VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.2% → 99.7%

Time: 9.5s

Alternatives: 20

Speedup: 1.6×

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: 77.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1e+154)
   (/ (- (fma (cos B) x 1.0)) (sin B))
   (if (<= F 5e+129)
     (- (/ (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) (sin B)) (/ x (tan B)))
     (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e+154) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else if (F <= 5e+129) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) / sin(B)) - (x / tan(B));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1e+154)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	elseif (F <= 5e+129)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) / sin(B)) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1e+154], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5e+129], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.00000000000000004e154

   1. Initial program 32.0%

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

   3. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -1.00000000000000004e154 < F < 5.0000000000000003e129

   1. Initial program 96.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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

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

   4. Applied rewrites 99.5%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. inv-pow N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-fma.f64 N/A

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

      17. +-commutative N/A

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

      18. associate-+l+ N/A

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

      19. lift-fma.f64 N/A

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

      20. +-commutative N/A

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

      21. lift-fma.f64 N/A

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

      22. lower-sqrt.f64 99.6

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

      23. lift-fma.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-fma.f64 99.6

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

   6. Applied rewrites 99.6%

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

   8. Applied rewrites 99.6%

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

   10. Applied rewrites 99.7%

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

   if 5.0000000000000003e129 < F

   1. Initial program 42.9%

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

   3. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+154}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{+129}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 53.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ t_1 := \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}}\\ t_2 := \frac{F}{\sin B}\\ t_3 := t\_0 + t\_2 \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+14}:\\ \;\;\;\;t\_0 + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;t\_3 \leq 5:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{+110}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, F, \mathsf{fma}\left(0.16666666666666666 \cdot F, t\_1, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right) - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B))))
        (t_1 (sqrt (/ 1.0 (+ (fma 2.0 x (* F F)) 2.0))))
        (t_2 (/ F (sin B)))
        (t_3 (+ t_0 (* t_2 (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (/ -1.0 2.0))))))
   (if (<= t_3 -5e+14)
     (+ t_0 (/ (- (* -0.16666666666666666 (* B B)) 1.0) B))
     (if (<= t_3 5.0)
       (* (sqrt (/ 1.0 (fma F F 2.0))) t_2)
       (if (<= t_3 1e+110)
         (/
          (fma
           t_1
           F
           (-
            (*
             (fma (* 0.16666666666666666 F) t_1 (* 0.3333333333333333 x))
             (* B B))
            x))
          B)
         (- (/ (fma (* B B) 0.16666666666666666 1.0) B) (/ x (tan B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double t_1 = sqrt((1.0 / (fma(2.0, x, (F * F)) + 2.0)));
	double t_2 = F / sin(B);
	double t_3 = t_0 + (t_2 * pow((((F * F) + 2.0) + (2.0 * x)), (-1.0 / 2.0)));
	double tmp;
	if (t_3 <= -5e+14) {
		tmp = t_0 + (((-0.16666666666666666 * (B * B)) - 1.0) / B);
	} else if (t_3 <= 5.0) {
		tmp = sqrt((1.0 / fma(F, F, 2.0))) * t_2;
	} else if (t_3 <= 1e+110) {
		tmp = fma(t_1, F, ((fma((0.16666666666666666 * F), t_1, (0.3333333333333333 * x)) * (B * B)) - x)) / B;
	} else {
		tmp = (fma((B * B), 0.16666666666666666, 1.0) / B) - (x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	t_1 = sqrt(Float64(1.0 / Float64(fma(2.0, x, Float64(F * F)) + 2.0)))
	t_2 = Float64(F / sin(B))
	t_3 = Float64(t_0 + Float64(t_2 * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-1.0 / 2.0))))
	tmp = 0.0
	if (t_3 <= -5e+14)
		tmp = Float64(t_0 + Float64(Float64(Float64(-0.16666666666666666 * Float64(B * B)) - 1.0) / B));
	elseif (t_3 <= 5.0)
		tmp = Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) * t_2);
	elseif (t_3 <= 1e+110)
		tmp = Float64(fma(t_1, F, Float64(Float64(fma(Float64(0.16666666666666666 * F), t_1, Float64(0.3333333333333333 * x)) * Float64(B * B)) - x)) / B);
	else
		tmp = Float64(Float64(fma(Float64(B * B), 0.16666666666666666, 1.0) / B) - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 / N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 + N[(t$95$2 * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], N[(-1.0 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+14], N[(t$95$0 + N[(N[(N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 5.0], N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$3, 1e+110], N[(N[(t$95$1 * F + N[(N[(N[(N[(0.16666666666666666 * F), $MachinePrecision] * t$95$1 + N[(0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(B * B), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 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)))))) < -5e14

   1. Initial program 93.8%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 62.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right)}{B}} \]

