VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.6% → 99.1%

Time: 11.6s

Alternatives: 24

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.6% 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: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -37000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{F}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -37000000.0)
   (/ (fma (- x) (cos B) -1.0) (sin B))
   (if (<= F 1.9e-10)
     (fma (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F (sin B)) (/ (- x) (tan B)))
     (/ (- 1.0 (* x (cos B))) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -37000000.0) {
		tmp = fma(-x, cos(B), -1.0) / sin(B);
	} else if (F <= 1.9e-10) {
		tmp = fma(pow(fma(2.0, x, fma(F, F, 2.0)), -0.5), (F / sin(B)), (-x / tan(B)));
	} else {
		tmp = (1.0 - (x * cos(B))) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -37000000.0)
		tmp = Float64(fma(Float64(-x), cos(B), -1.0) / sin(B));
	elseif (F <= 1.9e-10)
		tmp = fma((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5), Float64(F / sin(B)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(1.0 - Float64(x * cos(B))) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -37000000.0], N[(N[((-x) * N[Cos[B], $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.9e-10], N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.7e7

   1. Initial program 53.9%

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

   3. Taylor expanded in F around -inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -3.7e7 < F < 1.8999999999999999e-10

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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. *-commutative N/A

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

      5. lower-fma.f64 99.6

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

   4. Applied rewrites 99.7%

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

   if 1.8999999999999999e-10 < F

   1. Initial program 56.6%

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

   3. Taylor expanded in F around inf

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

      1. associate-/l* N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-sin.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -37000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{F}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -920000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -920000000.0)
   (/ (fma (- x) (cos B) -1.0) (sin B))
   (if (<= F 1.9e-10)
     (fma F (/ (sqrt (pow (fma F F 2.0) -1.0)) (sin B)) (/ (- x) (tan B)))
     (/ (- 1.0 (* x (cos B))) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -920000000.0) {
		tmp = fma(-x, cos(B), -1.0) / sin(B);
	} else if (F <= 1.9e-10) {
		tmp = fma(F, (sqrt(pow(fma(F, F, 2.0), -1.0)) / sin(B)), (-x / tan(B)));
	} else {
		tmp = (1.0 - (x * cos(B))) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -920000000.0)
		tmp = Float64(fma(Float64(-x), cos(B), -1.0) / sin(B));
	elseif (F <= 1.9e-10)
		tmp = fma(F, Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) / sin(B)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(1.0 - Float64(x * cos(B))) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -920000000.0], N[(N[((-x) * N[Cos[B], $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.9e-10], N[(F * N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.2e8

   1. Initial program 52.6%

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

   3. Taylor expanded in F around -inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -9.2e8 < F < 1.8999999999999999e-10

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 99.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 99.6

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

   7. Applied rewrites 99.6%

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

   if 1.8999999999999999e-10 < F

   1. Initial program 56.6%

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

   3. Taylor expanded in F around inf

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

      1. associate-/l* N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-sin.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -920000000:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}\\ \mathbf{if}\;B \leq 0.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right), \mathsf{fma}\left(\mathsf{fma}\left(\left(--0.019444444444444445\right) \cdot F, t\_0, 0.022222222222222223 \cdot x\right), B \cdot B, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0))))
   (if (<= B 0.8)
     (/
      (-
       (fma
        t_0
        (fma (* 0.16666666666666666 F) (* B B) F)
        (*
         (fma
          (fma (* (- -0.019444444444444445) F) t_0 (* 0.022222222222222223 x))
          (* B B)
          (* 0.3333333333333333 x))
         (* B B)))
       x)
      B)
     (+ (* x (/ -1.0 (tan B))) (pow B -1.0)))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0));
	double tmp;
	if (B <= 0.8) {
		tmp = (fma(t_0, fma((0.16666666666666666 * F), (B * B), F), (fma(fma((-(-0.019444444444444445) * F), t_0, (0.022222222222222223 * x)), (B * B), (0.3333333333333333 * x)) * (B * B))) - x) / B;
	} else {
		tmp = (x * (-1.0 / tan(B))) + pow(B, -1.0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0))
	tmp = 0.0
	if (B <= 0.8)
		tmp = Float64(Float64(fma(t_0, fma(Float64(0.16666666666666666 * F), Float64(B * B), F), Float64(fma(fma(Float64(Float64(-(-0.019444444444444445)) * F), t_0, Float64(0.022222222222222223 * x)), Float64(B * B), Float64(0.3333333333333333 * x)) * Float64(B * B))) - x) / B);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + (B ^ -1.0));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B, 0.8], N[(N[(N[(t$95$0 * N[(N[(0.16666666666666666 * F), $MachinePrecision] * N[(B * B), $MachinePrecision] + F), $MachinePrecision] + N[(N[(N[(N[((--0.019444444444444445) * F), $MachinePrecision] * t$95$0 + N[(0.022222222222222223 * x), $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[Power[B, -1.0], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 0.80000000000000004

   1. Initial program 70.2%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 65.1%

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

   if 0.80000000000000004 < B

   1. Initial program 84.0%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 57.4

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

   5. Applied rewrites 57.4%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 54.4%

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

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right), \mathsf{fma}\left(\mathsf{fma}\left(\left(--0.019444444444444445\right) \cdot F, \sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, 0.022222222222222223 \cdot x\right), B \cdot B, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.75:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\sqrt{{\left(\mathsf{fma}\left(1 + x, 2, F \cdot F\right)\right)}^{-1}}}{B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.75)
   (/ (fma (- x) (cos B) -1.0) (sin B))
   (if (<= F 1.9e-10)
     (fma
      F
      (/ (sqrt (pow (fma (+ 1.0 x) 2.0 (* F F)) -1.0)) B)
      (/ (- x) (tan B)))
     (/ (- 1.0 (* x (cos B))) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.75) {
		tmp = fma(-x, cos(B), -1.0) / sin(B);
	} else if (F <= 1.9e-10) {
		tmp = fma(F, (sqrt(pow(fma((1.0 + x), 2.0, (F * F)), -1.0)) / B), (-x / tan(B)));
	} else {
		tmp = (1.0 - (x * cos(B))) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.75)
		tmp = Float64(fma(Float64(-x), cos(B), -1.0) / sin(B));
	elseif (F <= 1.9e-10)
		tmp = fma(F, Float64(sqrt((fma(Float64(1.0 + x), 2.0, Float64(F * F)) ^ -1.0)) / B), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(1.0 - Float64(x * cos(B))) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.75], N[(N[((-x) * N[Cos[B], $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.9e-10], N[(F * N[(N[Sqrt[N[Power[N[(N[(1.0 + x), $MachinePrecision] * 2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.75

   1. Initial program 55.2%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-sin.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if -1.75 < F < 1.8999999999999999e-10

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 99.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in B around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-+r+ N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 88.5

