VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.2% → 99.7%

Time: 6.0s

Alternatives: 7

Speedup: N/A×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -1.5 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left({\sin B}^{-1}, -1, \frac{-1 \cdot t\_0}{\sin B}\right)\\ \mathbf{elif}\;F \leq 5.7 \cdot 10^{+100}:\\ \;\;\;\;-1 \cdot \frac{x}{\tan B} + \frac{F \cdot {\left(\mathsf{fma}\left({F}^{1}, {F}^{1}, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -1.5e+63)
     (fma (pow (sin B) -1.0) -1.0 (/ (* -1.0 t_0) (sin B)))
     (if (<= F 5.7e+100)
       (+
        (* -1.0 (/ x (tan B)))
        (/
         (* F (pow (fma (pow F 1.0) (pow F 1.0) (fma 2.0 x 2.0)) -0.5))
         (sin B)))
       (/ (- 1.0 t_0) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -1.5e+63) {
		tmp = fma(pow(sin(B), -1.0), -1.0, ((-1.0 * t_0) / sin(B)));
	} else if (F <= 5.7e+100) {
		tmp = (-1.0 * (x / tan(B))) + ((F * pow(fma(pow(F, 1.0), pow(F, 1.0), fma(2.0, x, 2.0)), -0.5)) / sin(B));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -1.5e+63)
		tmp = fma((sin(B) ^ -1.0), -1.0, Float64(Float64(-1.0 * t_0) / sin(B)));
	elseif (F <= 5.7e+100)
		tmp = Float64(Float64(-1.0 * Float64(x / tan(B))) + Float64(Float64(F * (fma((F ^ 1.0), (F ^ 1.0), fma(2.0, x, 2.0)) ^ -0.5)) / sin(B)));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -1.5e+63], N[(N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision] * -1.0 + N[(N[(-1.0 * t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.7e+100], N[(N[(-1.0 * N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F * N[Power[N[(N[Power[F, 1.0], $MachinePrecision] * N[Power[F, 1.0], $MachinePrecision] + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5e63

   1. Initial program 47.3%

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

   3. Taylor expanded in F around -inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lift-sin.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -1.5e63 < F < 5.69999999999999984e100

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 99.7

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

   6. Applied rewrites 99.7%

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

   if 5.69999999999999984e100 < F

   1. Initial program 55.9%

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

   3. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{+63}:\\ \;\;\;\;\mathsf{fma}\left({\sin B}^{-1}, -1, \frac{-1 \cdot \left(\cos B \cdot x\right)}{\sin B}\right)\\ \mathbf{elif}\;F \leq 5.7 \cdot 10^{+100}:\\ \;\;\;\;-1 \cdot \frac{x}{\tan B} + \frac{F \cdot {\left(\mathsf{fma}\left({F}^{1}, {F}^{1}, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -1 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left({\sin B}^{-1}, -1, \frac{-1 \cdot t\_0}{\sin B}\right)\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{+100}:\\ \;\;\;\;-1 \cdot \frac{x}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-1 \cdot \frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -1e+62)
     (fma (pow (sin B) -1.0) -1.0 (/ (* -1.0 t_0) (sin B)))
     (if (<= F 5.2e+100)
       (+
        (* -1.0 (/ x (tan B)))
        (*
         (/ F (sin B))
         (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (* -1.0 (/ 1.0 2.0)))))
       (/ (- 1.0 t_0) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -1e+62) {
		tmp = fma(pow(sin(B), -1.0), -1.0, ((-1.0 * t_0) / sin(B)));
	} else if (F <= 5.2e+100) {
		tmp = (-1.0 * (x / tan(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), (-1.0 * (1.0 / 2.0))));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -1e+62)
		tmp = fma((sin(B) ^ -1.0), -1.0, Float64(Float64(-1.0 * t_0) / sin(B)));
	elseif (F <= 5.2e+100)
		tmp = Float64(Float64(-1.0 * Float64(x / tan(B))) + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-1.0 * Float64(1.0 / 2.0)))));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -1e+62], N[(N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision] * -1.0 + N[(N[(-1.0 * t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.2e+100], N[(N[(-1.0 * N[(x / 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 * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.00000000000000004e62

   1. Initial program 47.3%

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

   3. Taylor expanded in F around -inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lift-sin.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -1.00000000000000004e62 < F < 5.2000000000000003e100

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if 5.2000000000000003e100 < F