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 48.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B} \]

   if -5e14 < (+.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)))))) < 5

   1. Initial program 76.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}} \cdot \frac{F}{\sin B} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}} \cdot \frac{F}{\sin B} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}} \cdot \frac{F}{\sin B} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}} \cdot \frac{F}{\sin B} \]

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 43.1

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

   5. Applied rewrites 43.1%

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

   if 5 < (+.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)))))) < 1e110

   1. Initial program 96.7%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 46.4%

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

   6. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]

   8. Applied rewrites 46.7%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}}, F, \mathsf{fma}\left(0.16666666666666666 \cdot F, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}}, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right) - x\right)}{B}} \]

   if 1e110 < (+.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 54.9%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 70.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right)}{B}} \]

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 69.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} + \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{1}{\tan B}}\right)\right) \]

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

         \[\leadsto \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{1}{\tan B}} \]

      6. fp-cancel-sub-sign N/A

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

   10. Applied rewrites 69.5%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} - \frac{x}{\tan B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)} \leq -5 \cdot 10^{+14}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-0.16666666666666666 \cdot \left(B \cdot B\right) - 1}{B}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)} \leq 5:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\frac{-1}{2}\right)} \leq 10^{+110}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}}, F, \mathsf{fma}\left(0.16666666666666666 \cdot F, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}}, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right) - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+144}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1150:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5e+144)
   (/ (- (fma (cos B) x 1.0)) (sin B))
   (if (<= F 1150.0)
     (- (/ F (* (sin B) (sqrt (fma 2.0 x (fma F F 2.0))))) (/ x (tan B)))
     (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5e+144) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else if (F <= 1150.0) {
		tmp = (F / (sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - (x / tan(B));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5e+144)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	elseif (F <= 1150.0)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5e+144], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1150.0], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.9999999999999999e144

   1. Initial program 32.0%

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

   3. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -4.9999999999999999e144 < F < 1150

   1. Initial program 97.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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

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

   4. Applied rewrites 99.5%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. inv-pow N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-fma.f64 N/A

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

      17. +-commutative N/A

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

      18. associate-+l+ N/A

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

      19. lift-fma.f64 N/A

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

      20. +-commutative N/A

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

      21. lift-fma.f64 N/A

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

      22. lower-sqrt.f64 99.6

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

      23. lift-fma.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-fma.f64 99.6

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

   6. Applied rewrites 99.6%

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

   8. Applied rewrites 99.6%

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

   if 1150 < F

   1. Initial program 58.9%

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

   3. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+144}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1150:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.5:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.75:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, 2\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.5)
   (/ (- (fma (cos B) x 1.0)) (sin B))
   (if (<= F 1.75)
     (- (/ F (* (sin B) (sqrt (fma 2.0 x 2.0)))) (/ x (tan B)))
     (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.5) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else if (F <= 1.75) {
		tmp = (F / (sin(B) * sqrt(fma(2.0, x, 2.0)))) - (x / tan(B));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.5)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	elseif (F <= 1.75)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(2.0, x, 2.0)))) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.5], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.75], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 * x + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5

   1. Initial program 62.8%

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

   3. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -1.5 < F < 1.75

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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

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

   4. Applied rewrites 99.5%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. inv-pow N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-fma.f64 N/A

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

      17. +-commutative N/A

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

      18. associate-+l+ N/A

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

      19. lift-fma.f64 N/A

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

      20. +-commutative N/A

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

      21. lift-fma.f64 N/A

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

      22. lower-sqrt.f64 99.6

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

      23. lift-fma.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-fma.f64 99.6

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

   6. Applied rewrites 99.6%

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in F around 0

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

   11. Applied rewrites 98.4%

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

   if 1.75 < F

   1. Initial program 58.9%

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

   3. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.5:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.75:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, 2\right)}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.05 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 14.6:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.05e-30)
   (/ (- (fma (cos B) x 1.0)) (sin B))
   (if (<= F 14.6)
     (- (/ F (* (sqrt (+ (fma 2.0 x (* F F)) 2.0)) B)) (/ x (tan B)))
     (+ (* x (/ -1.0 (tan B))) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.05e-30) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else if (F <= 14.6) {
		tmp = (F / (sqrt((fma(2.0, x, (F * F)) + 2.0)) * B)) - (x / tan(B));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.05e-30)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	elseif (F <= 14.6)
		tmp = Float64(Float64(F / Float64(sqrt(Float64(fma(2.0, x, Float64(F * F)) + 2.0)) * B)) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.05e-30], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 14.6], N[(N[(F / N[(N[Sqrt[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.0500000000000001e-30

   1. Initial program 66.0%

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

   3. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 96.5

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

   5. Applied rewrites 96.5%

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

   if -1.0500000000000001e-30 < F < 14.5999999999999996

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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