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

   10. Applied rewrites 88.5%

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

   if 1.8999999999999999e-10 < F

   1. Initial program 56.6%

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

   3. Taylor expanded in F around inf

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

      1. associate-/l* N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-sin.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.75:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\sqrt{{\left(\mathsf{fma}\left(1 + x, 2, F \cdot F\right)\right)}^{-1}}}{B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.75:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.46 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\sqrt{{\left(\mathsf{fma}\left(1 + x, 2, F \cdot F\right)\right)}^{-1}}}{B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, 1\right)}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.75)
   (/ (fma (- x) (cos B) -1.0) (sin B))
   (if (<= F 1.46e+32)
     (fma
      F
      (/ (sqrt (pow (fma (+ 1.0 x) 2.0 (* F F)) -1.0)) B)
      (/ (- x) (tan B)))
     (/ (fma -1.0 x 1.0) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.75) {
		tmp = fma(-x, cos(B), -1.0) / sin(B);
	} else if (F <= 1.46e+32) {
		tmp = fma(F, (sqrt(pow(fma((1.0 + x), 2.0, (F * F)), -1.0)) / B), (-x / tan(B)));
	} else {
		tmp = fma(-1.0, x, 1.0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.75)
		tmp = Float64(fma(Float64(-x), cos(B), -1.0) / sin(B));
	elseif (F <= 1.46e+32)
		tmp = fma(F, Float64(sqrt((fma(Float64(1.0 + x), 2.0, Float64(F * F)) ^ -1.0)) / B), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(fma(-1.0, x, 1.0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.75], N[(N[((-x) * N[Cos[B], $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.46e+32], N[(F * N[(N[Sqrt[N[Power[N[(N[(1.0 + x), $MachinePrecision] * 2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * x + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.75

   1. Initial program 55.2%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-sin.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if -1.75 < F < 1.46000000000000005e32

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 99.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in B around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-+r+ N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 89.1

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

   10. Applied rewrites 89.1%

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

   if 1.46000000000000005e32 < F

   1. Initial program 53.2%

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

   3. Taylor expanded in F around inf

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

      1. associate-/l* N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-sin.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 87.2%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.75:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.46 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\sqrt{{\left(\mathsf{fma}\left(1 + x, 2, F \cdot F\right)\right)}^{-1}}}{B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, 1\right)}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.3 \cdot 10^{+87}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.46 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\sqrt{{\left(\mathsf{fma}\left(1 + x, 2, F \cdot F\right)\right)}^{-1}}}{B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, 1\right)}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.3e+87)
   (+ (- (/ x B)) (/ -1.0 (sin B)))
   (if (<= F 1.46e+32)
     (fma
      F
      (/ (sqrt (pow (fma (+ 1.0 x) 2.0 (* F F)) -1.0)) B)
      (/ (- x) (tan B)))
     (/ (fma -1.0 x 1.0) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.3e+87) {
		tmp = -(x / B) + (-1.0 / sin(B));
	} else if (F <= 1.46e+32) {
		tmp = fma(F, (sqrt(pow(fma((1.0 + x), 2.0, (F * F)), -1.0)) / B), (-x / tan(B)));
	} else {
		tmp = fma(-1.0, x, 1.0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.3e+87)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(-1.0 / sin(B)));
	elseif (F <= 1.46e+32)
		tmp = fma(F, Float64(sqrt((fma(Float64(1.0 + x), 2.0, Float64(F * F)) ^ -1.0)) / B), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(fma(-1.0, x, 1.0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.3e+87], N[((-N[(x / B), $MachinePrecision]) + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.46e+32], N[(F * N[(N[Sqrt[N[Power[N[(N[(1.0 + x), $MachinePrecision] * 2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] / B), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * x + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.3e87

   1. Initial program 41.7%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 37.2

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

   5. Applied rewrites 37.2%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 19.5

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

   8. Applied rewrites 19.5%

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

   9. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot \left(B \cdot B\right) - \frac{1}{3}, B \cdot B, 1\right)}{B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. lower-sin.f64 68.5

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

   11. Applied rewrites 68.5%

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

   12. Taylor expanded in B around 0

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

       1. lower-/.f64 81.9

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

   14. Applied rewrites 81.9%

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

   if -6.3e87 < F < 1.46000000000000005e32

   1. Initial program 98.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 99.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in B around 0

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

      1. associate-*l/ N/A

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

      2. *-lft-identity N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-+r+ N/A

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

      7. *-commutative N/A

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

      8. distribute-rgt1-in N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-+.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 87.0

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

   10. Applied rewrites 87.0%

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

   if 1.46000000000000005e32 < F

   1. Initial program 53.2%

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

   3. Taylor expanded in F around inf

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

      1. associate-/l* N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-sin.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 87.2%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.3 \cdot 10^{+87}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.46 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\sqrt{{\left(\mathsf{fma}\left(1 + x, 2, F \cdot F\right)\right)}^{-1}}}{B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, 1\right)}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.46 \cdot 10^{+32}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, 1\right)}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e+18)
   (+ (- (/ x B)) (/ -1.0 (sin B)))
   (if (<= F 1.46e+32)
     (+
      (/ (- x) (tan B))
      (* (/ F B) (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0))))
     (/ (fma -1.0 x 1.0) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e+18) {
		tmp = -(x / B) + (-1.0 / sin(B));
	} else if (F <= 1.46e+32) {
		tmp = (-x / tan(B)) + ((F / B) * sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)));
	} else {
		tmp = fma(-1.0, x, 1.0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2e+18)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(-1.0 / sin(B)));
	elseif (F <= 1.46e+32)
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(Float64(F / B) * sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0))));
	else
		tmp = Float64(fma(-1.0, x, 1.0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2e+18], N[((-N[(x / B), $MachinePrecision]) + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.46e+32], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * x + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2e18

   1. Initial program 51.2%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 41.9

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

   5. Applied rewrites 41.9%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 21.8

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

   8. Applied rewrites 21.8%

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

   9. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot \left(B \cdot B\right) - \frac{1}{3}, B \cdot B, 1\right)}{B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. lower-sin.f64 66.2

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

   11. Applied rewrites 66.2%

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

   12. Taylor expanded in B around 0

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

       1. lower-/.f64 79.2

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

   14. Applied rewrites 79.2%

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

   if -2e18 < F < 1.46000000000000005e32

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 88.7

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

   5. Applied rewrites 88.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. *-rgt-identity N/A

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

      6. distribute-frac-neg N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 88.8

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

   7. Applied rewrites 88.8%

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

   if 1.46000000000000005e32 < F

   1. Initial program 53.2%

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

   3. Taylor expanded in F around inf

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

      1. associate-/l* N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-sin.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 87.2%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.46 \cdot 10^{+32}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, 1\right)}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.75:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+21}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, 1\right)}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.75)
   (+ (- (/ x B)) (/ -1.0 (sin B)))
   (if (<= F 6e+21)
     (+ (/ (- x) (tan B)) (* (/ F B) (sqrt (pow (fma 2.0 x 2.0) -1.0))))
     (/ (fma -1.0 x 1.0) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.75) {
		tmp = -(x / B) + (-1.0 / sin(B));
	} else if (F <= 6e+21) {
		tmp = (-x / tan(B)) + ((F / B) * sqrt(pow(fma(2.0, x, 2.0), -1.0)));
	} else {
		tmp = fma(-1.0, x, 1.0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.75)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(-1.0 / sin(B)));
	elseif (F <= 6e+21)
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(Float64(F / B) * sqrt((fma(2.0, x, 2.0) ^ -1.0))));
	else
		tmp = Float64(fma(-1.0, x, 1.0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.75], N[((-N[(x / B), $MachinePrecision]) + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6e+21], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * x + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.75

   1. Initial program 55.2%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 45.4

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

   5. Applied rewrites 45.4%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 25.5

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

   8. Applied rewrites 25.5%

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

   9. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot \left(B \cdot B\right) - \frac{1}{3}, B \cdot B, 1\right)}{B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

