}, x, \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}\right) \]

      12. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \frac{F}{\sin B}}\right) \]

      13. lower-*.f64 76.9

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \cdot \frac{F}{\sin B}}\right) \]

   3. Applied rewrites 76.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{\tan B}, x, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right)} \]
   4. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, \color{blue}{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot \frac{F}{\sin B}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot \color{blue}{\frac{F}{\sin B}}\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, \color{blue}{\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F}{\sin B}}\right) \]

      4. mult-flip N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, \color{blue}{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F\right) \cdot \frac{1}{\sin B}}\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, \color{blue}{\left(F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}\right)} \cdot \frac{1}{\sin B}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, \color{blue}{\left(F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}\right) \cdot \frac{1}{\sin B}}\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, \color{blue}{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F\right)} \cdot \frac{1}{\sin B}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, \color{blue}{\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F\right)} \cdot \frac{1}{\sin B}\right) \]

      9. lift-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, \left({\color{blue}{\left(2 \cdot x + \mathsf{fma}\left(F, F, 2\right)\right)}}^{\frac{-1}{2}} \cdot F\right) \cdot \frac{1}{\sin B}\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, \left({\left(\color{blue}{x \cdot 2} + \mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}} \cdot F\right) \cdot \frac{1}{\sin B}\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, \left({\color{blue}{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}}^{\frac{-1}{2}} \cdot F\right) \cdot \frac{1}{\sin B}\right) \]

      12. lower-/.f64 85.1

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right) \cdot \color{blue}{\frac{1}{\sin B}}\right) \]

   5. Applied rewrites 85.1%

      \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, \color{blue}{\left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right) \cdot \frac{1}{\sin B}}\right) \]

   6. Taylor expanded in F around inf

      \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, \color{blue}{1} \cdot \frac{1}{\sin B}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 56.3%

      \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, \color{blue}{1} \cdot \frac{1}{\sin B}\right) \]

   9. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, 1 \cdot \color{blue}{\frac{1}{B}}\right) \]
   10. Step-by-step derivation

       1. lower-/.f64 54.0

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, 1 \cdot \frac{1}{\color{blue}{B}}\right) \]

   11. Applied rewrites 54.0%

       \[\leadsto \mathsf{fma}\left(\frac{-1}{\tan B}, x, 1 \cdot \color{blue}{\frac{1}{B}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 55.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := F \cdot \sin B\\ t_1 := -1 \cdot \frac{x}{B}\\ \mathbf{if}\;F \leq -6.3 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{t\_0}, t\_1\right)\\ \mathbf{elif}\;F \leq 920000:\\ \;\;\;\;\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{t\_0}, t\_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* F (sin B))) (t_1 (* -1.0 (/ x B))))
   (if (<= F -6.3e-18)
     (fma F (/ -1.0 t_0) t_1)
     (if (<= F 920000.0)
       (/ (- (* F (pow (+ 2.0 (fma 2.0 x (pow F 2.0))) -0.5)) x) B)
       (fma F (/ 1.0 t_0) t_1)))))

C

double code(double F, double B, double x) {
	double t_0 = F * sin(B);
	double t_1 = -1.0 * (x / B);
	double tmp;
	if (F <= -6.3e-18) {
		tmp = fma(F, (-1.0 / t_0), t_1);
	} else if (F <= 920000.0) {
		tmp = ((F * pow((2.0 + fma(2.0, x, pow(F, 2.0))), -0.5)) - x) / B;
	} else {
		tmp = fma(F, (1.0 / t_0), t_1);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(F * sin(B))
	t_1 = Float64(-1.0 * Float64(x / B))
	tmp = 0.0
	if (F <= -6.3e-18)
		tmp = fma(F, Float64(-1.0 / t_0), t_1);
	elseif (F <= 920000.0)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + fma(2.0, x, (F ^ 2.0))) ^ -0.5)) - x) / B);
	else
		tmp = fma(F, Float64(1.0 / t_0), t_1);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -6.3e-18], N[(F * N[(-1.0 / t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[F, 920000.0], N[(N[(N[(F * N[Power[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(F * N[(1.0 / t$95$0), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.3000000000000004e-18

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. 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 85.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}, -1 \cdot \color{blue}{\frac{x}{B}}\right) \]

      2. lower-/.f64 58.0

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, -1 \cdot \frac{x}{\color{blue}{B}}\right) \]

   6. Applied rewrites 58.0%

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]