   1. Initial program 55.9%

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

   3. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left({\sin B}^{-1}, -1, \frac{-1 \cdot \left(\cos B \cdot x\right)}{\sin B}\right)\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{+100}:\\ \;\;\;\;-1 \cdot \frac{x}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-1 \cdot \frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -5 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left({\sin B}^{-1}, -1, \frac{-1 \cdot t\_0}{\sin B}\right)\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{+100}:\\ \;\;\;\;-1 \cdot \left(x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\left(\mathsf{fma}\left({F}^{1}, {F}^{1}, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -5e+24)
     (fma (pow (sin B) -1.0) -1.0 (/ (* -1.0 t_0) (sin B)))
     (if (<= F 5.5e+100)
       (+
        (* -1.0 (* x (/ 1.0 (tan B))))
        (/
         (* F (pow (fma (pow F 1.0) (pow F 1.0) (fma 2.0 x 2.0)) -0.5))
         (sin B)))
       (/ (- 1.0 t_0) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -5e+24) {
		tmp = fma(pow(sin(B), -1.0), -1.0, ((-1.0 * t_0) / sin(B)));
	} else if (F <= 5.5e+100) {
		tmp = (-1.0 * (x * (1.0 / tan(B)))) + ((F * pow(fma(pow(F, 1.0), pow(F, 1.0), fma(2.0, x, 2.0)), -0.5)) / sin(B));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -5e+24)
		tmp = fma((sin(B) ^ -1.0), -1.0, Float64(Float64(-1.0 * t_0) / sin(B)));
	elseif (F <= 5.5e+100)
		tmp = Float64(Float64(-1.0 * Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F * (fma((F ^ 1.0), (F ^ 1.0), fma(2.0, x, 2.0)) ^ -0.5)) / sin(B)));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -5e+24], N[(N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision] * -1.0 + N[(N[(-1.0 * t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.5e+100], N[(N[(-1.0 * N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F * N[Power[N[(N[Power[F, 1.0], $MachinePrecision] * N[Power[F, 1.0], $MachinePrecision] + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.00000000000000045e24

   1. Initial program 57.2%

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

   3. Taylor expanded in F around -inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lift-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -5.00000000000000045e24 < F < 5.5000000000000002e100

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   4. Applied rewrites 99.6%

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

   if 5.5000000000000002e100 < F

   1. Initial program 55.9%

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

   3. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left({\sin B}^{-1}, -1, \frac{-1 \cdot \left(\cos B \cdot x\right)}{\sin B}\right)\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{+100}:\\ \;\;\;\;-1 \cdot \left(x \cdot \frac{1}{\tan B}\right) + \frac{F \cdot {\left(\mathsf{fma}\left({F}^{1}, {F}^{1}, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -1 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left({\sin B}^{-1}, -1, \frac{-1 \cdot t\_0}{\sin B}\right)\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{+100}:\\ \;\;\;\;-1 \cdot \left(x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-1 \cdot \frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -1e+62)
     (fma (pow (sin B) -1.0) -1.0 (/ (* -1.0 t_0) (sin B)))
     (if (<= F 5.2e+100)
       (+
        (* -1.0 (* x (/ 1.0 (tan B))))
        (*
         (/ F (sin B))
         (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (* -1.0 (/ 1.0 2.0)))))
       (/ (- 1.0 t_0) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -1e+62) {
		tmp = fma(pow(sin(B), -1.0), -1.0, ((-1.0 * t_0) / sin(B)));
	} else if (F <= 5.2e+100) {
		tmp = (-1.0 * (x * (1.0 / tan(B)))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), (-1.0 * (1.0 / 2.0))));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -1e+62)
		tmp = fma((sin(B) ^ -1.0), -1.0, Float64(Float64(-1.0 * t_0) / sin(B)));
	elseif (F <= 5.2e+100)
		tmp = Float64(Float64(-1.0 * Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-1.0 * Float64(1.0 / 2.0)))));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -1e+62], N[(N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision] * -1.0 + N[(N[(-1.0 * t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.2e+100], N[(N[(-1.0 * N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $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 * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.00000000000000004e62

   1. Initial program 47.3%

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

   3. Taylor expanded in F around -inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lift-sin.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -1.00000000000000004e62 < F < 5.2000000000000003e100

