VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.1% → 99.7%

Time: 8.0s

Alternatives: 23

Speedup: 1.4×

Specification

Math

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

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: 76.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -36:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos B, -x, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.00018:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \left(\frac{F}{\sin B} \cdot \tan B\right) - x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -36.0)
  (/ (fma (cos B) (- x) -1.0) (sin B))
  (if (<= F 0.00018)
    (/
     (-
      (*
       (pow (fma x 2.0 (fma F F 2.0)) -0.5)
       (* (/ F (sin B)) (tan B)))
      x)
     (tan B))
    (- (/ 1.0 (sin B)) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -36.0) {
		tmp = fma(cos(B), -x, -1.0) / sin(B);
	} else if (F <= 0.00018) {
		tmp = ((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * ((F / sin(B)) * tan(B))) - x) / tan(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -36.0)
		tmp = Float64(fma(cos(B), Float64(-x), -1.0) / sin(B));
	elseif (F <= 0.00018)
		tmp = Float64(Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * Float64(Float64(F / sin(B)) * tan(B))) - x) / tan(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -36.0], N[(N[(N[Cos[B], $MachinePrecision] * (-x) + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.00018], N[(N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -36

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-tan.f64 N/A

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

      6. +-commutative N/A

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

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

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

      8. lift-neg.f64 N/A

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

      9. tan-quot N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. div-flip-rev N/A

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

      13. associate-/l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

   8. Applied rewrites 56.6%

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

   if -36 < F < 1.8000000000000001e-4

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   5. Applied rewrites 76.2%

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

   if 1.8000000000000001e-4 < F

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 56.5%

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

Alternative 2: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.45 \cdot 10^{+56}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos B, -x, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 2800000000000:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (/ x (tan B))))
  (if (<= F -1.45e+56)
    (/ (fma (cos B) (- x) -1.0) (sin B))
    (if (<= F 2800000000000.0)
      (- (* F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B))) t_0)
      (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.45e+56) {
		tmp = fma(cos(B), -x, -1.0) / sin(B);
	} else if (F <= 2800000000000.0) {
		tmp = (F * (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.45e+56)
		tmp = Float64(fma(cos(B), Float64(-x), -1.0) / sin(B));
	elseif (F <= 2800000000000.0)
		tmp = Float64(Float64(F * Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.45e+56], N[(N[(N[Cos[B], $MachinePrecision] * (-x) + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2800000000000.0], N[(N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.45e56

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-tan.f64 N/A

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

      6. +-commutative N/A

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

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

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

      8. lift-neg.f64 N/A

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

      9. tan-quot N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. div-flip-rev N/A

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

      13. associate-/l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

   8. Applied rewrites 56.6%

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

   if -1.45e56 < F < 2.8e12

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lift-/.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 84.6%

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

   5. Applied rewrites 84.6%

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

   if 2.8e12 < F

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 56.5%

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

Alternative 3: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+113}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos B, -x, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.00018:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (/ x (tan B))))
  (if (<= F -2e+113)
    (/ (fma (cos B) (- x) -1.0) (sin B))
    (if (<= F 0.00018)
      (- (/ (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F) (sin B)) t_0)
      (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2e+113) {
		tmp = fma(cos(B), -x, -1.0) / sin(B);
	} else if (F <= 0.00018) {
		tmp = ((pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F) / sin(B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2e+113)
		tmp = Float64(fma(cos(B), Float64(-x), -1.0) / sin(B));
	elseif (F <= 0.00018)
		tmp = Float64(Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F) / sin(B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2e+113], N[(N[(N[Cos[B], $MachinePrecision] * (-x) + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.00018], N[(N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2e113

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-tan.f64 N/A

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

      6. +-commutative N/A

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

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

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

      8. lift-neg.f64 N/A

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

      9. tan-quot N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. div-flip-rev N/A

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

      13. associate-/l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

   8. Applied rewrites 56.6%

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

   if -2e113 < F < 1.8000000000000001e-4

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   if 1.8000000000000001e-4 < F

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 56.5%

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

Alternative 4: 99.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos B, -x, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.00018:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, F, \left(-x\right) \cdot \cos B\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -1e+116)
  (/ (fma (cos B) (- x) -1.0) (sin B))
  (if (<= F 0.00018)
    (/
     (fma (pow (fma x 2.0 (fma F F 2.0)) -0.5) F (* (- x) (cos B)))
     (sin B))
    (- (/ 1.0 (sin B)) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e+116) {
		tmp = fma(cos(B), -x, -1.0) / sin(B);
	} else if (F <= 0.00018) {
		tmp = fma(pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5), F, (-x * cos(B))) / sin(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1e+116)
		tmp = Float64(fma(cos(B), Float64(-x), -1.0) / sin(B));
	elseif (F <= 0.00018)
		tmp = Float64(fma((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5), F, Float64(Float64(-x) * cos(B))) / sin(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1e+116], N[(N[(N[Cos[B], $MachinePrecision] * (-x) + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.00018], N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F + N[((-x) * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e116

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-tan.f64 N/A

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

      6. +-commutative N/A

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

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

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

      8. lift-neg.f64 N/A

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

      9. tan-quot N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. div-flip-rev N/A

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

      13. associate-/l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

   8. Applied rewrites 56.6%

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

   if -1e116 < F < 1.8000000000000001e-4

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   5. Applied rewrites 84.6%

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

   if 1.8000000000000001e-4 < F

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 56.5%

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

Alternative 5: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -36:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos B, -x, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.00018:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (/ x (tan B))))
  (if (<= F -36.0)
    (/ (fma (cos B) (- x) -1.0) (sin B))
    (if (<= F 0.00018)
      (- (* F (/ (pow (fma x 2.0 2.0) -0.5) (sin B))) t_0)
      (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -36.0) {
		tmp = fma(cos(B), -x, -1.0) / sin(B);
	} else if (F <= 0.00018) {
		tmp = (F * (pow(fma(x, 2.0, 2.0), -0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -36.0)
		tmp = Float64(fma(cos(B), Float64(-x), -1.0) / sin(B));
	elseif (F <= 0.00018)
		tmp = Float64(Float64(F * Float64((fma(x, 2.0, 2.0) ^ -0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -36.0], N[(N[(N[Cos[B], $MachinePrecision] * (-x) + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.00018], N[(N[(F * N[(N[Power[N[(x * 2.0 + 2.0), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -36

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-tan.f64 N/A

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

      6. +-commutative N/A

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

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

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

      8. lift-neg.f64 N/A

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

      9. tan-quot N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. div-flip-rev N/A

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

      13. associate-/l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

   8. Applied rewrites 56.6%

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

   if -36 < F < 1.8000000000000001e-4

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lift-/.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 84.6%

