VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.7% → 99.1%

Time: 13.1s

Alternatives: 25

Speedup: 2.2×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.7% 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, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\cos B, \frac{-x}{\sin B}, \frac{-1}{\sin B}\right)\\ \mathbf{elif}\;F \leq 3.65 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\mathsf{fma}\left(-1, \frac{1 + x}{F \cdot F}, 1\right)}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.1e+154)
   (fma (cos B) (/ (- x) (sin B)) (/ -1.0 (sin B)))
   (if (<= F 3.65e+164)
     (- (/ (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) (sin B)) (/ x (tan B)))
     (+
      (* x (/ -1.0 (tan B)))
      (/ (fma -1.0 (/ (+ 1.0 x) (* F F)) 1.0) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.1e+154) {
		tmp = fma(cos(B), (-x / sin(B)), (-1.0 / sin(B)));
	} else if (F <= 3.65e+164) {
		tmp = ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / sin(B)) - (x / tan(B));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (fma(-1.0, ((1.0 + x) / (F * F)), 1.0) / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.1e+154)
		tmp = fma(cos(B), Float64(Float64(-x) / sin(B)), Float64(-1.0 / sin(B)));
	elseif (F <= 3.65e+164)
		tmp = Float64(Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / sin(B)) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(fma(-1.0, Float64(Float64(1.0 + x) / Float64(F * F)), 1.0) / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.1e+154], N[(N[Cos[B], $MachinePrecision] * N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.65e+164], N[(N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 * N[(N[(1.0 + x), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.1000000000000001e154

   1. Initial program 32.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 99.8

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 100.0%

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

   if -3.1000000000000001e154 < F < 3.65000000000000024e164

   1. Initial program 94.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 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. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-/.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lift-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-fma.f64 N/A

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

      20. lower-sqrt.f64 99.6

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-fma.f64 99.6

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

   6. Applied rewrites 99.6%

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

   8. Applied rewrites 99.7%

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

   if 3.65000000000000024e164 < F

   1. Initial program 30.6%

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

   3. Taylor expanded in F around inf

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

      1. +-commutative N/A

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

      2. associate-/r* N/A

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

      3. associate-*r/ N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{+154}:\\ \;\;\;\;\mathsf{fma}\left(\cos B, \frac{-x}{\sin B}, \frac{-1}{\sin B}\right)\\ \mathbf{elif}\;F \leq 3.65 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\mathsf{fma}\left(-1, \frac{1 + x}{F \cdot F}, 1\right)}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.1 \cdot 10^{+43}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq -7.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 19:\\ \;\;\;\;\mathsf{fma}\left(F, \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot {B}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, 1\right)}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.1e+43)
   (/ (fma (- x) (cos B) -1.0) (sin B))
   (if (<= F -7.6e-107)
     (- (/ (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) (sin B)) (/ x B))
     (if (<= F 19.0)
       (fma
        F
        (* (sqrt (pow (+ (fma 2.0 x (* F F)) 2.0) -1.0)) (pow B -1.0))
        (/ (- x) (tan B)))
       (/ (fma (- x) (cos B) 1.0) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.1e+43) {
		tmp = fma(-x, cos(B), -1.0) / sin(B);
	} else if (F <= -7.6e-107) {
		tmp = ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / sin(B)) - (x / B);
	} else if (F <= 19.0) {
		tmp = fma(F, (sqrt(pow((fma(2.0, x, (F * F)) + 2.0), -1.0)) * pow(B, -1.0)), (-x / tan(B)));
	} else {
		tmp = fma(-x, cos(B), 1.0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.1e+43)
		tmp = Float64(fma(Float64(-x), cos(B), -1.0) / sin(B));
	elseif (F <= -7.6e-107)
		tmp = Float64(Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / sin(B)) - Float64(x / B));
	elseif (F <= 19.0)
		tmp = fma(F, Float64(sqrt((Float64(fma(2.0, x, Float64(F * F)) + 2.0) ^ -1.0)) * (B ^ -1.0)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(fma(Float64(-x), cos(B), 1.0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.1e+43], N[(N[((-x) * N[Cos[B], $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7.6e-107], N[(N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 19.0], N[(F * N[(N[Sqrt[N[Power[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) * N[Cos[B], $MachinePrecision] + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.1e43

   1. Initial program 49.5%

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

   3. Taylor expanded in 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 -4.1e43 < F < -7.6000000000000004e-107

   1. Initial program 99.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-/.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lift-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-fma.f64 N/A

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

      20. lower-sqrt.f64 99.4

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-fma.f64 99.4

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

   6. Applied rewrites 99.4%

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in B around 0

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

       1. lower-/.f64 88.8

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

   11. Applied rewrites 88.8%

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

   if -7.6000000000000004e-107 < F < 19

   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 99.7%

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

   5. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 89.1

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

   7. Applied rewrites 89.1%

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

   if 19 < F

   1. Initial program 56.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 71.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-/.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lift-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-fma.f64 N/A

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

      20. lower-sqrt.f64 71.7

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-fma.f64 71.7

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

   6. Applied rewrites 71.7%

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

   7. Taylor expanded in F around inf

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 99.2

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

   9. Applied rewrites 99.2%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.1 \cdot 10^{+43}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq -7.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 19:\\ \;\;\;\;\mathsf{fma}\left(F, \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot {B}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, 1\right)}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-73} \lor \neg \left(x \leq 1.5 \cdot 10^{-73}\right):\\ \;\;\;\;\mathsf{fma}\left(F, \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot {B}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -2.05e-73) (not (<= x 1.5e-73)))
   (fma
    F
    (* (sqrt (pow (+ (fma 2.0 x (* F F)) 2.0) -1.0)) (pow B -1.0))
    (/ (- x) (tan B)))
   (- (/ (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) (sin B)) (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -2.05e-73) || !(x <= 1.5e-73)) {
		tmp = fma(F, (sqrt(pow((fma(2.0, x, (F * F)) + 2.0), -1.0)) * pow(B, -1.0)), (-x / tan(B)));
	} else {
		tmp = ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / sin(B)) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -2.05e-73) || !(x <= 1.5e-73))
		tmp = fma(F, Float64(sqrt((Float64(fma(2.0, x, Float64(F * F)) + 2.0) ^ -1.0)) * (B ^ -1.0)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / sin(B)) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -2.05e-73], N[Not[LessEqual[x, 1.5e-73]], $MachinePrecision]], N[(F * N[(N[Sqrt[N[Power[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.05000000000000008e-73 or 1.5e-73 < x