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

       2. lower-sin.f64 66.3

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

   11. Applied rewrites 66.3%

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

   12. Taylor expanded in B around 0

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

       1. lower-/.f64 78.4

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

   14. Applied rewrites 78.4%

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

   if -1.75 < F < 6e21

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 88.7

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 87.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. distribute-neg-frac N/A

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

      6. *-rgt-identity N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 88.0

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

   10. Applied rewrites 88.0%

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

   if 6e21 < F

   1. Initial program 55.0%

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

   3. Taylor expanded in F around inf

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

      1. associate-/l* N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-sin.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 86.7%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.75:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+21}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{F}{B} \cdot \sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, 1\right)}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 98.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\sqrt{0.5}}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.4)
   (/ (fma (- x) (cos B) -1.0) (sin B))
   (if (<= F 1.9e-10)
     (fma F (/ (sqrt 0.5) (sin B)) (/ (- x) (tan B)))
     (/ (- 1.0 (* x (cos B))) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4) {
		tmp = fma(-x, cos(B), -1.0) / sin(B);
	} else if (F <= 1.9e-10) {
		tmp = fma(F, (sqrt(0.5) / sin(B)), (-x / tan(B)));
	} else {
		tmp = (1.0 - (x * cos(B))) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(fma(Float64(-x), cos(B), -1.0) / sin(B));
	elseif (F <= 1.9e-10)
		tmp = fma(F, Float64(sqrt(0.5) / sin(B)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(1.0 - Float64(x * cos(B))) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.4], N[(N[((-x) * N[Cos[B], $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.9e-10], N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 55.2%

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

   3. Taylor expanded in 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. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. metadata-eval N/A

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

      6. div-add-rev N/A

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

      7. lower-/.f64 N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-sin.f64 98.7

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

   5. Applied rewrites 98.7%

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

   if -1.3999999999999999 < F < 1.8999999999999999e-10

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 99.6%

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

   5. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 99.6

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in F around 0

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

   10. Applied rewrites 99.6%

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

   if 1.8999999999999999e-10 < F

   1. Initial program 56.6%

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

   3. Taylor expanded in F around inf

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

      1. associate-/l* N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-sin.f64 99.9

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

   5. Applied rewrites 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\sqrt{0.5}}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x \cdot \cos B}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B} + {B}^{-1}\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -2.45 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-64}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ (* x (/ -1.0 (tan B))) (pow B -1.0))))
   (if (<= x -5.8e-11)
     t_0
     (if (<= x -2.45e-83)
       (/ (- (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
       (if (<= x 8.6e-64)
         (* (sqrt (pow (fma F F 2.0) -1.0)) (/ F (sin B)))
         t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (x * (-1.0 / tan(B))) + pow(B, -1.0);
	double tmp;
	if (x <= -5.8e-11) {
		tmp = t_0;
	} else if (x <= -2.45e-83) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else if (x <= 8.6e-64) {
		tmp = sqrt(pow(fma(F, F, 2.0), -1.0)) * (F / sin(B));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(x * Float64(-1.0 / tan(B))) + (B ^ -1.0))
	tmp = 0.0
	if (x <= -5.8e-11)
		tmp = t_0;
	elseif (x <= -2.45e-83)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	elseif (x <= 8.6e-64)
		tmp = Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * Float64(F / sin(B)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.8e-11], t$95$0, If[LessEqual[x, -2.45e-83], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[x, 8.6e-64], N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.8e-11 or 8.59999999999999947e-64 < x

   1. Initial program 75.2%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 74.4

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

   5. Applied rewrites 74.4%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 96.4%

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

   if -5.8e-11 < x < -2.45e-83

   1. Initial program 65.1%

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

   3. 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

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 60.4

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

   5. Applied rewrites 60.4%

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

   7. Applied rewrites 60.6%

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

   if -2.45e-83 < x < 8.59999999999999947e-64

   1. Initial program 73.3%

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

   3. Taylor expanded in 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 56.7

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

   5. Applied rewrites 56.7%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-11}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {B}^{-1}\\ \mathbf{elif}\;x \leq -2.45 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-64}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right), \left(0.3333333333333333 \cdot \left(B \cdot B\right)\right) \cdot x\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.8)
   (/
    (-
     (fma
      (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0))
      (fma (* 0.16666666666666666 F) (* B B) F)
      (* (* 0.3333333333333333 (* B B)) x))
     x)
    B)
   (+ (* x (/ -1.0 (tan B))) (pow B -1.0))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.8) {
		tmp = (fma(sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)), fma((0.16666666666666666 * F), (B * B), F), ((0.3333333333333333 * (B * B)) * x)) - x) / B;
	} else {
		tmp = (x * (-1.0 / tan(B))) + pow(B, -1.0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.8)
		tmp = Float64(Float64(fma(sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0)), fma(Float64(0.16666666666666666 * F), Float64(B * B), F), Float64(Float64(0.3333333333333333 * Float64(B * B)) * x)) - x) / B);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + (B ^ -1.0));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.8], N[(N[(N[(N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[(0.16666666666666666 * F), $MachinePrecision] * N[(B * B), $MachinePrecision] + F), $MachinePrecision] + N[(N[(0.3333333333333333 * N[(B * B), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.80000000000000004

   1. Initial program 70.2%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 64.7%

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

   if 0.80000000000000004 < B

   1. Initial program 84.0%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. unpow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-fma.f64 57.4

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

   5. Applied rewrites 57.4%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 54.4%

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

4. Final simplification 62.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.8:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right), \left(0.3333333333333333 \cdot \left(B \cdot B\right)\right) \cdot x\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 57.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right), \left(0.3333333333333333 \cdot \left(B \cdot B\right)\right) \cdot x\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, 1\right)}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.2e+133)
   (/ (- -1.0 x) B)
   (if (<= F 1.9e-10)
     (/
      (-
       (fma
        (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0))
        (fma (* 0.16666666666666666 F) (* B B) F)
        (* (* 0.3333333333333333 (* B B)) x))
       x)
      B)
     (/ (fma -1.0 x 1.0) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.2e+133) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.9e-10) {
		tmp = (fma(sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)), fma((0.16666666666666666 * F), (B * B), F), ((0.3333333333333333 * (B * B)) * x)) - x) / B;
	} else {
		tmp = fma(-1.0, x, 1.0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.2e+133)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.9e-10)
		tmp = Float64(Float64(fma(sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0)), fma(Float64(0.16666666666666666 * F), Float64(B * B), F), Float64(Float64(0.3333333333333333 * Float64(B * B)) * x)) - x) / B);
	else
		tmp = Float64(fma(-1.0, x, 1.0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.2e+133], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.9e-10], N[(N[(N[(N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[(0.16666666666666666 * F), $MachinePrecision] * N[(B * B), $MachinePrecision] + F), $MachinePrecision] + N[(N[(0.3333333333333333 * N[(B * B), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(-1.0 * x + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.2e133