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 55.8%

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

   if 1.8000000000000001e-4 < F

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 56.5%

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

Alternative 6: 99.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -36:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos B, -x, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.00018:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(2 + 2 \cdot x\right)}^{-0.5}, F, \left(-x\right) \cdot \cos B\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -36.0)
  (/ (fma (cos B) (- x) -1.0) (sin B))
  (if (<= F 0.00018)
    (/ (fma (pow (+ 2.0 (* 2.0 x)) -0.5) F (* (- x) (cos B))) (sin B))
    (- (/ 1.0 (sin B)) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -36.0) {
		tmp = fma(cos(B), -x, -1.0) / sin(B);
	} else if (F <= 0.00018) {
		tmp = fma(pow((2.0 + (2.0 * x)), -0.5), F, (-x * cos(B))) / sin(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -36.0)
		tmp = Float64(fma(cos(B), Float64(-x), -1.0) / sin(B));
	elseif (F <= 0.00018)
		tmp = Float64(fma((Float64(2.0 + Float64(2.0 * x)) ^ -0.5), F, Float64(Float64(-x) * cos(B))) / sin(B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -36.0], N[(N[(N[Cos[B], $MachinePrecision] * (-x) + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.00018], N[(N[(N[Power[N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F + N[((-x) * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -36

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-tan.f64 N/A

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

      6. +-commutative N/A

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

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

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

      8. lift-neg.f64 N/A

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

      9. tan-quot N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. div-flip-rev N/A

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

      13. associate-/l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

   8. Applied rewrites 56.6%

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

   if -36 < F < 1.8000000000000001e-4

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in F around 0

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 56.2%

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

   8. Applied rewrites 56.2%

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

   if 1.8000000000000001e-4 < F

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 56.5%

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

Alternative 7: 92.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -36:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos B, -x, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.00018:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0 (/ x (tan B))))
  (if (<= F -36.0)
    (/ (fma (cos B) (- x) -1.0) (sin B))
    (if (<= F 0.00018)
      (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) (/ F B)) t_0)
      (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -36.0) {
		tmp = fma(cos(B), -x, -1.0) / sin(B);
	} else if (F <= 0.00018) {
		tmp = (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * (F / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -36.0)
		tmp = Float64(fma(cos(B), Float64(-x), -1.0) / sin(B));
	elseif (F <= 0.00018)
		tmp = Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -36.0], N[(N[(N[Cos[B], $MachinePrecision] * (-x) + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.00018], N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -36

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-tan.f64 N/A

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

      6. +-commutative N/A

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

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

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

      8. lift-neg.f64 N/A

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

      9. tan-quot N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. div-flip-rev N/A

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

      13. associate-/l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

   8. Applied rewrites 56.6%

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

   if -36 < F < 1.8000000000000001e-4

   1. Initial program 76.1%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.2%

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

   4. Applied rewrites 62.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lift-/.f64 N/A

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

      10. sub-flip-reverse N/A

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

      11. lower--.f64 62.3%

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

   6. Applied rewrites 62.3%

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

   if 1.8000000000000001e-4 < F

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around inf

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

   6. Applied rewrites 56.5%

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

Alternative 8: 82.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -36:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos B, -x, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-39}:\\ \;\;\;\;{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{+154}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{x}{1}\right) \cdot 1}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -36.0)
  (/ (fma (cos B) (- x) -1.0) (sin B))
  (if (<= F 5.2e-39)
    (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) (/ F B)) (/ x (tan B)))
    (if (<= F 1.75e+154)
      (-
       (/ (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F) (sin B))
       (/ x B))
      (/ (* (- 1.0 (/ x 1.0)) 1.0) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -36.0) {
		tmp = fma(cos(B), -x, -1.0) / sin(B);
	} else if (F <= 5.2e-39) {
		tmp = (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * (F / B)) - (x / tan(B));
	} else if (F <= 1.75e+154) {
		tmp = ((pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F) / sin(B)) - (x / B);
	} else {
		tmp = ((1.0 - (x / 1.0)) * 1.0) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -36.0)
		tmp = Float64(fma(cos(B), Float64(-x), -1.0) / sin(B));
	elseif (F <= 5.2e-39)
		tmp = Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / tan(B)));
	elseif (F <= 1.75e+154)
		tmp = Float64(Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F) / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(x / 1.0)) * 1.0) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -36.0], N[(N[(N[Cos[B], $MachinePrecision] * (-x) + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.2e-39], N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.75e+154], N[(N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(x / 1.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -36

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-tan.f64 N/A

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

      6. +-commutative N/A

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

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

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

      8. lift-neg.f64 N/A

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

      9. tan-quot N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. div-flip-rev N/A

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

      13. associate-/l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

   8. Applied rewrites 56.6%

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

   if -36 < F < 5.2e-39

   1. Initial program 76.1%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.2%

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

   4. Applied rewrites 62.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. lift-tan.f64 N/A

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

      7. lift-tan.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lift-/.f64 N/A

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

      10. sub-flip-reverse N/A

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

      11. lower--.f64 62.3%

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

   6. Applied rewrites 62.3%

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

   if 5.2e-39 < F < 1.7500000000000001e154

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 57.9%

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

   6. Applied rewrites 57.9%

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

   if 1.7500000000000001e154 < F

   1. Initial program 76.1%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.7%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

   6. Applied rewrites 33.4%

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

   7. Taylor expanded in F around inf

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

   9. Applied rewrites 19.3%

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

   10. Taylor expanded in F around inf

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

   12. Applied rewrites 30.1%

       \[\leadsto \frac{\left(1 - \frac{x}{1}\right) \cdot 1}{B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 77.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\cos B, -x, -1\right)}{\sin B}\\ \mathbf{elif}\;F \leq -4.1 \cdot 10^{-107}:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{-202}:\\ \;\;\;\;-1 \cdot \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{+154}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{x}{1}\right) \cdot 1}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -5.2e-83)
  (/ (fma (cos B) (- x) -1.0) (sin B))
  (if (<= F -4.1e-107)
    (- (* F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B))) (/ x B))
    (if (<= F 1.45e-202)
      (* -1.0 (/ (* x (cos B)) (sin B)))
      (if (<= F 1.75e+154)
        (-
         (/ (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F) (sin B))
         (/ x B))
        (/ (* (- 1.0 (/ x 1.0)) 1.0) B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.2e-83) {
		tmp = fma(cos(B), -x, -1.0) / sin(B);
	} else if (F <= -4.1e-107) {
		tmp = (F * (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B))) - (x / B);
	} else if (F <= 1.45e-202) {
		tmp = -1.0 * ((x * cos(B)) / sin(B));
	} else if (F <= 1.75e+154) {
		tmp = ((pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F) / sin(B)) - (x / B);
	} else {
		tmp = ((1.0 - (x / 1.0)) * 1.0) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.2e-83)
		tmp = Float64(fma(cos(B), Float64(-x), -1.0) / sin(B));
	elseif (F <= -4.1e-107)
		tmp = Float64(Float64(F * Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B))) - Float64(x / B));
	elseif (F <= 1.45e-202)
		tmp = Float64(-1.0 * Float64(Float64(x * cos(B)) / sin(B)));
	elseif (F <= 1.75e+154)
		tmp = Float64(Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F) / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(x / 1.0)) * 1.0) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.2e-83], N[(N[(N[Cos[B], $MachinePrecision] * (-x) + -1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -4.1e-107], N[(N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.45e-202], N[(-1.0 * N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.75e+154], N[(N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(x / 1.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] / B), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -5.2000000000000002e-83