   1. Initial program 79.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 94.4%

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

   5. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 89.6

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

   7. Applied rewrites 89.6%

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

   if -2.05000000000000008e-73 < x < 1.5e-73

   1. Initial program 70.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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 74.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-/.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lift-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-fma.f64 N/A

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

      20. lower-sqrt.f64 74.6

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-fma.f64 74.6

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

   6. Applied rewrites 74.6%

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

   8. Applied rewrites 74.8%

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

   9. Taylor expanded in B around 0

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

       1. lower-/.f64 66.0

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

   11. Applied rewrites 66.0%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.05 \cdot 10^{-73} \lor \neg \left(x \leq 1.5 \cdot 10^{-73}\right):\\ \;\;\;\;\mathsf{fma}\left(F, \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot {B}^{-1}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.1 \cdot 10^{+43}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq -7.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot {B}^{-1}, \frac{-x}{\tan B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.1e+43)
   (/ (fma (- x) (cos B) -1.0) (sin B))
   (if (<= F -7.6e-107)
     (- (/ (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) (sin B)) (/ x B))
     (fma
      F
      (* (sqrt (pow (+ (fma 2.0 x (* F F)) 2.0) -1.0)) (pow B -1.0))
      (/ (- x) (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.1e+43) {
		tmp = fma(-x, cos(B), -1.0) / sin(B);
	} else if (F <= -7.6e-107) {
		tmp = ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / sin(B)) - (x / B);
	} else {
		tmp = fma(F, (sqrt(pow((fma(2.0, x, (F * F)) + 2.0), -1.0)) * pow(B, -1.0)), (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.1e+43)
		tmp = Float64(fma(Float64(-x), cos(B), -1.0) / sin(B));
	elseif (F <= -7.6e-107)
		tmp = Float64(Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / sin(B)) - Float64(x / B));
	else
		tmp = fma(F, Float64(sqrt((Float64(fma(2.0, x, Float64(F * F)) + 2.0) ^ -1.0)) * (B ^ -1.0)), Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.1e+43], N[(N[((-x) * N[Cos[B], $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7.6e-107], N[(N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(F * N[(N[Sqrt[N[Power[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[Power[B, -1.0], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.1e43

   1. Initial program 49.5%

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

   3. Taylor expanded in 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 -4.1e43 < F < -7.6000000000000004e-107

   1. Initial program 99.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-/.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lift-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-fma.f64 N/A

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

      20. lower-sqrt.f64 99.4

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-fma.f64 99.4

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

   6. Applied rewrites 99.4%

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in B around 0

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

       1. lower-/.f64 88.8

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

   11. Applied rewrites 88.8%

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

   if -7.6000000000000004e-107 < F

   1. Initial program 80.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 87.2%

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

   5. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 75.9

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

   7. Applied rewrites 75.9%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.1 \cdot 10^{+43}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq -7.6 \cdot 10^{-107}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot {B}^{-1}, \frac{-x}{\tan B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.7:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.96:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}}, F, \left(-x\right) \cdot \cos B\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, 1\right)}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.7)
   (/ (fma (- x) (cos B) -1.0) (sin B))
   (if (<= F 0.96)
     (/ (fma (sqrt (pow (fma 2.0 x 2.0) -1.0)) F (* (- x) (cos B))) (sin B))
     (/ (fma (- x) (cos B) 1.0) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.7) {
		tmp = fma(-x, cos(B), -1.0) / sin(B);
	} else if (F <= 0.96) {
		tmp = fma(sqrt(pow(fma(2.0, x, 2.0), -1.0)), F, (-x * cos(B))) / sin(B);
	} else {
		tmp = fma(-x, cos(B), 1.0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.7)
		tmp = Float64(fma(Float64(-x), cos(B), -1.0) / sin(B));
	elseif (F <= 0.96)
		tmp = Float64(fma(sqrt((fma(2.0, x, 2.0) ^ -1.0)), F, Float64(Float64(-x) * cos(B))) / sin(B));
	else
		tmp = Float64(fma(Float64(-x), cos(B), 1.0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.7], N[(N[((-x) * N[Cos[B], $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.96], N[(N[(N[Sqrt[N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F + N[((-x) * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[((-x) * N[Cos[B], $MachinePrecision] + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.69999999999999996

   1. Initial program 55.8%

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

   3. Taylor expanded in F around -inf

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

      1. +-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 99.4

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

   5. Applied rewrites 99.4%

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

   if -1.69999999999999996 < F < 0.95999999999999996

   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 F around 0

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

      1. associate-/l* N/A

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

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

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

      3. associate-*l/ N/A

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

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

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

      5. associate-/l* N/A

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

      6. mul-1-neg N/A

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

      7. associate-*r/ N/A

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

      8. div-add-rev N/A

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

      9. lower-/.f64 N/A

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

   5. Applied rewrites 98.5%

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

   if 0.95999999999999996 < 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 72.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-/.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lift-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-fma.f64 N/A

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

      20. lower-sqrt.f64 72.4

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-fma.f64 72.4

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

   6. Applied rewrites 72.4%

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

   7. Taylor expanded in F around inf

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 98.3

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

   9. Applied rewrites 98.3%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.7:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.96:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}}, F, \left(-x\right) \cdot \cos B\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, 1\right)}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 3.65 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\mathsf{fma}\left(-1, \frac{1 + x}{F \cdot F}, 1\right)}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.1e+154)
   (/ (fma (- x) (cos B) -1.0) (sin B))
   (if (<= F 3.65e+164)
     (- (/ (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) (sin B)) (/ x (tan B)))
     (+
      (* x (/ -1.0 (tan B)))
      (/ (fma -1.0 (/ (+ 1.0 x) (* F F)) 1.0) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.1e+154) {
		tmp = fma(-x, cos(B), -1.0) / sin(B);
	} else if (F <= 3.65e+164) {
		tmp = ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / sin(B)) - (x / tan(B));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (fma(-1.0, ((1.0 + x) / (F * F)), 1.0) / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.1e+154)
		tmp = Float64(fma(Float64(-x), cos(B), -1.0) / sin(B));
	elseif (F <= 3.65e+164)
		tmp = Float64(Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / sin(B)) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(fma(-1.0, Float64(Float64(1.0 + x) / Float64(F * F)), 1.0) / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.1e+154], N[(N[((-x) * N[Cos[B], $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.65e+164], N[(N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 * N[(N[(1.0 + x), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.1000000000000001e154

   1. Initial program 32.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 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 -3.1000000000000001e154 < F < 3.65000000000000024e164