   1. Initial program 31.0%

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

   3. Taylor expanded in 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

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 36.7

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

   5. Applied rewrites 36.7%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 56.6%

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

   if -6.2e133 < F < 1.8999999999999999e-10

   1. Initial program 96.8%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 57.0%

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

   if 1.8999999999999999e-10 < F

   1. Initial program 56.6%

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

   3. Taylor expanded in F around inf

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

      1. associate-/l* N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-sin.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 85.9%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right), \left(0.3333333333333333 \cdot \left(B \cdot B\right)\right) \cdot x\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, 1\right)}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.16666666666666666, -x, 0.16666666666666666\right)\\ \mathbf{if}\;F \leq -6.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{+61}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right), \left(0.3333333333333333 \cdot \left(B \cdot B\right)\right) \cdot x\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(-0.041666666666666664 \cdot x - \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, t\_0\right), -0.16666666666666666, \mathsf{fma}\left(0.008333333333333333, -x, 0.008333333333333333\right)\right), B \cdot B, t\_0\right)\right), B \cdot B, -x\right) + 1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (fma 0.16666666666666666 (- x) 0.16666666666666666)))
   (if (<= F -6.2e+133)
     (/ (- -1.0 x) B)
     (if (<= F 2.2e+61)
       (/
        (-
         (fma
          (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0))
          (fma (* 0.16666666666666666 F) (* B B) F)
          (* (* 0.3333333333333333 (* B B)) x))
         x)
        B)
       (/
        (+
         (fma
          (fma
           0.5
           x
           (fma
            (-
             (* -0.041666666666666664 x)
             (fma
              (fma 0.5 x t_0)
              -0.16666666666666666
              (fma 0.008333333333333333 (- x) 0.008333333333333333)))
            (* B B)
            t_0))
          (* B B)
          (- x))
         1.0)
        B)))))

C

double code(double F, double B, double x) {
	double t_0 = fma(0.16666666666666666, -x, 0.16666666666666666);
	double tmp;
	if (F <= -6.2e+133) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.2e+61) {
		tmp = (fma(sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)), fma((0.16666666666666666 * F), (B * B), F), ((0.3333333333333333 * (B * B)) * x)) - x) / B;
	} else {
		tmp = (fma(fma(0.5, x, fma(((-0.041666666666666664 * x) - fma(fma(0.5, x, t_0), -0.16666666666666666, fma(0.008333333333333333, -x, 0.008333333333333333))), (B * B), t_0)), (B * B), -x) + 1.0) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(0.16666666666666666, Float64(-x), 0.16666666666666666)
	tmp = 0.0
	if (F <= -6.2e+133)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.2e+61)
		tmp = Float64(Float64(fma(sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0)), fma(Float64(0.16666666666666666 * F), Float64(B * B), F), Float64(Float64(0.3333333333333333 * Float64(B * B)) * x)) - x) / B);
	else
		tmp = Float64(Float64(fma(fma(0.5, x, fma(Float64(Float64(-0.041666666666666664 * x) - fma(fma(0.5, x, t_0), -0.16666666666666666, fma(0.008333333333333333, Float64(-x), 0.008333333333333333))), Float64(B * B), t_0)), Float64(B * B), Float64(-x)) + 1.0) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(0.16666666666666666 * (-x) + 0.16666666666666666), $MachinePrecision]}, If[LessEqual[F, -6.2e+133], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.2e+61], N[(N[(N[(N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[(0.16666666666666666 * F), $MachinePrecision] * N[(B * B), $MachinePrecision] + F), $MachinePrecision] + N[(N[(0.3333333333333333 * N[(B * B), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(0.5 * x + N[(N[(N[(-0.041666666666666664 * x), $MachinePrecision] - N[(N[(0.5 * x + t$95$0), $MachinePrecision] * -0.16666666666666666 + N[(0.008333333333333333 * (-x) + 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + (-x)), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.2e133

   1. Initial program 31.0%

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

   3. Taylor expanded in 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

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 36.7

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

   5. Applied rewrites 36.7%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 56.6%

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

   if -6.2e133 < F < 2.2e61

   1. Initial program 97.0%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 56.6%

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

   if 2.2e61 < F

   1. Initial program 49.1%

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

   3. Taylor expanded in F around inf

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

      1. associate-/l* N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in B around 0

      \[\leadsto \frac{1 + \left(-1 \cdot x + {B}^{2} \cdot \left(\left(\frac{1}{2} \cdot x + {B}^{2} \cdot \left(\frac{-1}{24} \cdot x - \left(\frac{-1}{6} \cdot \left(\frac{1}{2} \cdot x - \frac{-1}{6} \cdot \left(1 + -1 \cdot x\right)\right) + \frac{1}{120} \cdot \left(1 + -1 \cdot x\right)\right)\right)\right) - \frac{-1}{6} \cdot \left(1 + -1 \cdot x\right)\right)\right)}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.9%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(-0.041666666666666664 \cdot x - \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(0.16666666666666666, -x, 0.16666666666666666\right)\right), -0.16666666666666666, \mathsf{fma}\left(0.008333333333333333, -x, 0.008333333333333333\right)\right), B \cdot B, \mathsf{fma}\left(0.16666666666666666, -x, 0.16666666666666666\right)\right)\right), B \cdot B, -x\right) + 1}{\color{blue}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{+61}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right), \left(0.3333333333333333 \cdot \left(B \cdot B\right)\right) \cdot x\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(-0.041666666666666664 \cdot x - \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(0.16666666666666666, -x, 0.16666666666666666\right)\right), -0.16666666666666666, \mathsf{fma}\left(0.008333333333333333, -x, 0.008333333333333333\right)\right), B \cdot B, \mathsf{fma}\left(0.16666666666666666, -x, 0.16666666666666666\right)\right)\right), B \cdot B, -x\right) + 1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right), \left(0.3333333333333333 \cdot \left(B \cdot B\right)\right) \cdot x\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(0.16666666666666666, -x, 0.16666666666666666\right)\right), B \cdot B, -x\right) + 1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.2e+133)
   (/ (- -1.0 x) B)
   (if (<= F 1.9e-10)
     (/
      (-
       (fma
        (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0))
        (fma (* 0.16666666666666666 F) (* B B) F)
        (* (* 0.3333333333333333 (* B B)) x))
       x)
      B)
     (/
      (+
       (fma
        (fma 0.5 x (fma 0.16666666666666666 (- x) 0.16666666666666666))
        (* B B)
        (- x))
       1.0)
      B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.2e+133) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.9e-10) {
		tmp = (fma(sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)), fma((0.16666666666666666 * F), (B * B), F), ((0.3333333333333333 * (B * B)) * x)) - x) / B;
	} else {
		tmp = (fma(fma(0.5, x, fma(0.16666666666666666, -x, 0.16666666666666666)), (B * B), -x) + 1.0) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.2e+133)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.9e-10)
		tmp = Float64(Float64(fma(sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0)), fma(Float64(0.16666666666666666 * F), Float64(B * B), F), Float64(Float64(0.3333333333333333 * Float64(B * B)) * x)) - x) / B);
	else
		tmp = Float64(Float64(fma(fma(0.5, x, fma(0.16666666666666666, Float64(-x), 0.16666666666666666)), Float64(B * B), Float64(-x)) + 1.0) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.2e+133], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.9e-10], N[(N[(N[(N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(N[(0.16666666666666666 * F), $MachinePrecision] * N[(B * B), $MachinePrecision] + F), $MachinePrecision] + N[(N[(0.3333333333333333 * N[(B * B), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(0.5 * x + N[(0.16666666666666666 * (-x) + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + (-x)), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.2e133