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.6%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. lift-/.f64 N/A

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

      4. mult-flip-rev N/A

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

      5. lift-tan.f64 N/A

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

      6. +-commutative N/A

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

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

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

      8. lift-neg.f64 N/A

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

      9. tan-quot N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. div-flip-rev N/A

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

      13. associate-/l* N/A

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

      14. lift-*.f64 N/A

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

      15. lift-/.f64 N/A

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

   8. Applied rewrites 56.6%

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

   if -5.2000000000000002e-83 < F < -4.0999999999999999e-107

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lift-/.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 84.6%

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 57.9%

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

   8. Applied rewrites 57.9%

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

   if -4.0999999999999999e-107 < F < 1.4499999999999999e-202

   1. Initial program 76.1%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 56.2%

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

   4. Applied rewrites 56.2%

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

   if 1.4499999999999999e-202 < F < 1.7500000000000001e154

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 57.9%

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

   6. Applied rewrites 57.9%

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

   if 1.7500000000000001e154 < F

   1. Initial program 76.1%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.7%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

   6. Applied rewrites 33.4%

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

   7. Taylor expanded in F around inf

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

   9. Applied rewrites 19.3%

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

   10. Taylor expanded in F around inf

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

   12. Applied rewrites 30.1%

       \[\leadsto \frac{\left(1 - \frac{x}{1}\right) \cdot 1}{B} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 10: 76.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -5.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{2}\right)} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.1 \cdot 10^{-107}:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{-202}:\\ \;\;\;\;-1 \cdot \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{+154}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{x}{1}\right) \cdot 1}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -5.2e-83)
  (-
   (/ -1.0 (* B (+ 1.0 (* -0.16666666666666666 (pow B 2.0)))))
   (/ x (tan B)))
  (if (<= F -4.1e-107)
    (- (* F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B))) (/ x B))
    (if (<= F 1.45e-202)
      (* -1.0 (/ (* x (cos B)) (sin B)))
      (if (<= F 1.75e+154)
        (-
         (/ (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F) (sin B))
         (/ x B))
        (/ (* (- 1.0 (/ x 1.0)) 1.0) B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.2e-83) {
		tmp = (-1.0 / (B * (1.0 + (-0.16666666666666666 * pow(B, 2.0))))) - (x / tan(B));
	} else if (F <= -4.1e-107) {
		tmp = (F * (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B))) - (x / B);
	} else if (F <= 1.45e-202) {
		tmp = -1.0 * ((x * cos(B)) / sin(B));
	} else if (F <= 1.75e+154) {
		tmp = ((pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F) / sin(B)) - (x / B);
	} else {
		tmp = ((1.0 - (x / 1.0)) * 1.0) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.2e-83)
		tmp = Float64(Float64(-1.0 / Float64(B * Float64(1.0 + Float64(-0.16666666666666666 * (B ^ 2.0))))) - Float64(x / tan(B)));
	elseif (F <= -4.1e-107)
		tmp = Float64(Float64(F * Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B))) - Float64(x / B));
	elseif (F <= 1.45e-202)
		tmp = Float64(-1.0 * Float64(Float64(x * cos(B)) / sin(B)));
	elseif (F <= 1.75e+154)
		tmp = Float64(Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F) / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(x / 1.0)) * 1.0) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.2e-83], N[(N[(-1.0 / N[(B * N[(1.0 + N[(-0.16666666666666666 * N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -4.1e-107], N[(N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.45e-202], N[(-1.0 * N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.75e+154], N[(N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(x / 1.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] / B), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -5.2000000000000002e-83

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.6%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 55.6%

         \[\leadsto \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{\color{blue}{2}}\right)} - \frac{x}{\tan B} \]

   9. Applied rewrites 55.6%

      \[\leadsto \frac{-1}{\color{blue}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{2}\right)}} - \frac{x}{\tan B} \]

   if -5.2000000000000002e-83 < F < -4.0999999999999999e-107

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-fma.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. lift-/.f64 N/A

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

      14. lift-neg.f64 N/A

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

      15. associate-/l* N/A

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

      16. lower-*.f64 N/A

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

      17. lower-/.f64 84.6%

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 57.9%

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

   8. Applied rewrites 57.9%

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

   if -4.0999999999999999e-107 < F < 1.4499999999999999e-202

   1. Initial program 76.1%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 56.2%

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

   4. Applied rewrites 56.2%

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

   if 1.4499999999999999e-202 < F < 1.7500000000000001e154

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 57.9%

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

   6. Applied rewrites 57.9%

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

   if 1.7500000000000001e154 < F

   1. Initial program 76.1%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.7%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

   6. Applied rewrites 33.4%

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

   7. Taylor expanded in F around inf

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

   9. Applied rewrites 19.3%

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

   10. Taylor expanded in F around inf

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

   12. Applied rewrites 30.1%

       \[\leadsto \frac{\left(1 - \frac{x}{1}\right) \cdot 1}{B} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 11: 70.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{2}\right)} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -340000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-7}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (let* ((t_0
        (-
         (/ -1.0 (* B (+ 1.0 (* -0.16666666666666666 (pow B 2.0)))))
         (/ x (tan B)))))
  (if (<= x -340000000000.0)
    t_0
    (if (<= x 1.85e-7)
      (-
       (/ (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) F) (sin B))
       (/ x B))
      t_0))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 / (B * (1.0 + (-0.16666666666666666 * pow(B, 2.0))))) - (x / tan(B));
	double tmp;
	if (x <= -340000000000.0) {
		tmp = t_0;
	} else if (x <= 1.85e-7) {
		tmp = ((pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * F) / sin(B)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 / Float64(B * Float64(1.0 + Float64(-0.16666666666666666 * (B ^ 2.0))))) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -340000000000.0)
		tmp = t_0;
	elseif (x <= 1.85e-7)
		tmp = Float64(Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * F) / sin(B)) - Float64(x / B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 / N[(B * N[(1.0 + N[(-0.16666666666666666 * N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -340000000000.0], t$95$0, If[LessEqual[x, 1.85e-7], N[(N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.4e11 or 1.85e-7 < x

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.6%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 55.6%

         \[\leadsto \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{\color{blue}{2}}\right)} - \frac{x}{\tan B} \]