   1. Initial program 94.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 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. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-/.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lift-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-fma.f64 N/A

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

      20. lower-sqrt.f64 99.6

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-fma.f64 99.6

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

   6. Applied rewrites 99.6%

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

   8. Applied rewrites 99.7%

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

   if 3.65000000000000024e164 < F

   1. Initial program 30.6%

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

   3. Taylor expanded in F around inf

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

      1. +-commutative N/A

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

      2. associate-/r* N/A

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

      3. associate-*r/ N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 3.65 \cdot 10^{+164}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\mathsf{fma}\left(-1, \frac{1 + x}{F \cdot F}, 1\right)}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+37}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 200000:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot \sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\mathsf{fma}\left(-1, \frac{1 + x}{F \cdot F}, 1\right)}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e+37)
   (/ (fma (- x) (cos B) -1.0) (sin B))
   (if (<= F 200000.0)
     (- (/ F (* (sqrt (fma x 2.0 (fma F F 2.0))) (sin B))) (/ x (tan B)))
     (+
      (* x (/ -1.0 (tan B)))
      (/ (fma -1.0 (/ (+ 1.0 x) (* F F)) 1.0) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e+37) {
		tmp = fma(-x, cos(B), -1.0) / sin(B);
	} else if (F <= 200000.0) {
		tmp = (F / (sqrt(fma(x, 2.0, fma(F, F, 2.0))) * sin(B))) - (x / tan(B));
	} else {
		tmp = (x * (-1.0 / tan(B))) + (fma(-1.0, ((1.0 + x) / (F * F)), 1.0) / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2e+37)
		tmp = Float64(fma(Float64(-x), cos(B), -1.0) / sin(B));
	elseif (F <= 200000.0)
		tmp = Float64(Float64(F / Float64(sqrt(fma(x, 2.0, fma(F, F, 2.0))) * sin(B))) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(fma(-1.0, Float64(Float64(1.0 + x) / Float64(F * F)), 1.0) / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2e+37], N[(N[((-x) * N[Cos[B], $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 200000.0], N[(N[(F / N[(N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0 * N[(N[(1.0 + x), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.99999999999999991e37

   1. Initial program 51.0%

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

   3. Taylor expanded in F around -inf

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

      1. +-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 -1.99999999999999991e37 < F < 2e5

   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. Step-by-step derivation

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-/.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lift-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-fma.f64 N/A

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

      20. lower-sqrt.f64 99.6

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-fma.f64 99.6

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

   6. Applied rewrites 99.6%

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

   8. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-sqrt.f64 99.7

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-fma.f64 N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-fma.f64 99.7

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

   10. Applied rewrites 99.7%

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

   if 2e5 < F

   1. Initial program 55.9%

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

   3. Taylor expanded in F around inf

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

      1. +-commutative N/A

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

      2. associate-/r* N/A

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

      3. associate-*r/ N/A

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

      4. div-add-rev N/A

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

      5. lower-/.f64 N/A

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+37}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 200000:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot \sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\mathsf{fma}\left(-1, \frac{1 + x}{F \cdot F}, 1\right)}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+37}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot \sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, 1\right)}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e+37)
   (/ (fma (- x) (cos B) -1.0) (sin B))
   (if (<= F 100000000.0)
     (- (/ F (* (sqrt (fma x 2.0 (fma F F 2.0))) (sin B))) (/ x (tan B)))
     (/ (fma (- x) (cos B) 1.0) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e+37) {
		tmp = fma(-x, cos(B), -1.0) / sin(B);
	} else if (F <= 100000000.0) {
		tmp = (F / (sqrt(fma(x, 2.0, fma(F, F, 2.0))) * sin(B))) - (x / tan(B));
	} else {
		tmp = fma(-x, cos(B), 1.0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2e+37)
		tmp = Float64(fma(Float64(-x), cos(B), -1.0) / sin(B));
	elseif (F <= 100000000.0)
		tmp = Float64(Float64(F / Float64(sqrt(fma(x, 2.0, fma(F, F, 2.0))) * sin(B))) - Float64(x / tan(B)));
	else
		tmp = Float64(fma(Float64(-x), cos(B), 1.0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2e+37], N[(N[((-x) * N[Cos[B], $MachinePrecision] + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 100000000.0], N[(N[(F / N[(N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) * N[Cos[B], $MachinePrecision] + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.99999999999999991e37

   1. Initial program 51.0%

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

   3. Taylor expanded in F around -inf

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

      1. +-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 -1.99999999999999991e37 < F < 1e8

   1. Initial program 99.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.6%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-/.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lift-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-fma.f64 N/A

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

      20. lower-sqrt.f64 99.6

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-fma.f64 99.6

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

   6. Applied rewrites 99.6%

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

   8. Applied rewrites 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-sqrt.f64 99.7

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lift-fma.f64 N/A

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

      17. lower-fma.f64 N/A

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

      18. lift-fma.f64 99.7

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

   10. Applied rewrites 99.7%

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

   if 1e8 < F

   1. Initial program 55.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. 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 70.8%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-/.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lift-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-fma.f64 N/A

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

      20. lower-sqrt.f64 70.8

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-fma.f64 70.8

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

   6. Applied rewrites 70.8%

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

   7. Taylor expanded in F around inf

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 99.6

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

   9. Applied rewrites 99.6%

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

Alternative 9: 99.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.96:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, 1\right)}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.4)
   (/ (fma (- x) (cos B) -1.0) (sin B))
   (if (<= F 0.96)
     (+ (* x (/ -1.0 (tan B))) (* (/ F (sin B)) (sqrt 0.5)))
     (/ (fma (- x) (cos B) 1.0) (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 <= 0.96) {
		tmp = (x * (-1.0 / tan(B))) + ((F / sin(B)) * sqrt(0.5));
	} else {
		tmp = fma(-x, cos(B), 1.0) / 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 <= 0.96)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / sin(B)) * sqrt(0.5)));
	else
		tmp = Float64(fma(Float64(-x), cos(B), 1.0) / 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, 0.96], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) * N[Cos[B], $MachinePrecision] + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 55.8%

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

   3. Taylor expanded in F around -inf

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

      1. +-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 99.4

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

   5. Applied rewrites 99.4%

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

   if -1.3999999999999999 < F < 0.95999999999999996

   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 x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 99.4

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 98.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \sqrt{0.5} \]

   if 0.95999999999999996 < 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. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 72.4%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-/.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lift-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-fma.f64 N/A