   1. Initial program 99.5%

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

   if 5.2000000000000003e100 < F

   1. Initial program 55.9%

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

   3. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left({\sin B}^{-1}, -1, \frac{-1 \cdot \left(\cos B \cdot x\right)}{\sin B}\right)\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{+100}:\\ \;\;\;\;-1 \cdot \left(x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-1 \cdot \frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.7% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -2.6 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left({\sin B}^{-1}, -1, \frac{-1 \cdot t\_0}{\sin B}\right)\\ \mathbf{elif}\;F \leq 200000000:\\ \;\;\;\;\left(-1 \cdot x\right) \cdot \frac{\cos B}{\sin B} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-1 \cdot \frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -2.6e+19)
     (fma (pow (sin B) -1.0) -1.0 (/ (* -1.0 t_0) (sin B)))
     (if (<= F 200000000.0)
       (+
        (* (* -1.0 x) (/ (cos B) (sin B)))
        (*
         (/ F (sin B))
         (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (* -1.0 (/ 1.0 2.0)))))
       (/ (- 1.0 t_0) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -2.6e+19) {
		tmp = fma(pow(sin(B), -1.0), -1.0, ((-1.0 * t_0) / sin(B)));
	} else if (F <= 200000000.0) {
		tmp = ((-1.0 * x) * (cos(B) / sin(B))) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), (-1.0 * (1.0 / 2.0))));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -2.6e+19)
		tmp = fma((sin(B) ^ -1.0), -1.0, Float64(Float64(-1.0 * t_0) / sin(B)));
	elseif (F <= 200000000.0)
		tmp = Float64(Float64(Float64(-1.0 * x) * Float64(cos(B) / sin(B))) + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-1.0 * Float64(1.0 / 2.0)))));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -2.6e+19], N[(N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision] * -1.0 + N[(N[(-1.0 * t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 200000000.0], N[(N[(N[(-1.0 * x), $MachinePrecision] * N[(N[Cos[B], $MachinePrecision] / N[Sin[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 * N[(1.0 / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.6e19

   1. Initial program 57.2%

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

   3. Taylor expanded in F around -inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lift-sin.f64 99.8

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

   5. Applied rewrites 99.8%

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

   if -2.6e19 < F < 2e8

   1. Initial program 99.5%

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

   3. Taylor expanded in B around inf

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

      1. lower-/.f64 N/A

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

      2. lower-cos.f64 N/A

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

      3. lift-sin.f64 99.5

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

   5. Applied rewrites 99.5%

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

   if 2e8 < F

   1. Initial program 70.0%

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

   3. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 99.6

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

   5. Applied rewrites 99.6%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.6 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left({\sin B}^{-1}, -1, \frac{-1 \cdot \left(\cos B \cdot x\right)}{\sin B}\right)\\ \mathbf{elif}\;F \leq 200000000:\\ \;\;\;\;\left(-1 \cdot x\right) \cdot \frac{\cos B}{\sin B} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-1 \cdot \frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.1% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq 8 \cdot 10^{-204}:\\ \;\;\;\;\mathsf{fma}\left({\sin B}^{-1}, -1, \frac{-1 \cdot t\_0}{\sin B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F 8e-204)
     (fma (pow (sin B) -1.0) -1.0 (/ (* -1.0 t_0) (sin B)))
     (/ (- 1.0 t_0) (sin B)))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= 8e-204) {
		tmp = fma(pow(sin(B), -1.0), -1.0, ((-1.0 * t_0) / sin(B)));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= 8e-204)
		tmp = fma((sin(B) ^ -1.0), -1.0, Float64(Float64(-1.0 * t_0) / sin(B)));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, 8e-204], N[(N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision] * -1.0 + N[(N[(-1.0 * t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if F < 8.00000000000000001e-204

   1. Initial program 78.4%

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

   3. Taylor expanded in F around -inf

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

      1. distribute-rgt-in N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lift-sin.f64 71.5

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

   5. Applied rewrites 71.5%

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

   if 8.00000000000000001e-204 < F

   1. Initial program 81.7%

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

   3. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 78.4

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

   5. Applied rewrites 78.4%

      \[\leadsto \color{blue}{\frac{1 - \cos B \cdot x}{\sin B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 55.8% accurate, N/A× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \cos B \cdot x}{\sin B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ (- 1.0 (* (cos B) x)) (sin B)))

C

double code(double F, double B, double x) {
	return (1.0 - (cos(B) * x)) / sin(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 - (cos(b) * x)) / sin(b)
end function

Java

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

Python

def code(F, B, x):
	return (1.0 - (math.cos(B) * x)) / math.sin(B)

Julia

function code(F, B, x)
	return Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B))
end

MATLAB

function tmp = code(F, B, x)
	tmp = (1.0 - (cos(B) * x)) / sin(B);
end

Wolfram

code[F_, B_, x_] := N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.8%

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

3. Taylor expanded in F around inf

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

   1. sub-div N/A

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

   2. lower-/.f64 N/A

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

   3. lower--.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-cos.f64 N/A

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

   7. lift-sin.f64 60.9

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

5. Applied rewrites 60.9%

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

Reproduce

herbie shell --seed 2025066 
(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))))))