   9. Applied rewrites 55.6%

      \[\leadsto \frac{-1}{\color{blue}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{2}\right)}} - \frac{x}{\tan B} \]

   if -3.4e11 < x < 1.85e-7

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in B around 0

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

      1. lower-/.f64 57.9%

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

   6. Applied rewrites 57.9%

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

Alternative 12: 69.7% accurate, 1.4× speedup

Math

\[\mathsf{copysign}\left(1, B\right) \cdot \begin{array}{l} \mathbf{if}\;\left|B\right| \leq 0.0115:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{\left|B\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\left|B\right| \cdot \left(1 + -0.16666666666666666 \cdot {\left(\left|B\right|\right)}^{2}\right)} - \frac{x}{\tan \left(\left|B\right|\right)}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (*
 (copysign 1.0 B)
 (if (<= (fabs B) 0.0115)
   (/ (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F) x) (fabs B))
   (-
    (/
     -1.0
     (* (fabs B) (+ 1.0 (* -0.16666666666666666 (pow (fabs B) 2.0)))))
    (/ x (tan (fabs B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (fabs(B) <= 0.0115) {
		tmp = ((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F) - x) / fabs(B);
	} else {
		tmp = (-1.0 / (fabs(B) * (1.0 + (-0.16666666666666666 * pow(fabs(B), 2.0))))) - (x / tan(fabs(B)));
	}
	return copysign(1.0, B) * tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (abs(B) <= 0.0115)
		tmp = Float64(Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F) - x) / abs(B));
	else
		tmp = Float64(Float64(-1.0 / Float64(abs(B) * Float64(1.0 + Float64(-0.16666666666666666 * (abs(B) ^ 2.0))))) - Float64(x / tan(abs(B))));
	end
	return Float64(copysign(1.0, B) * tmp)
end

Wolfram

code[F_, B_, x_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[B]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[B], $MachinePrecision], 0.0115], N[(N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / N[Abs[B], $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(N[Abs[B], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[Power[N[Abs[B], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[N[Abs[B], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if B < 0.0115

   1. Initial program 76.1%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+r+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 44.7%

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 44.7%

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

   6. Applied rewrites 44.7%

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

   if 0.0115 < B

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.6%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 55.6%

         \[\leadsto \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{\color{blue}{2}}\right)} - \frac{x}{\tan B} \]

   9. Applied rewrites 55.6%

      \[\leadsto \frac{-1}{\color{blue}{B \cdot \left(1 + -0.16666666666666666 \cdot {B}^{2}\right)}} - \frac{x}{\tan B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 58.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{+23}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 45000:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{+168}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{x}{1}\right) \cdot 1}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -2.3e+23)
  (- (/ -1.0 (sin B)) (/ x B))
  (if (<= F 45000.0)
    (+
     (/ (- (* 0.3333333333333333 (* (pow B 2.0) x)) x) B)
     (* (/ F B) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
    (if (<= F 1.22e+168)
      (/ 1.0 (sin B))
      (/ (* (- 1.0 (/ x 1.0)) 1.0) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.3e+23) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 45000.0) {
		tmp = (((0.3333333333333333 * (pow(B, 2.0) * x)) - x) / B) + ((F / B) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else if (F <= 1.22e+168) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = ((1.0 - (x / 1.0)) * 1.0) / 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 <= (-2.3d+23)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 45000.0d0) then
        tmp = (((0.3333333333333333d0 * ((b ** 2.0d0) * x)) - x) / b) + ((f / b) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
    else if (f <= 1.22d+168) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = ((1.0d0 - (x / 1.0d0)) * 1.0d0) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.3e+23) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 45000.0) {
		tmp = (((0.3333333333333333 * (Math.pow(B, 2.0) * x)) - x) / B) + ((F / B) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else if (F <= 1.22e+168) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = ((1.0 - (x / 1.0)) * 1.0) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.3e+23:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 45000.0:
		tmp = (((0.3333333333333333 * (math.pow(B, 2.0) * x)) - x) / B) + ((F / B) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))
	elif F <= 1.22e+168:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = ((1.0 - (x / 1.0)) * 1.0) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.3e+23)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 45000.0)
		tmp = Float64(Float64(Float64(Float64(0.3333333333333333 * Float64((B ^ 2.0) * x)) - x) / B) + Float64(Float64(F / B) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))));
	elseif (F <= 1.22e+168)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(x / 1.0)) * 1.0) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.3e+23)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 45000.0)
		tmp = (((0.3333333333333333 * ((B ^ 2.0) * x)) - x) / B) + ((F / B) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
	elseif (F <= 1.22e+168)
		tmp = 1.0 / sin(B);
	else
		tmp = ((1.0 - (x / 1.0)) * 1.0) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.3e+23], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 45000.0], N[(N[(N[(N[(0.3333333333333333 * N[(N[Power[B, 2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.22e+168], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(x / 1.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.3e23

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.6%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 37.0%

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

   9. Applied rewrites 37.0%

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

   if -2.3e23 < F < 45000

   1. Initial program 76.1%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 62.2%

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

   4. Applied rewrites 62.2%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 36.2%

         \[\leadsto \frac{0.3333333333333333 \cdot \left({B}^{2} \cdot x\right) - x}{B} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   7. Applied rewrites 36.2%

      \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left({B}^{2} \cdot x\right) - x}{B}} + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if 45000 < F < 1.2199999999999999e168

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 16.7%

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

   if 1.2199999999999999e168 < F

   1. Initial program 76.1%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.7%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

   6. Applied rewrites 33.4%

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

   7. Taylor expanded in F around inf

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

   9. Applied rewrites 19.3%

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

   10. Taylor expanded in F around inf

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

   12. Applied rewrites 30.1%

       \[\leadsto \frac{\left(1 - \frac{x}{1}\right) \cdot 1}{B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 14: 58.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{+23}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 30000000000:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{B}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{+168}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{x}{1}\right) \cdot 1}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -2.3e+23)
  (- (/ -1.0 (sin B)) (/ x B))
  (if (<= F 30000000000.0)
    (/ (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F) x) B)
    (if (<= F 1.22e+168)
      (/ 1.0 (sin B))
      (/ (* (- 1.0 (/ x 1.0)) 1.0) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.3e+23) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 30000000000.0) {
		tmp = ((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F) - x) / B;
	} else if (F <= 1.22e+168) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = ((1.0 - (x / 1.0)) * 1.0) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.3e+23)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 30000000000.0)
		tmp = Float64(Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F) - x) / B);
	elseif (F <= 1.22e+168)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(x / 1.0)) * 1.0) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.3e+23], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 30000000000.0], N[(N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.22e+168], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(x / 1.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.3e23

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 56.6%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 37.0%

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

   9. Applied rewrites 37.0%

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

   if -2.3e23 < F < 3e10

   1. Initial program 76.1%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+r+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 44.7%

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 44.7%

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

   6. Applied rewrites 44.7%

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

   if 3e10 < F < 1.2199999999999999e168

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 16.7%

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

   if 1.2199999999999999e168 < F

   1. Initial program 76.1%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.7%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