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

      20. lower-sqrt.f64 72.4

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-fma.f64 72.4

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

   6. Applied rewrites 72.4%

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

   7. Taylor expanded in F around inf

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

      1. associate-*r/ N/A

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

      2. div-add-rev N/A

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

      3. lower-/.f64 N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-sin.f64 98.3

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

   9. Applied rewrites 98.3%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.96:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot \sqrt{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-x, \cos B, 1\right)}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.00115:\\ \;\;\;\;\frac{\left(\frac{0.5}{F} \cdot \frac{\mathsf{fma}\left(2, x, 2\right)}{F} - 1\right) - x}{B}\\ \mathbf{elif}\;F \leq 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}, F, \left(\mathsf{fma}\left(\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0.16666666666666666, 0.3333333333333333 \cdot x\right) \cdot B\right) \cdot B - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.00115)
   (/ (- (- (* (/ 0.5 F) (/ (fma 2.0 x 2.0) F)) 1.0) x) B)
   (if (<= F 1e+16)
     (/
      (fma
       (sqrt (pow (+ (fma 2.0 x (* F F)) 2.0) -1.0))
       F
       (-
        (*
         (*
          (fma
           (/ F (sqrt (fma x 2.0 (fma F F 2.0))))
           0.16666666666666666
           (* 0.3333333333333333 x))
          B)
         B)
        x))
      B)
     (/
      (- (fma (fma 0.3333333333333333 x 0.16666666666666666) (* B B) 1.0) x)
      B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.00115) {
		tmp = ((((0.5 / F) * (fma(2.0, x, 2.0) / F)) - 1.0) - x) / B;
	} else if (F <= 1e+16) {
		tmp = fma(sqrt(pow((fma(2.0, x, (F * F)) + 2.0), -1.0)), F, (((fma((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))), 0.16666666666666666, (0.3333333333333333 * x)) * B) * B) - x)) / B;
	} else {
		tmp = (fma(fma(0.3333333333333333, x, 0.16666666666666666), (B * B), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.00115)
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 / F) * Float64(fma(2.0, x, 2.0) / F)) - 1.0) - x) / B);
	elseif (F <= 1e+16)
		tmp = Float64(fma(sqrt((Float64(fma(2.0, x, Float64(F * F)) + 2.0) ^ -1.0)), F, Float64(Float64(Float64(fma(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))), 0.16666666666666666, Float64(0.3333333333333333 * x)) * B) * B) - x)) / B);
	else
		tmp = Float64(Float64(fma(fma(0.3333333333333333, x, 0.16666666666666666), Float64(B * B), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.00115], N[(N[(N[(N[(N[(0.5 / F), $MachinePrecision] * N[(N[(2.0 * x + 2.0), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1e+16], N[(N[(N[Sqrt[N[Power[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F + N[(N[(N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.16666666666666666 + N[(0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision] * B), $MachinePrecision] * B), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.00115

   1. Initial program 56.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 39.8

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

   5. Applied rewrites 39.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 50.2%

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

   if -0.00115 < F < 1e16

   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 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 B around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 50.9%

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

   10. Applied rewrites 51.0%

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

   if 1e16 < F

   1. Initial program 54.7%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      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 38.8

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

   5. Applied rewrites 38.8%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 39.7%

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

   9. Taylor expanded in F around inf

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

   11. Applied rewrites 49.9%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00115:\\ \;\;\;\;\frac{\left(\frac{0.5}{F} \cdot \frac{\mathsf{fma}\left(2, x, 2\right)}{F} - 1\right) - x}{B}\\ \mathbf{elif}\;F \leq 10^{+16}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}}, F, \left(\mathsf{fma}\left(\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0.16666666666666666, 0.3333333333333333 \cdot x\right) \cdot B\right) \cdot B - x\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.00115:\\ \;\;\;\;\frac{\left(\frac{0.5}{F} \cdot \frac{\mathsf{fma}\left(2, x, 2\right)}{F} - 1\right) - x}{B}\\ \mathbf{elif}\;F \leq 24.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right), \left(0.3333333333333333 \cdot \left(B \cdot B\right)\right) \cdot x\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.00115)
   (/ (- (- (* (/ 0.5 F) (/ (fma 2.0 x 2.0) F)) 1.0) x) B)
   (if (<= F 24.5)
     (/
      (-
       (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 0.3333333333333333 x 0.16666666666666666) (* B B) 1.0) x)
      B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.00115) {
		tmp = ((((0.5 / F) * (fma(2.0, x, 2.0) / F)) - 1.0) - x) / B;
	} else if (F <= 24.5) {
		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(0.3333333333333333, x, 0.16666666666666666), (B * B), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.00115)
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 / F) * Float64(fma(2.0, x, 2.0) / F)) - 1.0) - x) / B);
	elseif (F <= 24.5)
		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(0.3333333333333333, x, 0.16666666666666666), Float64(B * B), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.00115], N[(N[(N[(N[(N[(0.5 / F), $MachinePrecision] * N[(N[(2.0 * x + 2.0), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 24.5], 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[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.00115

   1. Initial program 56.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 39.8

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

   5. Applied rewrites 39.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 50.2%

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

   if -0.00115 < F < 24.5

   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

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   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 51.1%

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

   if 24.5 < F

   1. Initial program 56.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 38.7

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

   5. Applied rewrites 38.7%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 39.5%

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

   9. Taylor expanded in F around inf

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

   11. Applied rewrites 49.2%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.00115:\\ \;\;\;\;\frac{\left(\frac{0.5}{F} \cdot \frac{\mathsf{fma}\left(2, x, 2\right)}{F} - 1\right) - x}{B}\\ \mathbf{elif}\;F \leq 24.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{{\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-1}}, \mathsf{fma}\left(0.16666666666666666 \cdot F, B \cdot B, F\right), \left(0.3333333333333333 \cdot \left(B \cdot B\right)\right) \cdot x\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 77.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{-14} \lor \neg \left(x \leq 2.4 \cdot 10^{-42}\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -8.6e-14) (not (<= x 2.4e-42)))
   (/ (- x) (tan B))
   (- (/ (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) (sin B)) (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -8.6e-14) || !(x <= 2.4e-42)) {
		tmp = -x / tan(B);
	} else {
		tmp = ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / sin(B)) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -8.6e-14) || !(x <= 2.4e-42))
		tmp = Float64(Float64(-x) / tan(B));
	else
		tmp = Float64(Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / sin(B)) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -8.6e-14], N[Not[LessEqual[x, 2.4e-42]], $MachinePrecision]], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.59999999999999996e-14 or 2.40000000000000003e-42 < x