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

   4. Applied rewrites 99.5%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. inv-pow N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-fma.f64 N/A

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

      17. +-commutative N/A

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

      18. associate-+l+ N/A

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

      19. lift-fma.f64 N/A

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

      20. +-commutative N/A

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

      21. lift-fma.f64 N/A

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

      22. lower-sqrt.f64 99.6

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

      23. lift-fma.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-fma.f64 99.6

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

   6. Applied rewrites 99.6%

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in B around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 80.8

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

   11. Applied rewrites 80.8%

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

   if 14.5999999999999996 < F

   1. Initial program 58.9%

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

   3. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{1}{\sin B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.05 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 14.6:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.05 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 14.6:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \cos B \cdot x, 1\right)}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.05e-30)
   (/ (- (fma (cos B) x 1.0)) (sin B))
   (if (<= F 14.6)
     (- (/ F (* (sqrt (+ (fma 2.0 x (* F F)) 2.0)) B)) (/ x (tan B)))
     (/ (fma -1.0 (* (cos B) x) 1.0) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.05e-30) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else if (F <= 14.6) {
		tmp = (F / (sqrt((fma(2.0, x, (F * F)) + 2.0)) * B)) - (x / tan(B));
	} else {
		tmp = fma(-1.0, (cos(B) * x), 1.0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.05e-30)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	elseif (F <= 14.6)
		tmp = Float64(Float64(F / Float64(sqrt(Float64(fma(2.0, x, Float64(F * F)) + 2.0)) * B)) - Float64(x / tan(B)));
	else
		tmp = Float64(fma(-1.0, Float64(cos(B) * x), 1.0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.05e-30], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 14.6], N[(N[(F / N[(N[Sqrt[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.0500000000000001e-30

   1. Initial program 66.0%

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

   3. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 96.5

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

   5. Applied rewrites 96.5%

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

   if -1.0500000000000001e-30 < F < 14.5999999999999996

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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

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

   4. Applied rewrites 99.5%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. inv-pow N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-fma.f64 N/A

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

      17. +-commutative N/A

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

      18. associate-+l+ N/A

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

      19. lift-fma.f64 N/A

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

      20. +-commutative N/A

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

      21. lift-fma.f64 N/A

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

      22. lower-sqrt.f64 99.6

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

      23. lift-fma.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-fma.f64 99.6

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

   6. Applied rewrites 99.6%

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in B around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 80.8

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

   11. Applied rewrites 80.8%

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

   if 14.5999999999999996 < F

   1. Initial program 58.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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

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

   4. Applied rewrites 72.4%

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

   5. Taylor expanded in F around inf

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-sin.f64 99.7

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

   7. Applied rewrites 99.7%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.05 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 14.6:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \cos B \cdot x, 1\right)}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.05 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.05e-30)
   (/ (- (fma (cos B) x 1.0)) (sin B))
   (- (/ F (* (sqrt (+ (fma 2.0 x (* F F)) 2.0)) B)) (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.05e-30) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else {
		tmp = (F / (sqrt((fma(2.0, x, (F * F)) + 2.0)) * B)) - (x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.05e-30)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	else
		tmp = Float64(Float64(F / Float64(sqrt(Float64(fma(2.0, x, Float64(F * F)) + 2.0)) * B)) - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.05e-30], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(F / N[(N[Sqrt[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.0500000000000001e-30

   1. Initial program 66.0%

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

   3. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. +-commutative N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 96.5

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

   5. Applied rewrites 96.5%

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

   if -1.0500000000000001e-30 < F

   1. Initial program 83.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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

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

   4. Applied rewrites 89.1%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. inv-pow N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-fma.f64 N/A