   7. Taylor expanded in F around -inf

      \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{-1}{F \cdot \sin B}}, -1 \cdot \frac{x}{B}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{\color{blue}{F \cdot \sin B}}, -1 \cdot \frac{x}{B}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{F \cdot \color{blue}{\sin B}}, -1 \cdot \frac{x}{B}\right) \]

      3. lower-sin.f64 34.7

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, -1 \cdot \frac{x}{B}\right) \]

   9. Applied rewrites 34.7%

      \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{-1}{F \cdot \sin B}}, -1 \cdot \frac{x}{B}\right) \]

   if -6.3000000000000004e-18 < F < 9.2e5

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   if 9.2e5 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. 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 85.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}, -1 \cdot \color{blue}{\frac{x}{B}}\right) \]

      2. lower-/.f64 58.0

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, -1 \cdot \frac{x}{\color{blue}{B}}\right) \]

   6. Applied rewrites 58.0%

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]

   7. Taylor expanded in F around inf

      \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{1}{F \cdot \sin B}}, -1 \cdot \frac{x}{B}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{1}{\color{blue}{F \cdot \sin B}}, -1 \cdot \frac{x}{B}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{1}{F \cdot \color{blue}{\sin B}}, -1 \cdot \frac{x}{B}\right) \]

      3. lower-sin.f64 32.4

         \[\leadsto \mathsf{fma}\left(F, \frac{1}{F \cdot \sin B}, -1 \cdot \frac{x}{B}\right) \]

   9. Applied rewrites 32.4%

      \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{1}{F \cdot \sin B}}, -1 \cdot \frac{x}{B}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 55.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.3 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, -1 \cdot \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 6600000:\\ \;\;\;\;\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.3e-18)
   (fma F (/ -1.0 (* F (sin B))) (* -1.0 (/ x B)))
   (if (<= F 6600000.0)
     (/ (- (* F (pow (+ 2.0 (fma 2.0 x (pow F 2.0))) -0.5)) x) B)
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.3e-18) {
		tmp = fma(F, (-1.0 / (F * sin(B))), (-1.0 * (x / B)));
	} else if (F <= 6600000.0) {
		tmp = ((F * pow((2.0 + fma(2.0, x, pow(F, 2.0))), -0.5)) - x) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.3e-18)
		tmp = fma(F, Float64(-1.0 / Float64(F * sin(B))), Float64(-1.0 * Float64(x / B)));
	elseif (F <= 6600000.0)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + fma(2.0, x, (F ^ 2.0))) ^ -0.5)) - x) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.3e-18], N[(F * N[(-1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6600000.0], N[(N[(N[(F * N[Power[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.3000000000000004e-18

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. 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 85.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B}, -1 \cdot \color{blue}{\frac{x}{B}}\right) \]

      2. lower-/.f64 58.0

         \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, -1 \cdot \frac{x}{\color{blue}{B}}\right) \]

   6. Applied rewrites 58.0%

      \[\leadsto \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \color{blue}{-1 \cdot \frac{x}{B}}\right) \]

   7. Taylor expanded in F around -inf

      \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{-1}{F \cdot \sin B}}, -1 \cdot \frac{x}{B}\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{\color{blue}{F \cdot \sin B}}, -1 \cdot \frac{x}{B}\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{F \cdot \color{blue}{\sin B}}, -1 \cdot \frac{x}{B}\right) \]

      3. lower-sin.f64 34.7

         \[\leadsto \mathsf{fma}\left(F, \frac{-1}{F \cdot \sin B}, -1 \cdot \frac{x}{B}\right) \]

   9. Applied rewrites 34.7%

      \[\leadsto \mathsf{fma}\left(F, \color{blue}{\frac{-1}{F \cdot \sin B}}, -1 \cdot \frac{x}{B}\right) \]

   if -6.3000000000000004e-18 < F < 6.6e6

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   if 6.6e6 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 16.9

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 16.9%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 52.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{+153}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 6600000:\\ \;\;\;\;\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.4e+153)
   (/ -1.0 (sin B))
   (if (<= F 6600000.0)
     (/ (- (* F (pow (+ 2.0 (fma 2.0 x (pow F 2.0))) -0.5)) x) B)
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.4e+153) {
		tmp = -1.0 / sin(B);
	} else if (F <= 6600000.0) {
		tmp = ((F * pow((2.0 + fma(2.0, x, pow(F, 2.0))), -0.5)) - x) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.4e+153)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 6600000.0)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + fma(2.0, x, (F ^ 2.0))) ^ -0.5)) - x) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.4e+153], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6600000.0], N[(N[(N[(F * N[Power[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.3999999999999997e153