   1. Initial program 81.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-sin.f64 94.2

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

   7. Applied rewrites 94.2%

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

   9. Applied rewrites 94.4%

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

   if -8.59999999999999996e-14 < x < 2.40000000000000003e-42

   1. Initial program 70.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 75.1%

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

      1. lift-pow.f64 N/A

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

      2. metadata-eval N/A

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

      3. sqrt-pow2 N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. associate-+l+ N/A

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

      8. +-commutative N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. inv-pow N/A

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

      12. lower-/.f64 N/A

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

      13. lift-fma.f64 N/A

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

      14. lift-fma.f64 N/A

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

      15. +-commutative N/A

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

      16. associate-+l+ N/A

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

      17. lift-fma.f64 N/A

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

      18. +-commutative N/A

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

      19. lift-fma.f64 N/A

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

      20. lower-sqrt.f64 75.1

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

      21. lift-fma.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-fma.f64 75.1

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

   6. Applied rewrites 75.1%

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

   8. Applied rewrites 75.3%

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

   9. Taylor expanded in B around 0

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

       1. lower-/.f64 64.6

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

   11. Applied rewrites 64.6%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{-14} \lor \neg \left(x \leq 2.4 \cdot 10^{-42}\right):\\ \;\;\;\;\frac{-x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(0.3333333333333333 \cdot x - 0.16666666666666666\right) \cdot \left(B \cdot B\right) - \left(1 + x\right)}{B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.1e+154)
   (/
    (- (* (- (* 0.3333333333333333 x) 0.16666666666666666) (* B B)) (+ 1.0 x))
    B)
   (if (<= F 7.5e+29)
     (/ (- (* (sqrt (pow (fma F F 2.0) -1.0)) F) x) B)
     (/
      (- (fma (fma 0.3333333333333333 x 0.16666666666666666) (* B B) 1.0) x)
      B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.1e+154) {
		tmp = ((((0.3333333333333333 * x) - 0.16666666666666666) * (B * B)) - (1.0 + x)) / B;
	} else if (F <= 7.5e+29) {
		tmp = ((sqrt(pow(fma(F, F, 2.0), -1.0)) * F) - x) / B;
	} else {
		tmp = (fma(fma(0.3333333333333333, x, 0.16666666666666666), (B * B), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.1e+154)
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 * x) - 0.16666666666666666) * Float64(B * B)) - Float64(1.0 + x)) / B);
	elseif (F <= 7.5e+29)
		tmp = Float64(Float64(Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * F) - x) / B);
	else
		tmp = Float64(Float64(fma(fma(0.3333333333333333, x, 0.16666666666666666), Float64(B * B), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.1e+154], N[(N[(N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision]), $MachinePrecision] - N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 7.5e+29], N[(N[(N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.1000000000000001e154

   1. Initial program 32.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 31.3

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

   5. Applied rewrites 31.3%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 32.0%

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

   9. Taylor expanded in F around -inf

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

   11. Applied rewrites 50.5%

       \[\leadsto \frac{\left(0.3333333333333333 \cdot x - 0.16666666666666666\right) \cdot \left(B \cdot B\right) - \left(1 + x\right)}{B} \]

   if -3.1000000000000001e154 < F < 7.49999999999999945e29

   1. Initial program 96.2%

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

   3. 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 50.1

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

   5. Applied rewrites 50.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 49.8%

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

   if 7.49999999999999945e29 < F

   1. Initial program 54.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 39.4

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

   5. Applied rewrites 39.4%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 40.3%

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

   9. Taylor expanded in F around inf

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

   11. Applied rewrites 50.6%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(0.3333333333333333 \cdot x - 0.16666666666666666\right) \cdot \left(B \cdot B\right) - \left(1 + x\right)}{B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 51.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 0.96:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.05e-5)
   (/ (- -1.0 x) B)
   (if (<= F 0.96)
     (/ (- (* (sqrt (pow (fma 2.0 x 2.0) -1.0)) F) x) B)
     (/
      (- (fma (fma 0.3333333333333333 x 0.16666666666666666) (* B B) 1.0) x)
      B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.05e-5) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 0.96) {
		tmp = ((sqrt(pow(fma(2.0, x, 2.0), -1.0)) * F) - x) / B;
	} else {
		tmp = (fma(fma(0.3333333333333333, x, 0.16666666666666666), (B * B), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.05e-5)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 0.96)
		tmp = Float64(Float64(Float64(sqrt((fma(2.0, x, 2.0) ^ -1.0)) * F) - x) / B);
	else
		tmp = Float64(Float64(fma(fma(0.3333333333333333, x, 0.16666666666666666), Float64(B * B), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.05e-5], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 0.96], N[(N[(N[(N[Sqrt[N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.04999999999999994e-5

   1. Initial program 57.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 39.4

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

   5. Applied rewrites 39.4%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 49.6%

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

   if -3.04999999999999994e-5 < F < 0.95999999999999996

   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 50.4

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

   5. Applied rewrites 50.4%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 50.1%

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

   if 0.95999999999999996 < 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 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 40.3

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

   5. Applied rewrites 40.3%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 41.1%

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

   9. Taylor expanded in F around inf

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

   11. Applied rewrites 49.6%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.05 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 0.96:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-1}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 44.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -4.2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot F}{B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-44}:\\ \;\;\;\;\frac{\left(\left(B \cdot B\right) \cdot 0.3333333333333333 - 1\right) \cdot x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3e-18)
   (/ (- -1.0 x) B)
   (if (<= F -4.2e-133)
     (/ (* (sqrt (pow (fma F F 2.0) -1.0)) F) B)
     (if (<= F 8e-44)
       (/ (* (- (* (* B B) 0.3333333333333333) 1.0) x) B)
       (/
        (- (fma (fma 0.3333333333333333 x 0.16666666666666666) (* B B) 1.0) x)
        B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3e-18) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -4.2e-133) {
		tmp = (sqrt(pow(fma(F, F, 2.0), -1.0)) * F) / B;
	} else if (F <= 8e-44) {
		tmp = ((((B * B) * 0.3333333333333333) - 1.0) * x) / B;
	} else {
		tmp = (fma(fma(0.3333333333333333, x, 0.16666666666666666), (B * B), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3e-18)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -4.2e-133)
		tmp = Float64(Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * F) / B);
	elseif (F <= 8e-44)
		tmp = Float64(Float64(Float64(Float64(Float64(B * B) * 0.3333333333333333) - 1.0) * x) / B);
	else
		tmp = Float64(Float64(fma(fma(0.3333333333333333, x, 0.16666666666666666), Float64(B * B), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3e-18], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -4.2e-133], N[(N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 8e-44], N[(N[(N[(N[(N[(B * B), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.99999999999999983e-18