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

      17. +-commutative N/A

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

      18. associate-+l+ N/A

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

      19. lift-fma.f64 N/A

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

      20. +-commutative N/A

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

      21. lift-fma.f64 N/A

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

      22. lower-sqrt.f64 89.2

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

      23. lift-fma.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-fma.f64 89.2

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

   6. Applied rewrites 89.2%

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

   8. Applied rewrites 89.2%

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

   9. Taylor expanded in B around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 75.4

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

   11. Applied rewrites 75.4%

       \[\leadsto \frac{F}{\color{blue}{\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B}} - \frac{x}{\tan B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.05 \cdot 10^{-30}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 76.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(2, x, F \cdot F\right) + 2\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{-182}:\\ \;\;\;\;\mathsf{fma}\left(F, \sqrt{\frac{1}{t\_0}} \cdot \frac{1}{B}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-128}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sqrt{t\_0} \cdot B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ (fma 2.0 x (* F F)) 2.0)))
   (if (<= x -1.65e-182)
     (fma F (* (sqrt (/ 1.0 t_0)) (/ 1.0 B)) (/ (- x) (tan B)))
     (if (<= x 2.6e-128)
       (- (/ F (* (sin B) (sqrt (fma 2.0 x (fma F F 2.0))))) (/ x B))
       (- (/ F (* (sqrt t_0) B)) (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double t_0 = fma(2.0, x, (F * F)) + 2.0;
	double tmp;
	if (x <= -1.65e-182) {
		tmp = fma(F, (sqrt((1.0 / t_0)) * (1.0 / B)), (-x / tan(B)));
	} else if (x <= 2.6e-128) {
		tmp = (F / (sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - (x / B);
	} else {
		tmp = (F / (sqrt(t_0) * B)) - (x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(fma(2.0, x, Float64(F * F)) + 2.0)
	tmp = 0.0
	if (x <= -1.65e-182)
		tmp = fma(F, Float64(sqrt(Float64(1.0 / t_0)) * Float64(1.0 / B)), Float64(Float64(-x) / tan(B)));
	elseif (x <= 2.6e-128)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - Float64(x / B));
	else
		tmp = Float64(Float64(F / Float64(sqrt(t_0) * B)) - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]}, If[LessEqual[x, -1.65e-182], N[(F * N[(N[Sqrt[N[(1.0 / t$95$0), $MachinePrecision]], $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.6e-128], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(F / N[(N[Sqrt[t$95$0], $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.64999999999999998e-182

   1. Initial program 79.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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

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

   4. Applied rewrites 91.8%

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

   5. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 82.6

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

   7. Applied rewrites 82.6%

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

   if -1.64999999999999998e-182 < x < 2.59999999999999981e-128

   1. Initial program 70.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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

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

   4. Applied rewrites 73.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. inv-pow N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-fma.f64 N/A

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

      17. +-commutative N/A

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

      18. associate-+l+ N/A

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

      19. lift-fma.f64 N/A

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

      20. +-commutative N/A

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

      21. lift-fma.f64 N/A

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

      22. lower-sqrt.f64 73.3

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

      23. lift-fma.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-fma.f64 73.3

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

   6. Applied rewrites 73.3%

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

   8. Applied rewrites 73.5%

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

   9. Taylor expanded in B around 0

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

       1. lower-/.f64 67.0

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

   11. Applied rewrites 67.0%

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

   if 2.59999999999999981e-128 < x

   1. Initial program 83.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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

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

   4. Applied rewrites 89.1%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. inv-pow N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-fma.f64 N/A

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

      17. +-commutative N/A

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

      18. associate-+l+ N/A

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

      19. lift-fma.f64 N/A

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

      20. +-commutative N/A

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

      21. lift-fma.f64 N/A

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

      22. lower-sqrt.f64 89.1

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

      23. lift-fma.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-fma.f64 89.1

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

   6. Applied rewrites 89.1%

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

   8. Applied rewrites 89.2%

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

   9. Taylor expanded in B around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 85.8

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

   11. Applied rewrites 85.8%

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

Alternative 9: 76.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-182} \lor \neg \left(x \leq 2.6 \cdot 10^{-128}\right):\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -1.65e-182) (not (<= x 2.6e-128)))
   (- (/ F (* (sqrt (+ (fma 2.0 x (* F F)) 2.0)) B)) (/ x (tan B)))
   (- (/ F (* (sin B) (sqrt (fma 2.0 x (fma F F 2.0))))) (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -1.65e-182) || !(x <= 2.6e-128)) {
		tmp = (F / (sqrt((fma(2.0, x, (F * F)) + 2.0)) * B)) - (x / tan(B));
	} else {
		tmp = (F / (sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -1.65e-182) || !(x <= 2.6e-128))
		tmp = Float64(Float64(F / Float64(sqrt(Float64(fma(2.0, x, Float64(F * F)) + 2.0)) * B)) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -1.65e-182], N[Not[LessEqual[x, 2.6e-128]], $MachinePrecision]], N[(N[(F / N[(N[Sqrt[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.64999999999999998e-182 or 2.59999999999999981e-128 < x

   1. Initial program 81.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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

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

   4. Applied rewrites 90.2%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. inv-pow N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-fma.f64 N/A

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

      17. +-commutative N/A

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

      18. associate-+l+ N/A

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

      19. lift-fma.f64 N/A

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

      20. +-commutative N/A

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

      21. lift-fma.f64 N/A

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

      22. lower-sqrt.f64 90.2

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

      23. lift-fma.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-fma.f64 90.2

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

   6. Applied rewrites 90.2%

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

   8. Applied rewrites 90.2%

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

   9. Taylor expanded in B around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 84.5

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

   11. Applied rewrites 84.5%

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

   if -1.64999999999999998e-182 < x < 2.59999999999999981e-128

   1. Initial program 70.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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

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

   4. Applied rewrites 73.3%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. inv-pow N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-fma.f64 N/A