   6. Applied rewrites 33.4%

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

   7. Taylor expanded in F around inf

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

   9. Applied rewrites 19.3%

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

   10. Taylor expanded in F around inf

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

   12. Applied rewrites 30.1%

       \[\leadsto \frac{\left(1 - \frac{x}{1}\right) \cdot 1}{B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 15: 52.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.55 \cdot 10^{+143}:\\ \;\;\;\;\frac{\left(1 - \frac{x}{-1}\right) \cdot -1}{B}\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 30000000000:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{B}\\ \mathbf{elif}\;F \leq 1.22 \cdot 10^{+168}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{x}{1}\right) \cdot 1}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -1.55e+143)
  (/ (* (- 1.0 (/ x -1.0)) -1.0) B)
  (if (<= F -4.5e+21)
    (/ -1.0 (sin B))
    (if (<= F 30000000000.0)
      (/ (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F) x) B)
      (if (<= F 1.22e+168)
        (/ 1.0 (sin B))
        (/ (* (- 1.0 (/ x 1.0)) 1.0) B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.55e+143) {
		tmp = ((1.0 - (x / -1.0)) * -1.0) / B;
	} else if (F <= -4.5e+21) {
		tmp = -1.0 / sin(B);
	} else if (F <= 30000000000.0) {
		tmp = ((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F) - x) / B;
	} else if (F <= 1.22e+168) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = ((1.0 - (x / 1.0)) * 1.0) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.55e+143)
		tmp = Float64(Float64(Float64(1.0 - Float64(x / -1.0)) * -1.0) / B);
	elseif (F <= -4.5e+21)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 30000000000.0)
		tmp = Float64(Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F) - x) / B);
	elseif (F <= 1.22e+168)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(x / 1.0)) * 1.0) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.55e+143], N[(N[(N[(1.0 - N[(x / -1.0), $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -4.5e+21], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 30000000000.0], N[(N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.22e+168], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(x / 1.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] / B), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.55e143

   1. Initial program 76.1%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.7%

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

      1. lift--.f64 N/A

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

      2. sub-to-mult N/A

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

      3. lower-unsound-*.f64 N/A

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

   6. Applied rewrites 33.4%

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

   7. Taylor expanded in F around -inf

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

   9. Applied rewrites 19.3%

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

   10. Taylor expanded in F around -inf

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

   12. Applied rewrites 30.5%

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

   if -1.55e143 < F < -4.5e21

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 17.3%

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

   if -4.5e21 < F < 3e10

   1. Initial program 76.1%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 44.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-fma.f64 N/A

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

      6. lift-pow.f64 N/A

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

      7. pow2 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-+r+ N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 N/A

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

      12. lift-fma.f64 N/A

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

      13. lift-*.f64 44.7%

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

      14. lift-fma.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-fma.f64 44.7%

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

   6. Applied rewrites 44.7%

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

   if 3e10 < F < 1.2199999999999999e168

   1. Initial program 76.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.1%

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

   3. Applied rewrites 84.7%

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 16.7%

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

   if 1.2199999999999999e168 < F

   1. Initial program 76.1%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      2. sub-to-mult N/A

         \[\leadsto \frac{\left(1 - \frac{x}{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}\right) \cdot \left(F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}\right)}{B} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left(1 - \frac{x}{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}\right) \cdot \left(F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}\right)}{B} \]

   6. Applied rewrites 33.4%

      \[\leadsto \frac{\left(1 - \frac{x}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]

   7. Taylor expanded in F around inf

      \[\leadsto \frac{\left(1 - \frac{x}{1}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 19.3%

      \[\leadsto \frac{\left(1 - \frac{x}{1}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]

   10. Taylor expanded in F around inf

       \[\leadsto \frac{\left(1 - \frac{x}{1}\right) \cdot 1}{B} \]
   11. Step-by-step derivation

   12. Applied rewrites 30.1%

       \[\leadsto \frac{\left(1 - \frac{x}{1}\right) \cdot 1}{B} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 16: 52.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.55 \cdot 10^{+143}:\\ \;\;\;\;\frac{\left(1 - \frac{x}{-1}\right) \cdot -1}{B}\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 430000000:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{x}{1}\right) \cdot 1}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -1.55e+143)
  (/ (* (- 1.0 (/ x -1.0)) -1.0) B)
  (if (<= F -4.5e+21)
    (/ -1.0 (sin B))
    (if (<= F 430000000.0)
      (/ (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F) x) B)
      (/ (* (- 1.0 (/ x 1.0)) 1.0) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.55e+143) {
		tmp = ((1.0 - (x / -1.0)) * -1.0) / B;
	} else if (F <= -4.5e+21) {
		tmp = -1.0 / sin(B);
	} else if (F <= 430000000.0) {
		tmp = ((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F) - x) / B;
	} else {
		tmp = ((1.0 - (x / 1.0)) * 1.0) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.55e+143)
		tmp = Float64(Float64(Float64(1.0 - Float64(x / -1.0)) * -1.0) / B);
	elseif (F <= -4.5e+21)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 430000000.0)
		tmp = Float64(Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F) - x) / B);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(x / 1.0)) * 1.0) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.55e+143], N[(N[(N[(1.0 - N[(x / -1.0), $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -4.5e+21], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 430000000.0], N[(N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(1.0 - N[(x / 1.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.55e143

   1. Initial program 76.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      2. sub-to-mult N/A

         \[\leadsto \frac{\left(1 - \frac{x}{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}\right) \cdot \left(F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}\right)}{B} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left(1 - \frac{x}{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}\right) \cdot \left(F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}\right)}{B} \]

   6. Applied rewrites 33.4%

      \[\leadsto \frac{\left(1 - \frac{x}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{\left(1 - \frac{x}{-1}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 19.3%

      \[\leadsto \frac{\left(1 - \frac{x}{-1}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]

   10. Taylor expanded in F around -inf

       \[\leadsto \frac{\left(1 - \frac{x}{-1}\right) \cdot -1}{B} \]
   11. Step-by-step derivation

   12. Applied rewrites 30.5%

       \[\leadsto \frac{\left(1 - \frac{x}{-1}\right) \cdot -1}{B} \]

   if -1.55e143 < F < -4.5e21

   1. Initial program 76.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \color{blue}{\left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right)} \]

      4. sub-flip-reverse N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

      5. lower--.f64 76.1%

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} - x \cdot \frac{1}{\tan B}} \]

   3. Applied rewrites 84.7%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}{\sin B} - \frac{x}{\tan B}} \]

   5. Applied rewrites 84.6%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, F, \left(-x\right) \cdot \cos B\right)}{\sin B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \frac{\color{blue}{-1}}{\sin B} \]
   7. Step-by-step derivation

   8. Applied rewrites 17.3%

      \[\leadsto \frac{\color{blue}{-1}}{\sin B} \]

   if -4.5e21 < F < 4.3e8

   1. Initial program 76.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      2. *-commutative N/A

         \[\leadsto \frac{{\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{{\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      4. +-commutative N/A

         \[\leadsto \frac{{\left(\mathsf{fma}\left(2, x, {F}^{2}\right) + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{{\left(\left(2 \cdot x + {F}^{2}\right) + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{{\left(\left(2 \cdot x + {F}^{2}\right) + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      7. pow2 N/A

         \[\leadsto \frac{{\left(\left(2 \cdot x + F \cdot F\right) + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{{\left(\left(2 \cdot x + F \cdot F\right) + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      9. associate-+r+ N/A

         \[\leadsto \frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      11. lift-fma.f64 N/A

          \[\leadsto \frac{{\left(2 \cdot x + \mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      13. lift-*.f64 44.7%

          \[\leadsto \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{B} \]