   1. Initial program 31.0%

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

   3. Taylor expanded in 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

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 36.7

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

   5. Applied rewrites 36.7%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 56.6%

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

   if -6.2e133 < F < 1.8999999999999999e-10

   1. Initial program 96.8%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 57.0%

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

   if 1.8999999999999999e-10 < F

   1. Initial program 56.6%

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

   3. Taylor expanded in F around inf

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

      1. associate-/l* N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-sin.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 56.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(0.16666666666666666, -x, 0.16666666666666666\right)\right), B \cdot B, -x\right) + 1}{\color{blue}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.2 \cdot 10^{+133}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right), \left(0.3333333333333333 \cdot \left(B \cdot B\right)\right) \cdot x\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(0.16666666666666666, -x, 0.16666666666666666\right)\right), B \cdot B, -x\right) + 1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 43.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}\\ \mathbf{if}\;F \leq -1.15 \cdot 10^{-45}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -2.2 \cdot 10^{-167}:\\ \;\;\;\;t\_0 \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{-80}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 10^{-13}:\\ \;\;\;\;\frac{t\_0 \cdot F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(0.16666666666666666, -x, 0.16666666666666666\right)\right), B \cdot B, -x\right) + 1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (pow (fma F F 2.0) -1.0))))
   (if (<= F -1.15e-45)
     (/ (- -1.0 x) B)
     (if (<= F -2.2e-167)
       (* t_0 (/ F B))
       (if (<= F 1.22e-80)
         (/ (- x) B)
         (if (<= F 1e-13)
           (/ (* t_0 F) B)
           (/
            (+
             (fma
              (fma 0.5 x (fma 0.16666666666666666 (- x) 0.16666666666666666))
              (* B B)
              (- x))
             1.0)
            B)))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt(pow(fma(F, F, 2.0), -1.0));
	double tmp;
	if (F <= -1.15e-45) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -2.2e-167) {
		tmp = t_0 * (F / B);
	} else if (F <= 1.22e-80) {
		tmp = -x / B;
	} else if (F <= 1e-13) {
		tmp = (t_0 * F) / B;
	} else {
		tmp = (fma(fma(0.5, x, fma(0.16666666666666666, -x, 0.16666666666666666)), (B * B), -x) + 1.0) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = sqrt((fma(F, F, 2.0) ^ -1.0))
	tmp = 0.0
	if (F <= -1.15e-45)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -2.2e-167)
		tmp = Float64(t_0 * Float64(F / B));
	elseif (F <= 1.22e-80)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 1e-13)
		tmp = Float64(Float64(t_0 * F) / B);
	else
		tmp = Float64(Float64(fma(fma(0.5, x, fma(0.16666666666666666, Float64(-x), 0.16666666666666666)), Float64(B * B), Float64(-x)) + 1.0) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[F, -1.15e-45], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -2.2e-167], N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.22e-80], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 1e-13], N[(N[(t$95$0 * F), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(0.5 * x + N[(0.16666666666666666 * (-x) + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + (-x)), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.14999999999999996e-45

   1. Initial program 59.5%

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

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 47.2

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

   5. Applied rewrites 47.2%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 56.1%

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

   if -1.14999999999999996e-45 < F < -2.2e-167

   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

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 76.7

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.6%

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

   if -2.2e-167 < F < 1.22e-80

   1. Initial program 99.6%

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

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 49.7

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 44.2%

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

   if 1.22e-80 < F < 1e-13

   1. Initial program 99.2%

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

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 55.1

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

   5. Applied rewrites 55.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 47.8%

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

   if 1e-13 < F

   1. Initial program 57.7%

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

   3. Taylor expanded in F around inf

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

      1. associate-/l* N/A

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

      2. fp-cancel-sub-sign-inv N/A

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

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

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

      4. associate-/l* N/A

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

      5. mul-1-neg N/A

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

      6. associate-*r/ N/A

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

      7. div-add-rev N/A

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

      8. lower-/.f64 N/A

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

      9. lower-+.f64 N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-*.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. lower-cos.f64 N/A

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

      15. lower-sin.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 56.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(0.16666666666666666, -x, 0.16666666666666666\right)\right), B \cdot B, -x\right) + 1}{\color{blue}{B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.15 \cdot 10^{-45}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -2.2 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{-80}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 10^{-13}:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(0.16666666666666666, -x, 0.16666666666666666\right)\right), B \cdot B, -x\right) + 1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 43.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{B}\\ \mathbf{if}\;F \leq -1.15 \cdot 10^{-45}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -2.2 \cdot 10^{-167}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{-80}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 10^{-13}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(0.16666666666666666, -x, 0.16666666666666666\right)\right), B \cdot B, -x\right) + 1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (sqrt (pow (fma F F 2.0) -1.0)) (/ F B))))
   (if (<= F -1.15e-45)
     (/ (- -1.0 x) B)
     (if (<= F -2.2e-167)
       t_0
       (if (<= F 1.22e-80)
         (/ (- x) B)
         (if (<= F 1e-13)
           t_0
           (/
            (+
             (fma
              (fma 0.5 x (fma 0.16666666666666666 (- x) 0.16666666666666666))
              (* B B)
              (- x))
             1.0)
            B)))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt(pow(fma(F, F, 2.0), -1.0)) * (F / B);
	double tmp;
	if (F <= -1.15e-45) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -2.2e-167) {
		tmp = t_0;
	} else if (F <= 1.22e-80) {
		tmp = -x / B;
	} else if (F <= 1e-13) {
		tmp = t_0;
	} else {
		tmp = (fma(fma(0.5, x, fma(0.16666666666666666, -x, 0.16666666666666666)), (B * B), -x) + 1.0) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * Float64(F / B))
	tmp = 0.0
	if (F <= -1.15e-45)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -2.2e-167)
		tmp = t_0;
	elseif (F <= 1.22e-80)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 1e-13)
		tmp = t_0;
	else
		tmp = Float64(Float64(fma(fma(0.5, x, fma(0.16666666666666666, Float64(-x), 0.16666666666666666)), Float64(B * B), Float64(-x)) + 1.0) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.15e-45], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -2.2e-167], t$95$0, If[LessEqual[F, 1.22e-80], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 1e-13], t$95$0, N[(N[(N[(N[(0.5 * x + N[(0.16666666666666666 * (-x) + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + (-x)), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.14999999999999996e-45

   1. Initial program 59.5%

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

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 47.2

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

   5. Applied rewrites 47.2%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 56.1%

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

   if -1.14999999999999996e-45 < F < -2.2e-167 or 1.22e-80 < F < 1e-13

   1. Initial program 99.3%

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

   3. Taylor expanded in 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

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 67.3

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

   5. Applied rewrites 67.3%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.8%

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

   if -2.2e-167 < F < 1.22e-80

   1. Initial program 99.6%

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

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-sqrt.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. associate-+r+ N/A

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

      8. +-commutative N/A

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

      9. unpow2 N/A

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

      10. lower-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 49.7

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 44.2%

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

   if 1e-13 < F

   1. Initial program 57.7%

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

   3. Taylor expanded in F around inf

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

      1. associate-/l* N/A

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

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\frac{1}{\sin B} + \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{\sin B} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{\cos B}{\sin B}\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto \frac{1}{\sin B} + \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \cos B}{\sin B}}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \frac{1}{\sin B} + \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{1}{\sin B} + \color{blue}{\frac{-1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

      7. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{1 + -1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + -1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + -1 \cdot \left(x \cdot \cos B\right)}}{\sin B} \]

      10. associate-*r* N/A

          \[\leadsto \frac{1 + \color{blue}{\left(-1 \cdot x\right) \cdot \cos B}}{\sin B} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \cos B}{\sin B} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \cos B}}{\sin B} \]

      13. lower-neg.f64 N/A

          \[\leadsto \frac{1 + \color{blue}{\left(-x\right)} \cdot \cos B}{\sin B} \]

      14. lower-cos.f64 N/A

          \[\leadsto \frac{1 + \left(-x\right) \cdot \color{blue}{\cos B}}{\sin B} \]