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

      17. +-commutative N/A

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

      18. associate-+l+ N/A

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

      19. lift-fma.f64 N/A

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

      20. +-commutative N/A

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

      21. lift-fma.f64 N/A

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

      22. lower-sqrt.f64 73.3

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

      23. lift-fma.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-fma.f64 73.3

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

   6. Applied rewrites 73.3%

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

   8. Applied rewrites 73.5%

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

   9. Taylor expanded in B around 0

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

       1. lower-/.f64 67.0

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

   11. Applied rewrites 67.0%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-182} \lor \neg \left(x \leq 2.6 \cdot 10^{-128}\right):\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 72.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -560 \lor \neg \left(x \leq 5200000000\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{1}{F}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -560.0) (not (<= x 5200000000.0)))
   (+ (* x (/ -1.0 (tan B))) (* (/ F B) (/ 1.0 F)))
   (- (/ F (* (sin B) (sqrt (fma 2.0 x (fma F F 2.0))))) (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -560.0) || !(x <= 5200000000.0)) {
		tmp = (x * (-1.0 / tan(B))) + ((F / B) * (1.0 / F));
	} else {
		tmp = (F / (sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -560.0) || !(x <= 5200000000.0))
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / B) * Float64(1.0 / F)));
	else
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -560.0], N[Not[LessEqual[x, 5200000000.0]], $MachinePrecision]], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -560 or 5.2e9 < x

   1. Initial program 86.6%

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

   3. Taylor expanded in F around inf

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

      1. lower-/.f64 92.4

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

   5. Applied rewrites 92.4%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 92.6

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

   8. Applied rewrites 92.6%

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

   if -560 < x < 5.2e9

   1. Initial program 72.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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

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

   4. Applied rewrites 75.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow1 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. inv-pow N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-fma.f64 N/A

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

      17. +-commutative N/A

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

      18. associate-+l+ N/A

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

      19. lift-fma.f64 N/A

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

      20. +-commutative N/A

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

      21. lift-fma.f64 N/A

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

      22. lower-sqrt.f64 75.6

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

      23. lift-fma.f64 N/A

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

      24. *-commutative N/A

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

      25. lower-fma.f64 75.6

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

   6. Applied rewrites 75.6%

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

   8. Applied rewrites 75.7%

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

   9. Taylor expanded in B around 0

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

       1. lower-/.f64 58.8

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

   11. Applied rewrites 58.8%

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

4. Final simplification 72.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -560 \lor \neg \left(x \leq 5200000000\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{1}{F}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 54.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}\\ \mathbf{if}\;B \leq 440:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right), \mathsf{fma}\left(\mathsf{fma}\left(\left(-F\right) \cdot -0.019444444444444445, t\_0, x \cdot 0.022222222222222223\right), B \cdot B, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{1}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (fma F F (fma 2.0 x 2.0))))))
   (if (<= B 440.0)
     (/
      (-
       (fma
        t_0
        (fma (* (* B B) F) 0.16666666666666666 F)
        (*
         (fma
          (fma (* (- F) -0.019444444444444445) t_0 (* x 0.022222222222222223))
          (* B B)
          (* 0.3333333333333333 x))
         (* B B)))
       x)
      B)
     (+ (* x (/ -1.0 (tan B))) (* (/ F B) (/ 1.0 F))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt((1.0 / fma(F, F, fma(2.0, x, 2.0))));
	double tmp;
	if (B <= 440.0) {
		tmp = (fma(t_0, fma(((B * B) * F), 0.16666666666666666, F), (fma(fma((-F * -0.019444444444444445), t_0, (x * 0.022222222222222223)), (B * B), (0.3333333333333333 * x)) * (B * B))) - x) / B;
	} else {
		tmp = (x * (-1.0 / tan(B))) + ((F / B) * (1.0 / F));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = sqrt(Float64(1.0 / fma(F, F, fma(2.0, x, 2.0))))
	tmp = 0.0
	if (B <= 440.0)
		tmp = Float64(Float64(fma(t_0, fma(Float64(Float64(B * B) * F), 0.16666666666666666, F), Float64(fma(fma(Float64(Float64(-F) * -0.019444444444444445), t_0, Float64(x * 0.022222222222222223)), Float64(B * B), Float64(0.3333333333333333 * x)) * Float64(B * B))) - x) / B);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / B) * Float64(1.0 / F)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B, 440.0], N[(N[(N[(t$95$0 * N[(N[(N[(B * B), $MachinePrecision] * F), $MachinePrecision] * 0.16666666666666666 + F), $MachinePrecision] + N[(N[(N[(N[((-F) * -0.019444444444444445), $MachinePrecision] * t$95$0 + N[(x * 0.022222222222222223), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + N[(0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 440