   1. Initial program 57.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      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 40.2

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

   5. Applied rewrites 40.2%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 50.3%

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

   if -2.99999999999999983e-18 < F < -4.2000000000000002e-133

   1. Initial program 99.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 41.1%

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

   if -4.2000000000000002e-133 < F < 7.99999999999999962e-44

   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 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 50.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 37.4%

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

   if 7.99999999999999962e-44 < F

   1. Initial program 59.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 39.6

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

   5. Applied rewrites 39.6%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 40.4%

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

   9. Taylor expanded in F around inf

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

   11. Applied rewrites 47.4%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -4.2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot F}{B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-44}:\\ \;\;\;\;\frac{\left(\left(B \cdot B\right) \cdot 0.3333333333333333 - 1\right) \cdot x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -4.2 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-44}:\\ \;\;\;\;\frac{\left(\left(B \cdot B\right) \cdot 0.3333333333333333 - 1\right) \cdot x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3e-18)
   (/ (- -1.0 x) B)
   (if (<= F -4.2e-133)
     (* (sqrt (pow (fma F F 2.0) -1.0)) (/ F B))
     (if (<= F 8e-44)
       (/ (* (- (* (* B B) 0.3333333333333333) 1.0) x) B)
       (/
        (- (fma (fma 0.3333333333333333 x 0.16666666666666666) (* B B) 1.0) x)
        B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3e-18) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -4.2e-133) {
		tmp = sqrt(pow(fma(F, F, 2.0), -1.0)) * (F / B);
	} else if (F <= 8e-44) {
		tmp = ((((B * B) * 0.3333333333333333) - 1.0) * x) / B;
	} else {
		tmp = (fma(fma(0.3333333333333333, x, 0.16666666666666666), (B * B), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3e-18)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -4.2e-133)
		tmp = Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * Float64(F / B));
	elseif (F <= 8e-44)
		tmp = Float64(Float64(Float64(Float64(Float64(B * B) * 0.3333333333333333) - 1.0) * x) / B);
	else
		tmp = Float64(Float64(fma(fma(0.3333333333333333, x, 0.16666666666666666), Float64(B * B), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3e-18], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -4.2e-133], N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8e-44], N[(N[(N[(N[(N[(B * B), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.99999999999999983e-18

   1. Initial program 57.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      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 40.2

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

   5. Applied rewrites 40.2%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 50.3%

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

   if -2.99999999999999983e-18 < F < -4.2000000000000002e-133

   1. Initial program 99.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 41.0%

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

   if -4.2000000000000002e-133 < F < 7.99999999999999962e-44

   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 50.9

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

   5. Applied rewrites 50.9%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 50.9%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 37.4%

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

   if 7.99999999999999962e-44 < F

   1. Initial program 59.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 39.6

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

   5. Applied rewrites 39.6%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 40.4%

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

   9. Taylor expanded in F around inf

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

   11. Applied rewrites 47.4%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 44.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{-18}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -4.2 \cdot 10^{-133}:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-44}:\\ \;\;\;\;\frac{\left(\left(B \cdot B\right) \cdot 0.3333333333333333 - 1\right) \cdot x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 57.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{if}\;B \leq 0.0085:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(t\_0, 0.16666666666666666, 0.3333333333333333 \cdot x\right), t\_0\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ F (sqrt (fma x 2.0 (fma F F 2.0))))))
   (if (<= B 0.0085)
     (/
      (-
       (fma (* B B) (fma t_0 0.16666666666666666 (* 0.3333333333333333 x)) t_0)
       x)
      B)
     (/ (- x) (tan B)))))

C

double code(double F, double B, double x) {
	double t_0 = F / sqrt(fma(x, 2.0, fma(F, F, 2.0)));
	double tmp;
	if (B <= 0.0085) {
		tmp = (fma((B * B), fma(t_0, 0.16666666666666666, (0.3333333333333333 * x)), t_0) - x) / B;
	} else {
		tmp = -x / tan(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0))))
	tmp = 0.0
	if (B <= 0.0085)
		tmp = Float64(Float64(fma(Float64(B * B), fma(t_0, 0.16666666666666666, Float64(0.3333333333333333 * x)), t_0) - x) / B);
	else
		tmp = Float64(Float64(-x) / tan(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 0.0085], N[(N[(N[(N[(B * B), $MachinePrecision] * N[(t$95$0 * 0.16666666666666666 + N[(0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 0.0085000000000000006

   1. Initial program 72.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 61.0

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

   5. Applied rewrites 61.0%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 61.3%

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

   10. Applied rewrites 61.3%

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

   if 0.0085000000000000006 < B

   1. Initial program 82.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 82.8%

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

   5. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. associate-/l* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-sin.f64 55.7

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

   7. Applied rewrites 55.7%

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

   9. Applied rewrites 55.7%

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

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.0085:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}, 0.16666666666666666, 0.3333333333333333 \cdot x\right), \frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 51.5% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{if}\;F \leq -0.00115:\\ \;\;\;\;\frac{\left(\frac{0.5}{F} \cdot \frac{\mathsf{fma}\left(2, x, 2\right)}{F} - 1\right) - x}{B}\\ \mathbf{elif}\;F \leq 24.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(B \cdot B, \mathsf{fma}\left(t\_0, 0.16666666666666666, 0.3333333333333333 \cdot x\right), t\_0\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ F (sqrt (fma x 2.0 (fma F F 2.0))))))
   (if (<= F -0.00115)
     (/ (- (- (* (/ 0.5 F) (/ (fma 2.0 x 2.0) F)) 1.0) x) B)
     (if (<= F 24.5)
       (/
        (-
         (fma
          (* B B)
          (fma t_0 0.16666666666666666 (* 0.3333333333333333 x))
          t_0)
         x)
        B)
       (/
        (- (fma (fma 0.3333333333333333 x 0.16666666666666666) (* B B) 1.0) x)
        B)))))