      15. lower-sin.f64 98.8

          \[\leadsto \frac{1 + \left(-x\right) \cdot \cos B}{\color{blue}{\sin B}} \]

   5. Applied rewrites 98.8%

      \[\leadsto \color{blue}{\frac{1 + \left(-x\right) \cdot \cos B}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \frac{1 + \left(-1 \cdot x + {B}^{2} \cdot \left(\frac{1}{2} \cdot x - \frac{-1}{6} \cdot \left(1 + -1 \cdot x\right)\right)\right)}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(0.16666666666666666, -x, 0.16666666666666666\right)\right), B \cdot B, -x\right) + 1}{\color{blue}{B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 52.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.15 \cdot 10^{-45}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -2.2 \cdot 10^{-167}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{-80}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 10^{-13}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(0.16666666666666666, -x, 0.16666666666666666\right)\right), B \cdot B, -x\right) + 1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 64.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.75:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+21}:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, 1\right)}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.75)
   (+ (- (/ x B)) (/ -1.0 (sin B)))
   (if (<= F 6e+21)
     (/ (- (* (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0)) F) x) B)
     (/ (fma -1.0 x 1.0) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.75) {
		tmp = -(x / B) + (-1.0 / sin(B));
	} else if (F <= 6e+21) {
		tmp = ((sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)) * F) - x) / B;
	} else {
		tmp = fma(-1.0, x, 1.0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.75)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(-1.0 / sin(B)));
	elseif (F <= 6e+21)
		tmp = Float64(Float64(Float64(sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0)) * F) - x) / B);
	else
		tmp = Float64(fma(-1.0, x, 1.0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.75], N[((-N[(x / B), $MachinePrecision]) + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6e+21], N[(N[(N[(N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(-1.0 * x + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.75

   1. Initial program 55.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B}} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      5. associate-+r+ N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \]

      7. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \]

      9. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \]

      10. lower-fma.f64 45.4

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \]

   5. Applied rewrites 45.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \]

      2. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{{B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) + 1}}{B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \]

      3. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) \cdot {B}^{2}} + 1}{B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}, {B}^{2}, 1\right)}}{B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \]

      5. lower--.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}}, {B}^{2}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{45} \cdot {B}^{2}} - \frac{1}{3}, {B}^{2}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \]

      7. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot \color{blue}{\left(B \cdot B\right)} - \frac{1}{3}, {B}^{2}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \]

      8. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot \color{blue}{\left(B \cdot B\right)} - \frac{1}{3}, {B}^{2}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \]

      9. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot \left(B \cdot B\right) - \frac{1}{3}, \color{blue}{B \cdot B}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \]

      10. lower-*.f64 25.5

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(-0.022222222222222223 \cdot \left(B \cdot B\right) - 0.3333333333333333, \color{blue}{B \cdot B}, 1\right)}{B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \]

   8. Applied rewrites 25.5%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(-0.022222222222222223 \cdot \left(B \cdot B\right) - 0.3333333333333333, B \cdot B, 1\right)}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \]

   9. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot \left(B \cdot B\right) - \frac{1}{3}, B \cdot B, 1\right)}{B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot \left(B \cdot B\right) - \frac{1}{3}, B \cdot B, 1\right)}{B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

       2. lower-sin.f64 66.3

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(-0.022222222222222223 \cdot \left(B \cdot B\right) - 0.3333333333333333, B \cdot B, 1\right)}{B}\right) + \frac{-1}{\color{blue}{\sin B}} \]

   11. Applied rewrites 66.3%

       \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(-0.022222222222222223 \cdot \left(B \cdot B\right) - 0.3333333333333333, B \cdot B, 1\right)}{B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   12. Taylor expanded in B around 0

       \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]
   13. Step-by-step derivation

       1. lower-/.f64 78.4

          \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   14. Applied rewrites 78.4%

       \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   if -1.75 < F < 6e21

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 55.7

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   if 6e21 < F

   1. Initial program 55.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{1}{\sin B} - \color{blue}{x \cdot \frac{\cos B}{\sin B}} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\frac{1}{\sin B} + \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{\sin B} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{\cos B}{\sin B}\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto \frac{1}{\sin B} + \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \cos B}{\sin B}}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \frac{1}{\sin B} + \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{1}{\sin B} + \color{blue}{\frac{-1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

      7. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{1 + -1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + -1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + -1 \cdot \left(x \cdot \cos B\right)}}{\sin B} \]

      10. associate-*r* N/A

          \[\leadsto \frac{1 + \color{blue}{\left(-1 \cdot x\right) \cdot \cos B}}{\sin B} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \cos B}{\sin B} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \cos B}}{\sin B} \]

      13. lower-neg.f64 N/A

          \[\leadsto \frac{1 + \color{blue}{\left(-x\right)} \cdot \cos B}{\sin B} \]

      14. lower-cos.f64 N/A

          \[\leadsto \frac{1 + \left(-x\right) \cdot \color{blue}{\cos B}}{\sin B} \]

      15. lower-sin.f64 99.9

          \[\leadsto \frac{1 + \left(-x\right) \cdot \cos B}{\color{blue}{\sin B}} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{1 + \left(-x\right) \cdot \cos B}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \frac{1 + -1 \cdot x}{\sin \color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 86.7%

      \[\leadsto \frac{\mathsf{fma}\left(-1, x, 1\right)}{\sin \color{blue}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.75:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+21}:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, 1\right)}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 51.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+53}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{+21}:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(0.16666666666666666, -x, 0.16666666666666666\right)\right), B \cdot B, -x\right) + 1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e+53)
   (/ (- -1.0 x) B)
   (if (<= F 9.2e+21)
     (/ (- (* (sqrt (pow (fma F F (fma 2.0 x 2.0)) -1.0)) F) x) B)
     (/
      (+
       (fma
        (fma 0.5 x (fma 0.16666666666666666 (- x) 0.16666666666666666))
        (* B B)
        (- x))
       1.0)
      B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e+53) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 9.2e+21) {
		tmp = ((sqrt(pow(fma(F, F, fma(2.0, x, 2.0)), -1.0)) * F) - x) / B;
	} else {
		tmp = (fma(fma(0.5, x, fma(0.16666666666666666, -x, 0.16666666666666666)), (B * B), -x) + 1.0) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2e+53)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 9.2e+21)
		tmp = Float64(Float64(Float64(sqrt((fma(F, F, fma(2.0, x, 2.0)) ^ -1.0)) * F) - x) / B);
	else
		tmp = Float64(Float64(fma(fma(0.5, x, fma(0.16666666666666666, Float64(-x), 0.16666666666666666)), Float64(B * B), Float64(-x)) + 1.0) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2e+53], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 9.2e+21], N[(N[(N[(N[Sqrt[N[Power[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(0.5 * x + N[(0.16666666666666666 * (-x) + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + (-x)), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2e53

   1. Initial program 47.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 46.5

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 46.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 60.7%

      \[\leadsto \frac{-1 - x}{B} \]

   if -2e53 < F < 9.2e21

   1. Initial program 98.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. 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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 55.2

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 55.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   if 9.2e21 < F

   1. Initial program 55.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{1}{\sin B} - \color{blue}{x \cdot \frac{\cos B}{\sin B}} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\frac{1}{\sin B} + \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{\sin B} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{\cos B}{\sin B}\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto \frac{1}{\sin B} + \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \cos B}{\sin B}}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \frac{1}{\sin B} + \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{1}{\sin B} + \color{blue}{\frac{-1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