   1. Initial program 74.7%

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

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \left(\frac{1}{3} \cdot x + {B}^{2} \cdot \left(-1 \cdot \left(\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \left(\frac{-1}{36} \cdot F + \frac{1}{120} \cdot F\right)\right) + \left(\frac{-1}{9} \cdot x + \frac{2}{15} \cdot x\right)\right)\right)\right)\right) - x}{B}} \]

   5. Applied rewrites 55.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right), \mathsf{fma}\left(\mathsf{fma}\left(-F \cdot -0.019444444444444445, \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, x \cdot 0.022222222222222223\right), B \cdot B, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right)\right) - x}{B}} \]

   if 440 < B

   1. Initial program 87.3%

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

   3. Taylor expanded in F around inf

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

      1. lower-/.f64 48.8

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

   5. Applied rewrites 48.8%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 52.8

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

   8. Applied rewrites 52.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot \frac{1}{F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 440:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right), \mathsf{fma}\left(\mathsf{fma}\left(\left(-F\right) \cdot -0.019444444444444445, \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, x \cdot 0.022222222222222223\right), B \cdot B, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{1}{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 49.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.16:\\ \;\;\;\;\left(-x\right) \cdot \frac{\mathsf{fma}\left(-0.022222222222222223 \cdot \left(B \cdot B\right) - 0.3333333333333333, B \cdot B, 1\right)}{B} + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.16)
   (+
    (*
     (- x)
     (/
      (fma
       (- (* -0.022222222222222223 (* B B)) 0.3333333333333333)
       (* B B)
       1.0)
      B))
    (/
     (*
      (sqrt (/ 1.0 (fma F F (fma 2.0 x 2.0))))
      (fma (* (* B B) F) 0.16666666666666666 F))
     B))
   (* (sqrt (/ 1.0 (fma F F 2.0))) (/ F (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.16) {
		tmp = (-x * (fma(((-0.022222222222222223 * (B * B)) - 0.3333333333333333), (B * B), 1.0) / B)) + ((sqrt((1.0 / fma(F, F, fma(2.0, x, 2.0)))) * fma(((B * B) * F), 0.16666666666666666, F)) / B);
	} else {
		tmp = sqrt((1.0 / fma(F, F, 2.0))) * (F / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.16)
		tmp = Float64(Float64(Float64(-x) * Float64(fma(Float64(Float64(-0.022222222222222223 * Float64(B * B)) - 0.3333333333333333), Float64(B * B), 1.0) / B)) + Float64(Float64(sqrt(Float64(1.0 / fma(F, F, fma(2.0, x, 2.0)))) * fma(Float64(Float64(B * B) * F), 0.16666666666666666, F)) / B));
	else
		tmp = Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) * Float64(F / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.16], N[(N[((-x) * N[(N[(N[(N[(-0.022222222222222223 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Sqrt[N[(1.0 / N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(B * B), $MachinePrecision] * F), $MachinePrecision] * 0.16666666666666666 + F), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.160000000000000003

   1. Initial program 74.6%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 59.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right)}{B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, \frac{1}{6}, F\right)}{B} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, \frac{1}{6}, F\right)}{B} \]

      2. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{{B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) + 1}}{B}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, \frac{1}{6}, F\right)}{B} \]

      3. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) \cdot {B}^{2}} + 1}{B}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, \frac{1}{6}, F\right)}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}, {B}^{2}, 1\right)}}{B}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, \frac{1}{6}, F\right)}{B} \]

      5. lower--.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}}, {B}^{2}, 1\right)}{B}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, \frac{1}{6}, F\right)}{B} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{45} \cdot {B}^{2}} - \frac{1}{3}, {B}^{2}, 1\right)}{B}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, \frac{1}{6}, F\right)}{B} \]

      7. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot \color{blue}{\left(B \cdot B\right)} - \frac{1}{3}, {B}^{2}, 1\right)}{B}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, \frac{1}{6}, F\right)}{B} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot \color{blue}{\left(B \cdot B\right)} - \frac{1}{3}, {B}^{2}, 1\right)}{B}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, \frac{1}{6}, F\right)}{B} \]

      9. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot \left(B \cdot B\right) - \frac{1}{3}, \color{blue}{B \cdot B}, 1\right)}{B}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, \frac{1}{6}, F\right)}{B} \]

      10. lower-*.f64 55.5

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(-0.022222222222222223 \cdot \left(B \cdot B\right) - 0.3333333333333333, \color{blue}{B \cdot B}, 1\right)}{B}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right)}{B} \]

