VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.1% → 99.4%

Time: 9.3s

Alternatives: 17

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.1% 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: 99.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.1 \cdot 10^{+27}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \cos B \cdot x, 1\right)}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.1e+27)
   (/ (- (fma (cos B) x 1.0)) (sin B))
   (if (<= F 5.8e-6)
     (fma F (/ (sqrt (/ 1.0 (fma F F 2.0))) (sin B)) (/ (- x) (tan B)))
     (/ (fma -1.0 (* (cos B) x) 1.0) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.1e+27) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else if (F <= 5.8e-6) {
		tmp = fma(F, (sqrt((1.0 / fma(F, F, 2.0))) / sin(B)), (-x / tan(B)));
	} else {
		tmp = fma(-1.0, (cos(B) * x), 1.0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.1e+27)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	elseif (F <= 5.8e-6)
		tmp = fma(F, Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) / sin(B)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(fma(-1.0, Float64(cos(B) * x), 1.0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.1e+27], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.8e-6], N[(F * N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.0999999999999999e27

   1. Initial program 54.1%

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

   3. Taylor expanded in F around -inf

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

   5. Applied rewrites 99.7%

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

   if -1.0999999999999999e27 < F < 5.8000000000000004e-6

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.7%

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

   if 5.8000000000000004e-6 < F

   1. Initial program 60.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 82.7%

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

   5. Taylor expanded in F around inf

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.1 \cdot 10^{+27}:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \cos B \cdot x, 1\right)}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\sqrt{0.5}}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \cos B \cdot x, 1\right)}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.4)
   (/ (- (fma (cos B) x 1.0)) (sin B))
   (if (<= F 5.8e-6)
     (fma F (/ (sqrt 0.5) (sin B)) (/ (- x) (tan B)))
     (/ (fma -1.0 (* (cos B) x) 1.0) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else if (F <= 5.8e-6) {
		tmp = fma(F, (sqrt(0.5) / sin(B)), (-x / tan(B)));
	} else {
		tmp = fma(-1.0, (cos(B) * x), 1.0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	elseif (F <= 5.8e-6)
		tmp = fma(F, Float64(sqrt(0.5) / sin(B)), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(fma(-1.0, Float64(cos(B) * x), 1.0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.4], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.8e-6], N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 56.1%

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

   3. Taylor expanded in F around -inf

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

   5. Applied rewrites 99.7%

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

   if -1.3999999999999999 < F < 5.8000000000000004e-6

   1. Initial program 99.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.7%

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

   8. Taylor expanded in F around 0

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

   10. Applied rewrites 99.7%

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

   if 5.8000000000000004e-6 < F

   1. Initial program 60.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 82.7%

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

   5. Taylor expanded in F around inf

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\sqrt{0.5}}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \cos B \cdot x, 1\right)}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6300:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \cos B \cdot x, 1\right)}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6300.0)
   (/ (- (fma (cos B) x 1.0)) (sin B))
   (if (<= F 5.8e-6)
     (+ (/ (- x) (tan B)) (* (/ F B) (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0))))))
     (/ (fma -1.0 (* (cos B) x) 1.0) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6300.0) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else if (F <= 5.8e-6) {
		tmp = (-x / tan(B)) + ((F / B) * sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0)))));
	} else {
		tmp = fma(-1.0, (cos(B) * x), 1.0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6300.0)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	elseif (F <= 5.8e-6)
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(Float64(F / B) * sqrt(Float64(1.0 / fma(2.0, x, fma(F, F, 2.0))))));
	else
		tmp = Float64(fma(-1.0, Float64(cos(B) * x), 1.0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6300.0], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.8e-6], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 * N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6300

   1. Initial program 56.1%

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

   3. Taylor expanded in F around -inf

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

   5. Applied rewrites 99.7%

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

   if -6300 < F < 5.8000000000000004e-6

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 82.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. distribute-neg-frac N/A