C

double code(double F, double B, double x) {
	double t_0 = F / sqrt(fma(x, 2.0, fma(F, F, 2.0)));
	double tmp;
	if (F <= -0.00115) {
		tmp = ((((0.5 / F) * (fma(2.0, x, 2.0) / F)) - 1.0) - x) / B;
	} else if (F <= 24.5) {
		tmp = (fma((B * B), fma(t_0, 0.16666666666666666, (0.3333333333333333 * x)), t_0) - x) / B;
	} else {
		tmp = (fma(fma(0.3333333333333333, x, 0.16666666666666666), (B * B), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0))))
	tmp = 0.0
	if (F <= -0.00115)
		tmp = Float64(Float64(Float64(Float64(Float64(0.5 / F) * Float64(fma(2.0, x, 2.0) / F)) - 1.0) - x) / B);
	elseif (F <= 24.5)
		tmp = Float64(Float64(fma(Float64(B * B), fma(t_0, 0.16666666666666666, Float64(0.3333333333333333 * x)), t_0) - x) / B);
	else
		tmp = Float64(Float64(fma(fma(0.3333333333333333, x, 0.16666666666666666), Float64(B * B), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.00115], N[(N[(N[(N[(N[(0.5 / F), $MachinePrecision] * N[(N[(2.0 * x + 2.0), $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 24.5], N[(N[(N[(N[(B * B), $MachinePrecision] * N[(t$95$0 * 0.16666666666666666 + N[(0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.00115

   1. Initial program 56.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 39.8

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

   5. Applied rewrites 39.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 50.2%

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

   if -0.00115 < F < 24.5

   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 51.0

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

   5. Applied rewrites 51.0%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 51.3%

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

   10. Applied rewrites 51.4%

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

   if 24.5 < F

   1. Initial program 56.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 38.7

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

   5. Applied rewrites 38.7%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 39.5%

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

   9. Taylor expanded in F around inf

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

   11. Applied rewrites 49.2%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 51.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(0.3333333333333333 \cdot x - 0.16666666666666666\right) \cdot \left(B \cdot B\right) - \left(1 + x\right)}{B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.1e+154)
   (/
    (- (* (- (* 0.3333333333333333 x) 0.16666666666666666) (* B B)) (+ 1.0 x))
    B)
   (if (<= F 7.5e+29)
     (/ (- (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
     (/
      (- (fma (fma 0.3333333333333333 x 0.16666666666666666) (* B B) 1.0) x)
      B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.1e+154) {
		tmp = ((((0.3333333333333333 * x) - 0.16666666666666666) * (B * B)) - (1.0 + x)) / B;
	} else if (F <= 7.5e+29) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = (fma(fma(0.3333333333333333, x, 0.16666666666666666), (B * B), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.1e+154)
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 * x) - 0.16666666666666666) * Float64(B * B)) - Float64(1.0 + x)) / B);
	elseif (F <= 7.5e+29)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(fma(fma(0.3333333333333333, x, 0.16666666666666666), Float64(B * B), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.1e+154], N[(N[(N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision]), $MachinePrecision] - N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 7.5e+29], 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[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.1000000000000001e154

   1. Initial program 32.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 31.3

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

   5. Applied rewrites 31.3%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 32.0%

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

   9. Taylor expanded in F around -inf

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

   11. Applied rewrites 50.5%

       \[\leadsto \frac{\left(0.3333333333333333 \cdot x - 0.16666666666666666\right) \cdot \left(B \cdot B\right) - \left(1 + x\right)}{B} \]

   if -3.1000000000000001e154 < F < 7.49999999999999945e29

   1. Initial program 96.2%

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

   3. 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 50.1

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

   5. Applied rewrites 50.1%

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

   7. Applied rewrites 50.1%

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

   9. Applied rewrites 50.1%

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

   if 7.49999999999999945e29 < F

   1. Initial program 54.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 39.4

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

   5. Applied rewrites 39.4%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 40.3%

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

   9. Taylor expanded in F around inf

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

   11. Applied rewrites 50.6%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 51.8% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(0.3333333333333333 \cdot x - 0.16666666666666666\right) \cdot \left(B \cdot B\right) - \left(1 + x\right)}{B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.1e+154)
   (/
    (- (* (- (* 0.3333333333333333 x) 0.16666666666666666) (* B B)) (+ 1.0 x))
    B)
   (if (<= F 7.5e+29)
     (/ (- (/ F (sqrt (fma F F 2.0))) x) B)
     (/
      (- (fma (fma 0.3333333333333333 x 0.16666666666666666) (* B B) 1.0) x)
      B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.1e+154) {
		tmp = ((((0.3333333333333333 * x) - 0.16666666666666666) * (B * B)) - (1.0 + x)) / B;
	} else if (F <= 7.5e+29) {
		tmp = ((F / sqrt(fma(F, F, 2.0))) - x) / B;
	} else {
		tmp = (fma(fma(0.3333333333333333, x, 0.16666666666666666), (B * B), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.1e+154)
		tmp = Float64(Float64(Float64(Float64(Float64(0.3333333333333333 * x) - 0.16666666666666666) * Float64(B * B)) - Float64(1.0 + x)) / B);
	elseif (F <= 7.5e+29)
		tmp = Float64(Float64(Float64(F / sqrt(fma(F, F, 2.0))) - x) / B);
	else
		tmp = Float64(Float64(fma(fma(0.3333333333333333, x, 0.16666666666666666), Float64(B * B), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.1e+154], N[(N[(N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision]), $MachinePrecision] - N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 7.5e+29], N[(N[(N[(F / N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.1000000000000001e154

   1. Initial program 32.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 31.3

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

   5. Applied rewrites 31.3%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 32.0%

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

   9. Taylor expanded in F around -inf

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

   11. Applied rewrites 50.5%

       \[\leadsto \frac{\left(0.3333333333333333 \cdot x - 0.16666666666666666\right) \cdot \left(B \cdot B\right) - \left(1 + x\right)}{B} \]

   if -3.1000000000000001e154 < F < 7.49999999999999945e29

   1. Initial program 96.2%

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

   3. 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 50.1

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

   5. Applied rewrites 50.1%

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

   7. Applied rewrites 50.1%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 49.9%

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

   if 7.49999999999999945e29 < F

   1. Initial program 54.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 39.4

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

   5. Applied rewrites 39.4%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 40.3%

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

   9. Taylor expanded in F around inf

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

   11. Applied rewrites 50.6%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{\left(0.3333333333333333 \cdot x - 0.16666666666666666\right) \cdot \left(B \cdot B\right) - \left(1 + x\right)}{B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 44.5% accurate, 8.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.3 \cdot 10^{-93}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-44}:\\ \;\;\;\;\frac{\left(\left(B \cdot B\right) \cdot 0.3333333333333333 - 1\right) \cdot x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.3e-93)
   (/ (- -1.0 x) B)
   (if (<= F 8e-44)
     (/ (* (- (* (* B B) 0.3333333333333333) 1.0) x) B)
     (/
      (- (fma (fma 0.3333333333333333 x 0.16666666666666666) (* B B) 1.0) x)
      B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.3e-93) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 8e-44) {
		tmp = ((((B * B) * 0.3333333333333333) - 1.0) * x) / B;
	} else {
		tmp = (fma(fma(0.3333333333333333, x, 0.16666666666666666), (B * B), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.3e-93)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 8e-44)
		tmp = Float64(Float64(Float64(Float64(Float64(B * B) * 0.3333333333333333) - 1.0) * x) / B);
	else
		tmp = Float64(Float64(fma(fma(0.3333333333333333, x, 0.16666666666666666), Float64(B * B), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.3e-93], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 8e-44], N[(N[(N[(N[(N[(B * B), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(0.3333333333333333 * x + 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.3000000000000001e-93