      14. lift-fma.f64 N/A

          \[\leadsto \frac{{\left(2 \cdot x + \mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      15. *-commutative N/A

          \[\leadsto \frac{{\left(x \cdot 2 + \mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      16. lower-fma.f64 44.7%

          \[\leadsto \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{B} \]

   6. Applied rewrites 44.7%

      \[\leadsto \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{B} \]

   if 4.3e8 < F

   1. Initial program 76.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      2. sub-to-mult N/A

         \[\leadsto \frac{\left(1 - \frac{x}{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}\right) \cdot \left(F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}\right)}{B} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left(1 - \frac{x}{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}\right) \cdot \left(F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}\right)}{B} \]

   6. Applied rewrites 33.4%

      \[\leadsto \frac{\left(1 - \frac{x}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]

   7. Taylor expanded in F around inf

      \[\leadsto \frac{\left(1 - \frac{x}{1}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 19.3%

      \[\leadsto \frac{\left(1 - \frac{x}{1}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]

   10. Taylor expanded in F around inf

       \[\leadsto \frac{\left(1 - \frac{x}{1}\right) \cdot 1}{B} \]
   11. Step-by-step derivation

   12. Applied rewrites 30.1%

       \[\leadsto \frac{\left(1 - \frac{x}{1}\right) \cdot 1}{B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 17: 52.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+116}:\\ \;\;\;\;\frac{\left(1 - \frac{x}{-1}\right) \cdot -1}{B}\\ \mathbf{elif}\;F \leq 430000000:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{x}{1}\right) \cdot 1}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -1e+116)
  (/ (* (- 1.0 (/ x -1.0)) -1.0) B)
  (if (<= F 430000000.0)
    (/ (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F) x) B)
    (/ (* (- 1.0 (/ x 1.0)) 1.0) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e+116) {
		tmp = ((1.0 - (x / -1.0)) * -1.0) / B;
	} else if (F <= 430000000.0) {
		tmp = ((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F) - x) / B;
	} else {
		tmp = ((1.0 - (x / 1.0)) * 1.0) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1e+116)
		tmp = Float64(Float64(Float64(1.0 - Float64(x / -1.0)) * -1.0) / B);
	elseif (F <= 430000000.0)
		tmp = Float64(Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F) - x) / B);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(x / 1.0)) * 1.0) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1e+116], N[(N[(N[(1.0 - N[(x / -1.0), $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 430000000.0], N[(N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(1.0 - N[(x / 1.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e116

   1. Initial program 76.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      2. sub-to-mult N/A

         \[\leadsto \frac{\left(1 - \frac{x}{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}\right) \cdot \left(F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}\right)}{B} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left(1 - \frac{x}{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}\right) \cdot \left(F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}\right)}{B} \]

   6. Applied rewrites 33.4%

      \[\leadsto \frac{\left(1 - \frac{x}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{\left(1 - \frac{x}{-1}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 19.3%

      \[\leadsto \frac{\left(1 - \frac{x}{-1}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]

   10. Taylor expanded in F around -inf

       \[\leadsto \frac{\left(1 - \frac{x}{-1}\right) \cdot -1}{B} \]
   11. Step-by-step derivation

   12. Applied rewrites 30.5%

       \[\leadsto \frac{\left(1 - \frac{x}{-1}\right) \cdot -1}{B} \]

   if -1e116 < F < 4.3e8

   1. Initial program 76.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      2. *-commutative N/A

         \[\leadsto \frac{{\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      3. lift-+.f64 N/A

         \[\leadsto \frac{{\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      4. +-commutative N/A

         \[\leadsto \frac{{\left(\mathsf{fma}\left(2, x, {F}^{2}\right) + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{{\left(\left(2 \cdot x + {F}^{2}\right) + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      6. lift-pow.f64 N/A

         \[\leadsto \frac{{\left(\left(2 \cdot x + {F}^{2}\right) + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      7. pow2 N/A

         \[\leadsto \frac{{\left(\left(2 \cdot x + F \cdot F\right) + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{{\left(\left(2 \cdot x + F \cdot F\right) + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      9. associate-+r+ N/A

         \[\leadsto \frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{{\left(2 \cdot x + \left(F \cdot F + 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      11. lift-fma.f64 N/A

          \[\leadsto \frac{{\left(2 \cdot x + \mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      12. lift-fma.f64 N/A

          \[\leadsto \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      13. lift-*.f64 44.7%

          \[\leadsto \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{B} \]

      14. lift-fma.f64 N/A

          \[\leadsto \frac{{\left(2 \cdot x + \mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      15. *-commutative N/A

          \[\leadsto \frac{{\left(x \cdot 2 + \mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      16. lower-fma.f64 44.7%

          \[\leadsto \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{B} \]

   6. Applied rewrites 44.7%

      \[\leadsto \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{B} \]

   if 4.3e8 < F

   1. Initial program 76.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      2. sub-to-mult N/A

         \[\leadsto \frac{\left(1 - \frac{x}{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}\right) \cdot \left(F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}\right)}{B} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left(1 - \frac{x}{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}\right) \cdot \left(F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}\right)}{B} \]

   6. Applied rewrites 33.4%

      \[\leadsto \frac{\left(1 - \frac{x}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]

   7. Taylor expanded in F around inf

      \[\leadsto \frac{\left(1 - \frac{x}{1}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 19.3%

      \[\leadsto \frac{\left(1 - \frac{x}{1}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]

   10. Taylor expanded in F around inf

       \[\leadsto \frac{\left(1 - \frac{x}{1}\right) \cdot 1}{B} \]
   11. Step-by-step derivation

   12. Applied rewrites 30.1%

       \[\leadsto \frac{\left(1 - \frac{x}{1}\right) \cdot 1}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 51.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -36:\\ \;\;\;\;\frac{\left(1 - \frac{x}{-1}\right) \cdot -1}{B}\\ \mathbf{elif}\;F \leq 2700000000000:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{x}{1}\right) \cdot 1}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -36.0)
  (/ (* (- 1.0 (/ x -1.0)) -1.0) B)
  (if (<= F 2700000000000.0)
    (/ (- (* (pow (fma 2.0 x 2.0) -0.5) F) x) B)
    (/ (* (- 1.0 (/ x 1.0)) 1.0) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -36.0) {
		tmp = ((1.0 - (x / -1.0)) * -1.0) / B;
	} else if (F <= 2700000000000.0) {
		tmp = ((pow(fma(2.0, x, 2.0), -0.5) * F) - x) / B;
	} else {
		tmp = ((1.0 - (x / 1.0)) * 1.0) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -36.0)
		tmp = Float64(Float64(Float64(1.0 - Float64(x / -1.0)) * -1.0) / B);
	elseif (F <= 2700000000000.0)
		tmp = Float64(Float64(Float64((fma(2.0, x, 2.0) ^ -0.5) * F) - x) / B);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(x / 1.0)) * 1.0) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -36.0], N[(N[(N[(1.0 - N[(x / -1.0), $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2700000000000.0], N[(N[(N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(1.0 - N[(x / 1.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -36