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.2

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -3.3999999999999997e153 < F < 6.6e6

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 43.9%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   if 6.6e6 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 16.9

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 16.9%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 51.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.3 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 6600000:\\ \;\;\;\;\frac{F}{B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.3e-18)
   (/ -1.0 (sin B))
   (if (<= F 6600000.0)
     (- (* (/ F B) (pow (fma x 2.0 (fma F F 2.0)) -0.5)) (/ x B))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.3e-18) {
		tmp = -1.0 / sin(B);
	} else if (F <= 6600000.0) {
		tmp = ((F / B) * pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5)) - (x / B);
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.3e-18)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 6600000.0)
		tmp = Float64(Float64(Float64(F / B) * (fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5)) - Float64(x / B));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.3e-18], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6600000.0], N[(N[(N[(F / B), $MachinePrecision] * N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.3000000000000004e-18

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.2

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -6.3000000000000004e-18 < F < 6.6e6

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   3. Step-by-step derivation

      1. lower-/.f64 62.0

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\color{blue}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 62.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \left(\mathsf{neg}\left(\left(-x \cdot \frac{1}{\tan B}\right)\right)\right)} \]

      4. lift-neg.f64 N/A

         \[\leadsto \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)}\right)\right) \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{1}{\tan B}}\right)\right)\right)\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x \cdot \color{blue}{\frac{1}{\tan B}}\right)\right)\right)\right) \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\frac{x}{\tan B}}\right)\right)\right)\right) \]

      8. distribute-frac-neg N/A

         \[\leadsto \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \left(\mathsf{neg}\left(\color{blue}{\frac{\mathsf{neg}\left(x\right)}{\tan B}}\right)\right) \]

      9. lift-neg.f64 N/A

         \[\leadsto \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \left(\mathsf{neg}\left(\frac{\color{blue}{-x}}{\tan B}\right)\right) \]

      10. lift-/.f64 N/A

          \[\leadsto \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \left(\mathsf{neg}\left(\color{blue}{\frac{-x}{\tan B}}\right)\right) \]

      11. lower--.f64 N/A

          \[\leadsto \color{blue}{\frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - \left(\mathsf{neg}\left(\frac{-x}{\tan B}\right)\right)} \]

   6. Applied rewrites 62.1%

      \[\leadsto \color{blue}{\frac{F}{B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{\tan B}} \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{F}{B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} - \frac{x}{\color{blue}{B}} \]
   8. Step-by-step derivation

   9. Applied rewrites 35.7%

      \[\leadsto \frac{F}{B} \cdot {\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{\color{blue}{B}} \]

   if 6.6e6 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 16.9

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 16.9%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 36.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{-22}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.4e-22)
   (/ -1.0 (sin B))
   (/ (* F (fma -1.0 (/ x F) (/ 1.0 F))) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.4e-22) {
		tmp = -1.0 / sin(B);
	} else {
		tmp = (F * fma(-1.0, (x / F), (1.0 / F))) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.4e-22)
		tmp = Float64(-1.0 / sin(B));
	else
		tmp = Float64(Float64(F * fma(-1.0, Float64(x / F), Float64(1.0 / F))) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.4e-22], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(F * N[(-1.0 * N[(x / F), $MachinePrecision] + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -3.3999999999999998e-22

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.2

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -3.3999999999999998e-22 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   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 47.7

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.7%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      5. lower-/.f64 28.7

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   7. Applied rewrites 28.7%

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{\color{blue}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 30.7% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -215:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot {B}^{2} - 1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -215.0)
   (/ (- (* -0.16666666666666666 (pow B 2.0)) 1.0) B)
   (/ (* F (fma -1.0 (/ x F) (/ 1.0 F))) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -215.0) {
		tmp = ((-0.16666666666666666 * pow(B, 2.0)) - 1.0) / B;
	} else {
		tmp = (F * fma(-1.0, (x / F), (1.0 / F))) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -215.0)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * (B ^ 2.0)) - 1.0) / B);
	else
		tmp = Float64(Float64(F * fma(-1.0, Float64(x / F), Float64(1.0 / F))) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -215.0], N[(N[(N[(-0.16666666666666666 * N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / B), $MachinePrecision], N[(N[(F * N[(-1.0 * N[(x / F), $MachinePrecision] + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -215

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.2

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]

      4. lower-pow.f64 9.9

         \[\leadsto \frac{-0.16666666666666666 \cdot {B}^{2} - 1}{B} \]