   8. Applied rewrites 55.5%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(-0.022222222222222223 \cdot \left(B \cdot B\right) - 0.3333333333333333, B \cdot B, 1\right)}{B}}\right) + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right)}{B} \]

   if 0.160000000000000003 < B

   1. Initial program 87.5%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}} \cdot \frac{F}{\sin B} \]

      4. lower-/.f64 N/A

         \[\leadsto \sqrt{\color{blue}{\frac{1}{2 + {F}^{2}}}} \cdot \frac{F}{\sin B} \]

      5. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}} \cdot \frac{F}{\sin B} \]

      6. unpow2 N/A

         \[\leadsto \sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}} \cdot \frac{F}{\sin B} \]

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 30.2

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

   5. Applied rewrites 30.2%

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

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.16:\\ \;\;\;\;\left(-x\right) \cdot \frac{\mathsf{fma}\left(-0.022222222222222223 \cdot \left(B \cdot B\right) - 0.3333333333333333, B \cdot B, 1\right)}{B} + \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.1 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{+35}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.1e-28)
   (/ (- -1.0 x) B)
   (if (<= F 2.8e+35)
     (/ (- (* (sqrt (/ 1.0 (fma F F (fma 2.0 x 2.0)))) F) x) B)
     (- (/ (fma (* B B) 0.16666666666666666 1.0) B) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.1e-28) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.8e+35) {
		tmp = ((sqrt((1.0 / fma(F, F, fma(2.0, x, 2.0)))) * F) - x) / B;
	} else {
		tmp = (fma((B * B), 0.16666666666666666, 1.0) / B) - (x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.1e-28)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.8e+35)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(F, F, fma(2.0, x, 2.0)))) * F) - x) / B);
	else
		tmp = Float64(Float64(fma(Float64(B * B), 0.16666666666666666, 1.0) / B) - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.1e-28], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.8e+35], N[(N[(N[(N[Sqrt[N[(1.0 / N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(B * B), $MachinePrecision] * 0.16666666666666666 + 1.0), $MachinePrecision] / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.09999999999999981e-28

   1. Initial program 65.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 41.5%

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

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.7%

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

   if -7.09999999999999981e-28 < F < 2.79999999999999999e35

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 42.9%

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

   if 2.79999999999999999e35 < F

   1. Initial program 56.2%

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

   3. Taylor expanded in B around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

   5. Applied rewrites 42.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right)}{B}} \]

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 59.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} + \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{1}{\tan B}}\right)\right) \]

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

         \[\leadsto \frac{\mathsf{fma}\left(B \cdot B, \frac{1}{6}, 1\right)}{B} + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{1}{\tan B}} \]

      6. fp-cancel-sub-sign N/A

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

   10. Applied rewrites 59.9%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} - \frac{x}{\tan B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.1 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{+35}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, 0.16666666666666666, 1\right)}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.1 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1150:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.5}{F}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F}, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.1e-28)
   (/ (- -1.0 x) B)
   (if (<= F 1150.0)
     (/ (- (* (sqrt (/ 1.0 (fma F F (fma 2.0 x 2.0)))) F) x) B)
     (/ (- (fma (/ -0.5 F) (/ (fma 2.0 x 2.0) F) 1.0) x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.1e-28) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1150.0) {
		tmp = ((sqrt((1.0 / fma(F, F, fma(2.0, x, 2.0)))) * F) - x) / B;
	} else {
		tmp = (fma((-0.5 / F), (fma(2.0, x, 2.0) / F), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.1e-28)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1150.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(F, F, fma(2.0, x, 2.0)))) * F) - x) / B);
	else
		tmp = Float64(Float64(fma(Float64(-0.5 / F), Float64(fma(2.0, x, 2.0) / F), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.1e-28], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1150.0], N[(N[(N[(N[Sqrt[N[(1.0 / N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(-0.5 / F), $MachinePrecision] * N[(N[(2.0 * x + 2.0), $MachinePrecision] / F), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.09999999999999981e-28

   1. Initial program 65.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 41.5%

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

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.7%

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

   if -7.09999999999999981e-28 < F < 1150

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 43.5%

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

   if 1150 < F

   1. Initial program 58.9%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 37.7%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 49.6%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-0.5}{F}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F}, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.1 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1150:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.5}{F}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F}, 1\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 49.2% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.1 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 11.2:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.5}{F}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F}, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.1e-28)
   (/ (- -1.0 x) B)
   (if (<= F 11.2)
     (/ (- (* (sqrt (/ 1.0 (fma 2.0 x 2.0))) F) x) B)
     (/ (- (fma (/ -0.5 F) (/ (fma 2.0 x 2.0) F) 1.0) x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.1e-28) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 11.2) {
		tmp = ((sqrt((1.0 / fma(2.0, x, 2.0))) * F) - x) / B;
	} else {
		tmp = (fma((-0.5 / F), (fma(2.0, x, 2.0) / F), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.1e-28)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 11.2)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, 2.0))) * F) - x) / B);
	else
		tmp = Float64(Float64(fma(Float64(-0.5 / F), Float64(fma(2.0, x, 2.0) / F), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.1e-28], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 11.2], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(-0.5 / F), $MachinePrecision] * N[(N[(2.0 * x + 2.0), $MachinePrecision] / F), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.09999999999999981e-28