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

      6. *-rgt-identity N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 82.3

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

   7. Applied rewrites 82.3%

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

   if 5.8000000000000004e-6 < F

   1. Initial program 60.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 82.7%

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

   5. Taylor expanded in F around inf

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

   7. Applied rewrites 99.9%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6300:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, \cos B \cdot x, 1\right)}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6300:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{+152}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6300.0)
   (/ (- (fma (cos B) x 1.0)) (sin B))
   (if (<= F 1.55e+152)
     (+ (* x (/ -1.0 (tan B))) (/ F (* B (sqrt (fma x 2.0 (fma F F 2.0))))))
     (+ (- (/ x B)) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6300.0) {
		tmp = -fma(cos(B), x, 1.0) / sin(B);
	} else if (F <= 1.55e+152) {
		tmp = (x * (-1.0 / tan(B))) + (F / (B * sqrt(fma(x, 2.0, fma(F, F, 2.0)))));
	} else {
		tmp = -(x / B) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6300.0)
		tmp = Float64(Float64(-fma(cos(B), x, 1.0)) / sin(B));
	elseif (F <= 1.55e+152)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(F / Float64(B * sqrt(fma(x, 2.0, fma(F, F, 2.0))))));
	else
		tmp = Float64(Float64(-Float64(x / B)) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6300.0], N[((-N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]) / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.55e+152], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(B * N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[(x / B), $MachinePrecision]) + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6300

   1. Initial program 56.1%

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

   3. Taylor expanded in F around -inf

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

   5. Applied rewrites 99.7%

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

   if -6300 < F < 1.55e152

   1. Initial program 97.1%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 81.0%

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

   7. Applied rewrites 83.4%

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

   if 1.55e152 < F

   1. Initial program 20.1%

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

   3. Taylor expanded in F around inf

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 79.1%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6300:\\ \;\;\;\;\frac{-\mathsf{fma}\left(\cos B, x, 1\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{+152}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.9 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{B \cdot F}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq -9 \cdot 10^{+88}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{+152}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.9e+126)
   (fma F (/ -1.0 (* B F)) (/ (- x) (tan B)))
   (if (<= F -9e+88)
     (* (sqrt (/ 1.0 (fma F F 2.0))) (/ F (sin B)))
     (if (<= F 1.55e+152)
       (+ (* x (/ -1.0 (tan B))) (/ F (* B (sqrt (fma x 2.0 (fma F F 2.0))))))
       (+ (- (/ x B)) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.9e+126) {
		tmp = fma(F, (-1.0 / (B * F)), (-x / tan(B)));
	} else if (F <= -9e+88) {
		tmp = sqrt((1.0 / fma(F, F, 2.0))) * (F / sin(B));
	} else if (F <= 1.55e+152) {
		tmp = (x * (-1.0 / tan(B))) + (F / (B * sqrt(fma(x, 2.0, fma(F, F, 2.0)))));
	} else {
		tmp = -(x / B) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.9e+126)
		tmp = fma(F, Float64(-1.0 / Float64(B * F)), Float64(Float64(-x) / tan(B)));
	elseif (F <= -9e+88)
		tmp = Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) * Float64(F / sin(B)));
	elseif (F <= 1.55e+152)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(F / Float64(B * sqrt(fma(x, 2.0, fma(F, F, 2.0))))));
	else
		tmp = Float64(Float64(-Float64(x / B)) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.9e+126], N[(F * N[(-1.0 / N[(B * F), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -9e+88], N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.55e+152], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(B * N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[(x / B), $MachinePrecision]) + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.90000000000000008e126

   1. Initial program 41.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 58.8%

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

   5. Taylor expanded in F around -inf

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

   7. Applied rewrites 99.6%

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

   8. Taylor expanded in B around 0

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

   10. Applied rewrites 79.3%

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

   if -1.90000000000000008e126 < F < -9e88

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 82.2%

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

   if -9e88 < F < 1.55e152

   1. Initial program 96.3%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 80.1%

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

   7. Applied rewrites 83.3%

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

   if 1.55e152 < F

   1. Initial program 20.1%

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

   3. Taylor expanded in F around inf

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 79.1%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.9 \cdot 10^{+126}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{B \cdot F}, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq -9 \cdot 10^{+88}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;F \leq 1.55 \cdot 10^{+152}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 56.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(B \cdot B\right) \cdot F\\ t_1 := \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{if}\;B \leq 1.24 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_1, \mathsf{fma}\left(t\_0, 0.16666666666666666, F\right), \mathsf{fma}\left(\mathsf{fma}\left(\left(-F\right) \cdot -0.019444444444444445, t\_1, x \cdot 0.022222222222222223\right), B \cdot B, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{\mathsf{fma}\left(t\_0, -0.16666666666666666, F\right) \cdot B}, \frac{-x}{\tan B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (* B B) F)) (t_1 (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0))))))
   (if (<= B 1.24e+14)
     (/
      (-
       (fma
        t_1
        (fma t_0 0.16666666666666666 F)
        (*
         (fma
          (fma (* (- F) -0.019444444444444445) t_1 (* x 0.022222222222222223))
          (* B B)
          (* 0.3333333333333333 x))
         (* B B)))
       x)
      B)
     (fma
      F
      (/ -1.0 (* (fma t_0 -0.16666666666666666 F) B))
      (/ (- x) (tan B))))))