   1. Initial program 76.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      2. sub-to-mult N/A

         \[\leadsto \frac{\left(1 - \frac{x}{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}\right) \cdot \left(F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}\right)}{B} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left(1 - \frac{x}{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}\right) \cdot \left(F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}\right)}{B} \]

   6. Applied rewrites 33.4%

      \[\leadsto \frac{\left(1 - \frac{x}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{\left(1 - \frac{x}{-1}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 19.3%

      \[\leadsto \frac{\left(1 - \frac{x}{-1}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]

   10. Taylor expanded in F around -inf

       \[\leadsto \frac{\left(1 - \frac{x}{-1}\right) \cdot -1}{B} \]
   11. Step-by-step derivation

   12. Applied rewrites 30.5%

       \[\leadsto \frac{\left(1 - \frac{x}{-1}\right) \cdot -1}{B} \]

   if -36 < F < 2.7e12

   1. Initial program 76.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      2. sub-to-mult N/A

         \[\leadsto \frac{\left(1 - \frac{x}{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}\right) \cdot \left(F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}\right)}{B} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left(1 - \frac{x}{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}\right) \cdot \left(F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}\right)}{B} \]

   6. Applied rewrites 33.4%

      \[\leadsto \frac{\left(1 - \frac{x}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]

   7. Taylor expanded in F around 0

      \[\leadsto \frac{\left(1 - \frac{x}{{\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5} \cdot F}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 30.4%

      \[\leadsto \frac{\left(1 - \frac{x}{{\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5} \cdot F}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]

   10. Taylor expanded in F around 0

       \[\leadsto \frac{\left(1 - \frac{x}{{\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5} \cdot F}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5} \cdot F\right)}{B} \]
   11. Step-by-step derivation

   12. Applied rewrites 27.1%

       \[\leadsto \frac{\left(1 - \frac{x}{{\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5} \cdot F}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5} \cdot F\right)}{B} \]
   13. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \frac{\left(1 - \frac{x}{{\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{\frac{-1}{2}} \cdot F}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{\frac{-1}{2}} \cdot F\right)}{B} \]

       2. lift--.f64 N/A

          \[\leadsto \frac{\left(1 - \frac{x}{{\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{\frac{-1}{2}} \cdot F}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{\frac{-1}{2}} \cdot F\right)}{B} \]

       3. lift-/.f64 N/A

          \[\leadsto \frac{\left(1 - \frac{x}{{\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{\frac{-1}{2}} \cdot F}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{\frac{-1}{2}} \cdot F\right)}{B} \]

       4. sub-to-mult-rev N/A

          \[\leadsto \frac{{\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       5. lower--.f64 30.1%

          \[\leadsto \frac{{\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5} \cdot F - x}{B} \]

       6. lift-fma.f64 N/A

          \[\leadsto \frac{{\left(x \cdot 2 + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       7. *-commutative N/A

          \[\leadsto \frac{{\left(2 \cdot x + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

       8. lower-fma.f64 30.1%

          \[\leadsto \frac{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5} \cdot F - x}{B} \]

   14. Applied rewrites 30.1%

       \[\leadsto \frac{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5} \cdot F - x}{B} \]

   if 2.7e12 < F

   1. Initial program 76.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      2. sub-to-mult N/A

         \[\leadsto \frac{\left(1 - \frac{x}{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}\right) \cdot \left(F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}\right)}{B} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left(1 - \frac{x}{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}\right) \cdot \left(F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}\right)}{B} \]

   6. Applied rewrites 33.4%

      \[\leadsto \frac{\left(1 - \frac{x}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]

   7. Taylor expanded in F around inf

      \[\leadsto \frac{\left(1 - \frac{x}{1}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 19.3%

      \[\leadsto \frac{\left(1 - \frac{x}{1}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]

   10. Taylor expanded in F around inf

       \[\leadsto \frac{\left(1 - \frac{x}{1}\right) \cdot 1}{B} \]
   11. Step-by-step derivation

   12. Applied rewrites 30.1%

       \[\leadsto \frac{\left(1 - \frac{x}{1}\right) \cdot 1}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 44.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -7 \cdot 10^{-107}:\\ \;\;\;\;\frac{\left(1 - \frac{x}{-1}\right) \cdot -1}{B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-111}:\\ \;\;\;\;\frac{-1 \cdot x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 - \frac{x}{1}\right) \cdot 1}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -7e-107)
  (/ (* (- 1.0 (/ x -1.0)) -1.0) B)
  (if (<= F 5.2e-111)
    (/ (* -1.0 x) B)
    (/ (* (- 1.0 (/ x 1.0)) 1.0) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7e-107) {
		tmp = ((1.0 - (x / -1.0)) * -1.0) / B;
	} else if (F <= 5.2e-111) {
		tmp = (-1.0 * x) / B;
	} else {
		tmp = ((1.0 - (x / 1.0)) * 1.0) / 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 <= (-7d-107)) then
        tmp = ((1.0d0 - (x / (-1.0d0))) * (-1.0d0)) / b
    else if (f <= 5.2d-111) then
        tmp = ((-1.0d0) * x) / b
    else
        tmp = ((1.0d0 - (x / 1.0d0)) * 1.0d0) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -7e-107) {
		tmp = ((1.0 - (x / -1.0)) * -1.0) / B;
	} else if (F <= 5.2e-111) {
		tmp = (-1.0 * x) / B;
	} else {
		tmp = ((1.0 - (x / 1.0)) * 1.0) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -7e-107:
		tmp = ((1.0 - (x / -1.0)) * -1.0) / B
	elif F <= 5.2e-111:
		tmp = (-1.0 * x) / B
	else:
		tmp = ((1.0 - (x / 1.0)) * 1.0) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7e-107)
		tmp = Float64(Float64(Float64(1.0 - Float64(x / -1.0)) * -1.0) / B);
	elseif (F <= 5.2e-111)
		tmp = Float64(Float64(-1.0 * x) / B);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(x / 1.0)) * 1.0) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -7e-107)
		tmp = ((1.0 - (x / -1.0)) * -1.0) / B;
	elseif (F <= 5.2e-111)
		tmp = (-1.0 * x) / B;
	else
		tmp = ((1.0 - (x / 1.0)) * 1.0) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7e-107], N[(N[(N[(1.0 - N[(x / -1.0), $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 5.2e-111], N[(N[(-1.0 * x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(1.0 - N[(x / 1.0), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.9999999999999997e-107

   1. Initial program 76.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      2. sub-to-mult N/A

         \[\leadsto \frac{\left(1 - \frac{x}{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}\right) \cdot \left(F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}\right)}{B} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left(1 - \frac{x}{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}\right) \cdot \left(F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}\right)}{B} \]

   6. Applied rewrites 33.4%

      \[\leadsto \frac{\left(1 - \frac{x}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{\left(1 - \frac{x}{-1}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 19.3%

      \[\leadsto \frac{\left(1 - \frac{x}{-1}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]

   10. Taylor expanded in F around -inf

       \[\leadsto \frac{\left(1 - \frac{x}{-1}\right) \cdot -1}{B} \]
   11. Step-by-step derivation