      7. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{1 + -1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + -1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + -1 \cdot \left(x \cdot \cos B\right)}}{\sin B} \]

      10. associate-*r* N/A

          \[\leadsto \frac{1 + \color{blue}{\left(-1 \cdot x\right) \cdot \cos B}}{\sin B} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \cos B}{\sin B} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \cos B}}{\sin B} \]

      13. lower-neg.f64 N/A

          \[\leadsto \frac{1 + \color{blue}{\left(-x\right)} \cdot \cos B}{\sin B} \]

      14. lower-cos.f64 N/A

          \[\leadsto \frac{1 + \left(-x\right) \cdot \color{blue}{\cos B}}{\sin B} \]

      15. lower-sin.f64 99.9

          \[\leadsto \frac{1 + \left(-x\right) \cdot \cos B}{\color{blue}{\sin B}} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{1 + \left(-x\right) \cdot \cos B}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \frac{1 + \left(-1 \cdot x + {B}^{2} \cdot \left(\frac{1}{2} \cdot x - \frac{-1}{6} \cdot \left(1 + -1 \cdot x\right)\right)\right)}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.8%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(0.16666666666666666, -x, 0.16666666666666666\right)\right), B \cdot B, -x\right) + 1}{\color{blue}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+53}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{+21}:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(0.16666666666666666, -x, 0.16666666666666666\right)\right), B \cdot B, -x\right) + 1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 50.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(0.16666666666666666, -x, 0.16666666666666666\right)\right), B \cdot B, -x\right) + 1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.4)
   (/ (- -1.0 x) B)
   (if (<= F 1.9e-10)
     (/ (fma (sqrt (pow (fma 2.0 x 2.0) -1.0)) F (- x)) B)
     (/
      (+
       (fma
        (fma 0.5 x (fma 0.16666666666666666 (- x) 0.16666666666666666))
        (* B B)
        (- x))
       1.0)
      B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.9e-10) {
		tmp = fma(sqrt(pow(fma(2.0, x, 2.0), -1.0)), F, -x) / B;
	} else {
		tmp = (fma(fma(0.5, x, fma(0.16666666666666666, -x, 0.16666666666666666)), (B * B), -x) + 1.0) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.9e-10)
		tmp = Float64(fma(sqrt((fma(2.0, x, 2.0) ^ -1.0)), F, Float64(-x)) / B);
	else
		tmp = Float64(Float64(fma(fma(0.5, x, fma(0.16666666666666666, Float64(-x), 0.16666666666666666)), Float64(B * B), Float64(-x)) + 1.0) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.4], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.9e-10], N[(N[(N[Sqrt[N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F + (-x)), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(0.5 * x + N[(0.16666666666666666 * (-x) + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + (-x)), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 55.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 47.6

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 47.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 59.0%

      \[\leadsto \frac{-1 - x}{B} \]

   if -1.3999999999999999 < F < 1.8999999999999999e-10

   1. Initial program 99.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 55.3

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 55.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto -1 \cdot \frac{x}{B} + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + 2 \cdot x}}} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.3%

      \[\leadsto \frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}}, F, -x\right)}{\color{blue}{B}} \]

   if 1.8999999999999999e-10 < F

   1. Initial program 56.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{1}{\sin B} - \color{blue}{x \cdot \frac{\cos B}{\sin B}} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\frac{1}{\sin B} + \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{\sin B} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{\cos B}{\sin B}\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto \frac{1}{\sin B} + \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \cos B}{\sin B}}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \frac{1}{\sin B} + \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{1}{\sin B} + \color{blue}{\frac{-1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

      7. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{1 + -1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + -1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + -1 \cdot \left(x \cdot \cos B\right)}}{\sin B} \]

      10. associate-*r* N/A

          \[\leadsto \frac{1 + \color{blue}{\left(-1 \cdot x\right) \cdot \cos B}}{\sin B} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \cos B}{\sin B} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \cos B}}{\sin B} \]

      13. lower-neg.f64 N/A

          \[\leadsto \frac{1 + \color{blue}{\left(-x\right)} \cdot \cos B}{\sin B} \]

      14. lower-cos.f64 N/A

          \[\leadsto \frac{1 + \left(-x\right) \cdot \color{blue}{\cos B}}{\sin B} \]

      15. lower-sin.f64 99.9

          \[\leadsto \frac{1 + \left(-x\right) \cdot \cos B}{\color{blue}{\sin B}} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{1 + \left(-x\right) \cdot \cos B}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \frac{1 + \left(-1 \cdot x + {B}^{2} \cdot \left(\frac{1}{2} \cdot x - \frac{-1}{6} \cdot \left(1 + -1 \cdot x\right)\right)\right)}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(0.16666666666666666, -x, 0.16666666666666666\right)\right), B \cdot B, -x\right) + 1}{\color{blue}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.9 \cdot 10^{-10}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}}, F, -x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(0.16666666666666666, -x, 0.16666666666666666\right)\right), B \cdot B, -x\right) + 1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 50.3% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{+184}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(0.16666666666666666, -x, 0.16666666666666666\right)\right), B \cdot B, -x\right) + 1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.2e+184)
   (/ (- -1.0 x) B)
   (if (<= F 9.2e+21)
     (/ (- (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
     (/
      (+
       (fma
        (fma 0.5 x (fma 0.16666666666666666 (- x) 0.16666666666666666))
        (* B B)
        (- x))
       1.0)
      B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.2e+184) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 9.2e+21) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = (fma(fma(0.5, x, fma(0.16666666666666666, -x, 0.16666666666666666)), (B * B), -x) + 1.0) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.2e+184)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 9.2e+21)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(fma(fma(0.5, x, fma(0.16666666666666666, Float64(-x), 0.16666666666666666)), Float64(B * B), Float64(-x)) + 1.0) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.2e+184], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 9.2e+21], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(0.5 * x + N[(0.16666666666666666 * (-x) + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + (-x)), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.19999999999999986e184

   1. Initial program 30.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 26.1

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 26.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 52.2%

      \[\leadsto \frac{-1 - x}{B} \]

   if -5.19999999999999986e184 < F < 9.2e21

   1. Initial program 92.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 58.1

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 58.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]
   6. Step-by-step derivation

   7. Applied rewrites 58.1%

      \[\leadsto \frac{\frac{F \cdot 1}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B} \]

   if 9.2e21 < F

   1. Initial program 55.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{1}{\sin B} - \color{blue}{x \cdot \frac{\cos B}{\sin B}} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\frac{1}{\sin B} + \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{\sin B} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{\cos B}{\sin B}\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto \frac{1}{\sin B} + \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \cos B}{\sin B}}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \frac{1}{\sin B} + \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{1}{\sin B} + \color{blue}{\frac{-1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

      7. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{1 + -1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + -1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + -1 \cdot \left(x \cdot \cos B\right)}}{\sin B} \]

      10. associate-*r* N/A

          \[\leadsto \frac{1 + \color{blue}{\left(-1 \cdot x\right) \cdot \cos B}}{\sin B} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \cos B}{\sin B} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \cos B}}{\sin B} \]

      13. lower-neg.f64 N/A

          \[\leadsto \frac{1 + \color{blue}{\left(-x\right)} \cdot \cos B}{\sin B} \]

      14. lower-cos.f64 N/A

          \[\leadsto \frac{1 + \left(-x\right) \cdot \color{blue}{\cos B}}{\sin B} \]