   1. Initial program 65.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 41.7

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

   5. Applied rewrites 41.7%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 42.9%

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

   if -3.3000000000000001e-93 < F < 7.99999999999999962e-44

   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 51.4

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

   5. Applied rewrites 51.4%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   8. Applied rewrites 51.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 37.1%

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

   if 7.99999999999999962e-44 < F

   1. Initial program 59.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 39.6

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

   5. Applied rewrites 39.6%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]

   8. Applied rewrites 40.4%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}}, F, \mathsf{fma}\left(0.16666666666666666 \cdot F, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}}, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right) - x\right)}{B}} \]

   9. Taylor expanded in F around inf

      \[\leadsto \frac{\left(1 + {B}^{2} \cdot \left(\frac{1}{6} + \frac{1}{3} \cdot x\right)\right) - x}{B} \]
   10. Step-by-step derivation

   11. Applied rewrites 47.4%

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(0.3333333333333333, x, 0.16666666666666666\right), B \cdot B, 1\right) - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 44.5% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.3 \cdot 10^{-93}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.56 \cdot 10^{-24}:\\ \;\;\;\;\frac{\left(\left(B \cdot B\right) \cdot 0.3333333333333333 - 1\right) \cdot x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.3e-93)
   (/ (- -1.0 x) B)
   (if (<= F 1.56e-24)
     (/ (* (- (* (* B B) 0.3333333333333333) 1.0) x) B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.3e-93) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.56e-24) {
		tmp = ((((B * B) * 0.3333333333333333) - 1.0) * x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-3.3d-93)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.56d-24) then
        tmp = ((((b * b) * 0.3333333333333333d0) - 1.0d0) * x) / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.3e-93) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.56e-24) {
		tmp = ((((B * B) * 0.3333333333333333) - 1.0) * x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.3e-93:
		tmp = (-1.0 - x) / B
	elif F <= 1.56e-24:
		tmp = ((((B * B) * 0.3333333333333333) - 1.0) * x) / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.3e-93)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.56e-24)
		tmp = Float64(Float64(Float64(Float64(Float64(B * B) * 0.3333333333333333) - 1.0) * 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 <= -3.3e-93)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.56e-24)
		tmp = ((((B * B) * 0.3333333333333333) - 1.0) * x) / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.3e-93], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.56e-24], N[(N[(N[(N[(N[(B * B), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] - 1.0), $MachinePrecision] * x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.3000000000000001e-93

   1. Initial program 65.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 41.7

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.9%

      \[\leadsto \frac{-1 - x}{B} \]

   if -3.3000000000000001e-93 < F < 1.56e-24

   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 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 B around 0

      \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} + {B}^{2} \cdot \left(\frac{1}{6} \cdot \left(F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}\right) + \frac{1}{3} \cdot x\right)\right) - x}{B}} \]

   8. Applied rewrites 50.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}}, F, \mathsf{fma}\left(0.16666666666666666 \cdot F, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}}, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right) - x\right)}{B}} \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{x \cdot \left(\frac{1}{3} \cdot {B}^{2} - 1\right)}{B} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.9%

       \[\leadsto \frac{\left(\left(B \cdot B\right) \cdot 0.3333333333333333 - 1\right) \cdot x}{B} \]

   if 1.56e-24 < F

   1. Initial program 58.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 41.1

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 41.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 48.4%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 44.5% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.3 \cdot 10^{-93}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{-34}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.3e-93)
   (/ (- -1.0 x) B)
   (if (<= F 2.2e-34) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.3e-93) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.2e-34) {
		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 <= (-3.3d-93)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 2.2d-34) 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 <= -3.3e-93) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 2.2e-34) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.3e-93:
		tmp = (-1.0 - x) / B
	elif F <= 2.2e-34:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.3e-93)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 2.2e-34)
		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 <= -3.3e-93)
		tmp = (-1.0 - x) / B;
	elseif (F <= 2.2e-34)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.3e-93], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.2e-34], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.3000000000000001e-93

   1. Initial program 65.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 41.7

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.9%

      \[\leadsto \frac{-1 - x}{B} \]

   if -3.3000000000000001e-93 < F < 2.1999999999999999e-34

   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 50.8

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 50.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 36.2%

      \[\leadsto \frac{-x}{B} \]

   if 2.1999999999999999e-34 < F

   1. Initial program 59.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 40.1

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 40.1%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.2%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 37.3% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.3 \cdot 10^{-93}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.3e-93) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.3e-93) {
		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 <= (-3.3d-93)) 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 <= -3.3e-93) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.3e-93:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.3e-93)
		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 <= -3.3e-93)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.3e-93], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -3.3000000000000001e-93

   1. Initial program 65.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   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 41.7

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 41.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 42.9%

      \[\leadsto \frac{-1 - x}{B} \]

   if -3.3000000000000001e-93 < F

   1. Initial program 80.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   4. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]

      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 45.8

          \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

   5. Applied rewrites 45.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

   6. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   7. Step-by-step derivation

   8. Applied rewrites 29.5%

      \[\leadsto \frac{-x}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 29.9% 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 75.3%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
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 44.4

       \[\leadsto \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \color{blue}{\mathsf{fma}\left(2, x, 2\right)}\right)}} \cdot F - x}{B} \]

5. Applied rewrites 44.4%

   \[\leadsto \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} \cdot F - x}{B}} \]

6. Taylor expanded in F around 0

   \[\leadsto \frac{-1 \cdot x}{B} \]
7. Step-by-step derivation

8. Applied rewrites 27.0%

   \[\leadsto \frac{-x}{B} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024354 
(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))))))