   12. Applied rewrites 30.5%

       \[\leadsto \frac{\left(1 - \frac{x}{-1}\right) \cdot -1}{B} \]

   if -6.9999999999999997e-107 < F < 5.1999999999999996e-111

   1. Initial program 76.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. lower-*.f64 30.2%

         \[\leadsto \frac{-1 \cdot x}{B} \]

   7. Applied rewrites 30.2%

      \[\leadsto \frac{-1 \cdot x}{B} \]

   if 5.1999999999999996e-111 < F

   1. Initial program 76.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      2. sub-to-mult N/A

         \[\leadsto \frac{\left(1 - \frac{x}{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}\right) \cdot \left(F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}\right)}{B} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left(1 - \frac{x}{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}\right) \cdot \left(F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}\right)}{B} \]

   6. Applied rewrites 33.4%

      \[\leadsto \frac{\left(1 - \frac{x}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]

   7. Taylor expanded in F around inf

      \[\leadsto \frac{\left(1 - \frac{x}{1}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 19.3%

      \[\leadsto \frac{\left(1 - \frac{x}{1}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]

   10. Taylor expanded in F around inf

       \[\leadsto \frac{\left(1 - \frac{x}{1}\right) \cdot 1}{B} \]
   11. Step-by-step derivation

   12. Applied rewrites 30.1%

       \[\leadsto \frac{\left(1 - \frac{x}{1}\right) \cdot 1}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 37.4% accurate, 6.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -7 \cdot 10^{-107}:\\ \;\;\;\;\frac{\left(1 - \frac{x}{-1}\right) \cdot -1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -7e-107) (/ (* (- 1.0 (/ x -1.0)) -1.0) B) (/ (* -1.0 x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7e-107) {
		tmp = ((1.0 - (x / -1.0)) * -1.0) / B;
	} else {
		tmp = (-1.0 * x) / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-7d-107)) then
        tmp = ((1.0d0 - (x / (-1.0d0))) * (-1.0d0)) / b
    else
        tmp = ((-1.0d0) * x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -7e-107) {
		tmp = ((1.0 - (x / -1.0)) * -1.0) / B;
	} else {
		tmp = (-1.0 * x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -7e-107:
		tmp = ((1.0 - (x / -1.0)) * -1.0) / B
	else:
		tmp = (-1.0 * x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7e-107)
		tmp = Float64(Float64(Float64(1.0 - Float64(x / -1.0)) * -1.0) / B);
	else
		tmp = Float64(Float64(-1.0 * x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -7e-107)
		tmp = ((1.0 - (x / -1.0)) * -1.0) / B;
	else
		tmp = (-1.0 * x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7e-107], N[(N[(N[(1.0 - N[(x / -1.0), $MachinePrecision]), $MachinePrecision] * -1.0), $MachinePrecision] / B), $MachinePrecision], N[(N[(-1.0 * x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -6.9999999999999997e-107

   1. Initial program 76.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]
   5. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B} \]

      2. sub-to-mult N/A

         \[\leadsto \frac{\left(1 - \frac{x}{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}\right) \cdot \left(F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}\right)}{B} \]

      3. lower-unsound-*.f64 N/A

         \[\leadsto \frac{\left(1 - \frac{x}{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}}\right) \cdot \left(F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{\frac{-1}{2}}\right)}{B} \]

   6. Applied rewrites 33.4%

      \[\leadsto \frac{\left(1 - \frac{x}{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{\left(1 - \frac{x}{-1}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 19.3%

      \[\leadsto \frac{\left(1 - \frac{x}{-1}\right) \cdot \left({\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F\right)}{B} \]

   10. Taylor expanded in F around -inf

       \[\leadsto \frac{\left(1 - \frac{x}{-1}\right) \cdot -1}{B} \]
   11. Step-by-step derivation

   12. Applied rewrites 30.5%

       \[\leadsto \frac{\left(1 - \frac{x}{-1}\right) \cdot -1}{B} \]

   if -6.9999999999999997e-107 < F

   1. Initial program 76.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. lower-*.f64 30.2%

         \[\leadsto \frac{-1 \cdot x}{B} \]

   7. Applied rewrites 30.2%

      \[\leadsto \frac{-1 \cdot x}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 30.8% accurate, 10.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -410000000000:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1 \cdot x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F -410000000000.0) (/ -1.0 B) (/ (* -1.0 x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -410000000000.0) {
		tmp = -1.0 / B;
	} else {
		tmp = (-1.0 * x) / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-410000000000.0d0)) then
        tmp = (-1.0d0) / b
    else
        tmp = ((-1.0d0) * x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -410000000000.0) {
		tmp = -1.0 / B;
	} else {
		tmp = (-1.0 * x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -410000000000.0:
		tmp = -1.0 / B
	else:
		tmp = (-1.0 * x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -410000000000.0)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(Float64(-1.0 * x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -410000000000.0)
		tmp = -1.0 / B;
	else
		tmp = (-1.0 * x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -410000000000.0], N[(-1.0 / B), $MachinePrecision], N[(N[(-1.0 * x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -4.1e11

   1. Initial program 76.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \frac{-1}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 10.7%

         \[\leadsto \frac{-1}{B} \]

   7. Applied rewrites 10.7%

      \[\leadsto \frac{-1}{\color{blue}{B}} \]

   if -4.1e11 < F

   1. Initial program 76.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. lower-*.f64 30.2%

         \[\leadsto \frac{-1 \cdot x}{B} \]

   7. Applied rewrites 30.2%

      \[\leadsto \frac{-1 \cdot x}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 17.8% accurate, 14.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq 1.2 \cdot 10^{-180}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (if (<= F 1.2e-180) (/ -1.0 B) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 1.2e-180) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= 1.2d-180) then
        tmp = (-1.0d0) / b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 1.2e-180) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 1.2e-180:
		tmp = -1.0 / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 1.2e-180)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 1.2e-180)
		tmp = -1.0 / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 1.2e-180], N[(-1.0 / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.1999999999999999e-180

   1. Initial program 76.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \frac{-1}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 10.7%

         \[\leadsto \frac{-1}{B} \]

   7. Applied rewrites 10.7%

      \[\leadsto \frac{-1}{\color{blue}{B}} \]

   if 1.1999999999999999e-180 < F

   1. Initial program 76.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 44.7%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{1}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 9.9%

         \[\leadsto \frac{1}{B} \]

   7. Applied rewrites 9.9%

      \[\leadsto \frac{1}{\color{blue}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 10.7% accurate, 25.4× speedup

Math

\[\frac{-1}{B} \]

FPCore

(FPCore (F B x)
  :precision binary64
  (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 76.1%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
3. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

   2. metadata-eval N/A

      \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

   3. lower-/.f64 N/A

      \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

4. Applied rewrites 44.7%

   \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{B}} \]

5. Taylor expanded in F around -inf

   \[\leadsto \frac{-1}{\color{blue}{B}} \]
6. Step-by-step derivation

   1. lower-/.f64 10.7%

      \[\leadsto \frac{-1}{B} \]

7. Applied rewrites 10.7%

   \[\leadsto \frac{-1}{\color{blue}{B}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(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))))))