   1. Initial program 65.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 41.5%

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

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.7%

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

   if -7.09999999999999981e-28 < F < 11.199999999999999

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 43.5%

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

   6. Taylor expanded in F around 0

      \[\leadsto \frac{\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot F - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.2%

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

   if 11.199999999999999 < F

   1. Initial program 58.9%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 37.7%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 49.6%

      \[\leadsto \frac{\mathsf{fma}\left(\frac{-0.5}{F}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F}, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.1 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 11.2:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.5}{F}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F}, 1\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 49.0% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.1 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 350000000000:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.1e-28)
   (/ (- -1.0 x) B)
   (if (<= F 350000000000.0)
     (/ (- (* (sqrt (/ 1.0 (fma 2.0 x 2.0))) F) x) B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.1e-28) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 350000000000.0) {
		tmp = ((sqrt((1.0 / fma(2.0, x, 2.0))) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.1e-28)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 350000000000.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, 2.0))) * F) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.1e-28], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 350000000000.0], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.09999999999999981e-28

   1. Initial program 65.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 41.5%

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

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.7%

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

   if -7.09999999999999981e-28 < F < 3.5e11

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 42.7%

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

   6. Taylor expanded in F around 0

      \[\leadsto \frac{\sqrt{\frac{1}{2 + 2 \cdot x}} \cdot F - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 41.5%

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

   if 3.5e11 < F

   1. Initial program 57.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 38.8%

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

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 51.0%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.1 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 350000000000:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 49.2% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.1 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.35:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.1e-28)
   (/ (- -1.0 x) B)
   (if (<= F 1.35) (/ (- (/ F (sqrt (fma x 2.0 2.0))) x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.1e-28) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.35) {
		tmp = ((F / sqrt(fma(x, 2.0, 2.0))) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.1e-28)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.35)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, 2.0))) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.1e-28], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.35], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.09999999999999981e-28

   1. Initial program 65.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 41.5%

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

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.7%

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

   if -7.09999999999999981e-28 < F < 1.3500000000000001

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 43.5%

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

   7. Applied rewrites 43.5%

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

   8. Taylor expanded in F around 0

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

   10. Applied rewrites 42.2%

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

   12. Applied rewrites 42.2%

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

   if 1.3500000000000001 < F

   1. Initial program 58.9%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 37.7%

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

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 49.6%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.1 \cdot 10^{-28}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.35:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 42.5% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-33}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.8e-65)
   (/ (- -1.0 x) B)
   (if (<= F 2.7e-33) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.8e-65) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.7e-33) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-4.8d-65)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 2.7d-33) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.8e-65) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.7e-33) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.8e-65:
		tmp = (-1.0 - x) / B
	elif F <= 2.7e-33:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.8e-65)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.7e-33)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -4.8e-65)
		tmp = (-1.0 - x) / B;
	elseif (F <= 2.7e-33)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.8e-65], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.7e-33], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.8000000000000003e-65

   1. Initial program 67.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 41.6%

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

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.0%

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

   if -4.8000000000000003e-65 < F < 2.7000000000000001e-33

   1. Initial program 99.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 42.2%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 33.4%

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

   if 2.7000000000000001e-33 < F

   1. Initial program 63.7%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 40.0%

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

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.2%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-33}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 35.0% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.8e-65) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.8e-65) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -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 <= (-4.8d-65)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.8e-65) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.8e-65:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.8e-65)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -4.8e-65)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.8e-65], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -4.8000000000000003e-65

   1. Initial program 67.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 41.6%

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

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 54.0%

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

   if -4.8000000000000003e-65 < F

   1. Initial program 83.5%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 41.2%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 27.9%

      \[\leadsto \frac{-x}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 28.0% accurate, 26.3× speedup

Math

\[\begin{array}{l} \\ \frac{-x}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ (- x) B))

C

double code(double F, double B, double x) {
	return -x / B;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -x / b
end function

Java

public static double code(double F, double B, double x) {
	return -x / B;
}

Python

def code(F, B, x):
	return -x / B

Julia

function code(F, B, x)
	return Float64(Float64(-x) / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -x / B;
end

Wolfram

code[F_, B_, x_] := N[((-x) / B), $MachinePrecision]

Derivation

1. Initial program 78.2%

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

3. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

5. Applied rewrites 41.3%

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

6. Taylor expanded in F around 0

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

8. Applied rewrites 27.3%

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

9. Final simplification 27.3%

   \[\leadsto \frac{-x}{B} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025017 
(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))))))