   7. Applied rewrites 9.9%

      \[\leadsto \frac{-0.16666666666666666 \cdot {B}^{2} - 1}{\color{blue}{B}} \]

   if -215 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   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 47.7

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.7%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      5. lower-/.f64 28.7

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   7. Applied rewrites 28.7%

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{\color{blue}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 29.7% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3100000000:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3100000000.0) (/ -1.0 B) (/ (* F (fma -1.0 (/ x F) (/ 1.0 F))) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3100000000.0) {
		tmp = -1.0 / B;
	} else {
		tmp = (F * fma(-1.0, (x / F), (1.0 / F))) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3100000000.0)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(Float64(F * fma(-1.0, Float64(x / F), Float64(1.0 / F))) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3100000000.0], N[(-1.0 / B), $MachinePrecision], N[(N[(F * N[(-1.0 * N[(x / F), $MachinePrecision] + N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -3.1e9

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.2

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-1}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 10.2%

      \[\leadsto \frac{-1}{B} \]

   if -3.1e9 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   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 47.7

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.7%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot \left(-1 \cdot \frac{x}{F} + \frac{1}{F}\right)}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

      5. lower-/.f64 28.7

         \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{B} \]

   7. Applied rewrites 28.7%

      \[\leadsto \frac{F \cdot \mathsf{fma}\left(-1, \frac{x}{F}, \frac{1}{F}\right)}{\color{blue}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 29.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3100000000:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3100000000.0) (/ -1.0 B) (/ (- 1.0 x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3100000000.0) {
		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 <= (-3100000000.0d0)) 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 <= -3100000000.0) {
		tmp = -1.0 / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3100000000.0:
		tmp = -1.0 / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3100000000.0)
		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 <= -3100000000.0)
		tmp = -1.0 / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3100000000.0], N[(-1.0 / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -3.1e9

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lower-sin.f64 17.2

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-1}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 10.2%

      \[\leadsto \frac{-1}{B} \]

   if -3.1e9 < F

   1. Initial program 76.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \frac{1}{F \cdot \sin B}\right)} \]
   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 47.7

          \[\leadsto F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right) \]

   4. Applied rewrites 47.7%

      \[\leadsto \color{blue}{F \cdot \mathsf{fma}\left(-1, \frac{x \cdot \cos B}{F \cdot \sin B}, \frac{1}{F \cdot \sin B}\right)} \]
   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. lift-fma.f64 N/A

         \[\leadsto F \cdot \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B} + \color{blue}{\frac{1}{F \cdot \sin B}}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}\right) \cdot F + \color{blue}{\frac{1}{F \cdot \sin B} \cdot F} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-1 \cdot \frac{x \cdot \cos B}{F \cdot \sin B}, \color{blue}{F}, \frac{1}{F \cdot \sin B} \cdot F\right) \]

   6. Applied rewrites 48.0%

      \[\leadsto \mathsf{fma}\left(-\frac{\frac{x}{\tan B}}{F}, \color{blue}{F}, \frac{F}{\sin B \cdot F}\right) \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{1 + -1 \cdot x}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1 + -1 \cdot x}{B} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1 + -1 \cdot x}{B} \]

      3. lower-*.f64 29.7

         \[\leadsto \frac{1 + -1 \cdot x}{B} \]

   9. Applied rewrites 29.7%

      \[\leadsto \frac{1 + -1 \cdot x}{\color{blue}{B}} \]
   10. Step-by-step derivation

       1. lift-+.f64 N/A

          \[\leadsto \frac{1 + -1 \cdot x}{B} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{1 + -1 \cdot x}{B} \]

       3. mul-1-neg N/A

          \[\leadsto \frac{1 + \left(\mathsf{neg}\left(x\right)\right)}{B} \]

       4. sub-flip-reverse N/A

          \[\leadsto \frac{1 - x}{B} \]

       5. lower--.f64 29.7

          \[\leadsto \frac{1 - x}{B} \]

   11. Applied rewrites 29.7%

       \[\leadsto \frac{1 - x}{\color{blue}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 10.2% 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 76.9%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in F around -inf

   \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

   2. lower-sin.f64 17.2

      \[\leadsto \frac{-1}{\sin B} \]

4. Applied rewrites 17.2%

   \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

5. Taylor expanded in B around 0

   \[\leadsto \frac{-1}{B} \]
6. Step-by-step derivation

7. Applied rewrites 10.2%

   \[\leadsto \frac{-1}{B} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025150 
(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))))))