      15. lower-sin.f64 99.9

          \[\leadsto \frac{1 + \left(-x\right) \cdot \cos B}{\color{blue}{\sin B}} \]

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\frac{1 + \left(-x\right) \cdot \cos B}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \frac{1 + \left(-1 \cdot x + {B}^{2} \cdot \left(\frac{1}{2} \cdot x - \frac{-1}{6} \cdot \left(1 + -1 \cdot x\right)\right)\right)}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 55.8%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(0.16666666666666666, -x, 0.16666666666666666\right)\right), B \cdot B, -x\right) + 1}{\color{blue}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{+184}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 9.2 \cdot 10^{+21}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(0.16666666666666666, -x, 0.16666666666666666\right)\right), B \cdot B, -x\right) + 1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 42.5% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{-167}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(0.16666666666666666, -x, 0.16666666666666666\right)\right), B \cdot B, -x\right) + 1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.4e-167)
   (/ (- -1.0 x) B)
   (if (<= F 4.2e-65)
     (/ (- x) B)
     (/
      (+
       (fma
        (fma 0.5 x (fma 0.16666666666666666 (- x) 0.16666666666666666))
        (* B B)
        (- x))
       1.0)
      B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.4e-167) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.2e-65) {
		tmp = -x / B;
	} else {
		tmp = (fma(fma(0.5, x, fma(0.16666666666666666, -x, 0.16666666666666666)), (B * B), -x) + 1.0) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.4e-167)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 4.2e-65)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(fma(fma(0.5, x, fma(0.16666666666666666, Float64(-x), 0.16666666666666666)), Float64(B * B), Float64(-x)) + 1.0) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.4e-167], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4.2e-65], N[((-x) / B), $MachinePrecision], N[(N[(N[(N[(0.5 * x + N[(0.16666666666666666 * (-x) + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + (-x)), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.3999999999999997e-167

   1. Initial program 66.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 52.3

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 52.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.1%

      \[\leadsto \frac{-1 - x}{B} \]

   if -3.3999999999999997e-167 < F < 4.20000000000000006e-65

   1. Initial program 99.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 50.5

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 50.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.8%

      \[\leadsto \frac{-x}{B} \]

   if 4.20000000000000006e-65 < F

   1. Initial program 62.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   4. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \frac{1}{\sin B} - \color{blue}{x \cdot \frac{\cos B}{\sin B}} \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\frac{1}{\sin B} + \left(\mathsf{neg}\left(x\right)\right) \cdot \frac{\cos B}{\sin B}} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{1}{\sin B} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{\cos B}{\sin B}\right)\right)} \]

      4. associate-/l* N/A

         \[\leadsto \frac{1}{\sin B} + \left(\mathsf{neg}\left(\color{blue}{\frac{x \cdot \cos B}{\sin B}}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \frac{1}{\sin B} + \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]

      6. associate-*r/ N/A

         \[\leadsto \frac{1}{\sin B} + \color{blue}{\frac{-1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

      7. div-add-rev N/A

         \[\leadsto \color{blue}{\frac{1 + -1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1 + -1 \cdot \left(x \cdot \cos B\right)}{\sin B}} \]

      9. lower-+.f64 N/A

         \[\leadsto \frac{\color{blue}{1 + -1 \cdot \left(x \cdot \cos B\right)}}{\sin B} \]

      10. associate-*r* N/A

          \[\leadsto \frac{1 + \color{blue}{\left(-1 \cdot x\right) \cdot \cos B}}{\sin B} \]

      11. mul-1-neg N/A

          \[\leadsto \frac{1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \cdot \cos B}{\sin B} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{1 + \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \cos B}}{\sin B} \]

      13. lower-neg.f64 N/A

          \[\leadsto \frac{1 + \color{blue}{\left(-x\right)} \cdot \cos B}{\sin B} \]

      14. lower-cos.f64 N/A

          \[\leadsto \frac{1 + \left(-x\right) \cdot \color{blue}{\cos B}}{\sin B} \]

      15. lower-sin.f64 92.8

          \[\leadsto \frac{1 + \left(-x\right) \cdot \cos B}{\color{blue}{\sin B}} \]

   5. Applied rewrites 92.8%

      \[\leadsto \color{blue}{\frac{1 + \left(-x\right) \cdot \cos B}{\sin B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \frac{1 + \left(-1 \cdot x + {B}^{2} \cdot \left(\frac{1}{2} \cdot x - \frac{-1}{6} \cdot \left(1 + -1 \cdot x\right)\right)\right)}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.6%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.5, x, \mathsf{fma}\left(0.16666666666666666, -x, 0.16666666666666666\right)\right), B \cdot B, -x\right) + 1}{\color{blue}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 42.5% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{-167}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-65}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.4e-167)
   (/ (- -1.0 x) B)
   (if (<= F 4.2e-65) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.4e-167) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.2e-65) {
		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 <= (-3.4d-167)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 4.2d-65) 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 <= -3.4e-167) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.2e-65) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.4e-167:
		tmp = (-1.0 - x) / B
	elif F <= 4.2e-65:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.4e-167)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 4.2e-65)
		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 <= -3.4e-167)
		tmp = (-1.0 - x) / B;
	elseif (F <= 4.2e-65)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.4e-167], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4.2e-65], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.3999999999999997e-167

   1. Initial program 66.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 52.3

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 52.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.1%

      \[\leadsto \frac{-1 - x}{B} \]

   if -3.3999999999999997e-167 < F < 4.20000000000000006e-65

   1. Initial program 99.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 50.5

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 50.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 43.8%

      \[\leadsto \frac{-x}{B} \]

   if 4.20000000000000006e-65 < F

   1. Initial program 62.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 47.0

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 47.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 49.9%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 35.5% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{-167}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.4e-167) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.4e-167) {
		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 <= (-3.4d-167)) 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 <= -3.4e-167) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.4e-167:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.4e-167)
		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 <= -3.4e-167)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.4e-167], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -3.3999999999999997e-167

   1. Initial program 66.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 52.3

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 52.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 50.1%

      \[\leadsto \frac{-1 - x}{B} \]

   if -3.3999999999999997e-167 < F

   1. Initial program 77.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

      5. lower-sqrt.f64 N/A

         \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      6. lower-/.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

      7. associate-+r+ N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      9. unpow2 N/A

         \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

      11. +-commutative N/A

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

      12. lower-fma.f64 48.5

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 48.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 36.5%

      \[\leadsto \frac{-x}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 29.6% 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 73.5%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
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

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

   2. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}}{B} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot F} - x}{B} \]

   5. lower-sqrt.f64 N/A

      \[\leadsto \frac{\color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

   6. lower-/.f64 N/A

      \[\leadsto \frac{\sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot F - x}{B} \]

   7. associate-+r+ N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \cdot F - x}{B} \]

   8. +-commutative N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + \left(2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

   9. unpow2 N/A

      \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + \left(2 + 2 \cdot x\right)}} \cdot F - x}{B} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2 + 2 \cdot x\right)}}} \cdot F - x}{B} \]

   11. +-commutative N/A

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{2 \cdot x + 2}\right)}} \cdot F - x}{B} \]

   12. lower-fma.f64 49.9

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

5. Applied rewrites 49.9%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

6. Taylor expanded in F around 0

   \[\leadsto \frac{-1 \cdot x}{B} \]
7. Step-by-step derivation

8. Applied rewrites 32.8%

   \[\leadsto \frac{-x}{B} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024346 
(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))))))