C

double code(double F, double B, double x) {
	double t_0 = (B * B) * F;
	double t_1 = sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0))));
	double tmp;
	if (B <= 1.24e+14) {
		tmp = (fma(t_1, fma(t_0, 0.16666666666666666, F), (fma(fma((-F * -0.019444444444444445), t_1, (x * 0.022222222222222223)), (B * B), (0.3333333333333333 * x)) * (B * B))) - x) / B;
	} else {
		tmp = fma(F, (-1.0 / (fma(t_0, -0.16666666666666666, F) * B)), (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(B * B) * F)
	t_1 = sqrt(Float64(1.0 / fma(2.0, x, fma(F, F, 2.0))))
	tmp = 0.0
	if (B <= 1.24e+14)
		tmp = Float64(Float64(fma(t_1, fma(t_0, 0.16666666666666666, F), Float64(fma(fma(Float64(Float64(-F) * -0.019444444444444445), t_1, Float64(x * 0.022222222222222223)), Float64(B * B), Float64(0.3333333333333333 * x)) * Float64(B * B))) - x) / B);
	else
		tmp = fma(F, Float64(-1.0 / Float64(fma(t_0, -0.16666666666666666, F) * B)), Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(B * B), $MachinePrecision] * F), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B, 1.24e+14], N[(N[(N[(t$95$1 * N[(t$95$0 * 0.16666666666666666 + F), $MachinePrecision] + N[(N[(N[(N[((-F) * -0.019444444444444445), $MachinePrecision] * t$95$1 + N[(x * 0.022222222222222223), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + N[(0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(F * N[(-1.0 / N[(N[(t$95$0 * -0.16666666666666666 + F), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if B < 1.24e14

   1. Initial program 75.8%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 63.5%

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

   if 1.24e14 < B

   1. Initial program 85.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

   4. Applied rewrites 85.9%

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

   5. Taylor expanded in F around -inf

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

   7. Applied rewrites 54.8%

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

   8. Taylor expanded in B around 0

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

   10. Applied rewrites 52.4%

       \[\leadsto \mathsf{fma}\left(F, \frac{-1}{\mathsf{fma}\left(\left(B \cdot B\right) \cdot F, -0.16666666666666666, F\right) \cdot \color{blue}{B}}, \frac{-x}{\tan B}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.24 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}, \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right), \mathsf{fma}\left(\mathsf{fma}\left(\left(-F\right) \cdot -0.019444444444444445, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}, x \cdot 0.022222222222222223\right), B \cdot B, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{-1}{\mathsf{fma}\left(\left(B \cdot B\right) \cdot F, -0.16666666666666666, F\right) \cdot B}, \frac{-x}{\tan B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{if}\;B \leq 1.24 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right), \mathsf{fma}\left(\mathsf{fma}\left(\left(-F\right) \cdot -0.019444444444444445, t\_0, x \cdot 0.022222222222222223\right), B \cdot B, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{1}{F}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0))))))
   (if (<= B 1.24e+14)
     (/
      (-
       (fma
        t_0
        (fma (* (* B B) F) 0.16666666666666666 F)
        (*
         (fma
          (fma (* (- F) -0.019444444444444445) t_0 (* x 0.022222222222222223))
          (* B B)
          (* 0.3333333333333333 x))
         (* B B)))
       x)
      B)
     (+ (* x (/ -1.0 (tan B))) (* (/ F B) (/ 1.0 F))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0))));
	double tmp;
	if (B <= 1.24e+14) {
		tmp = (fma(t_0, fma(((B * B) * F), 0.16666666666666666, F), (fma(fma((-F * -0.019444444444444445), t_0, (x * 0.022222222222222223)), (B * B), (0.3333333333333333 * x)) * (B * B))) - x) / B;
	} else {
		tmp = (x * (-1.0 / tan(B))) + ((F / B) * (1.0 / F));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = sqrt(Float64(1.0 / fma(2.0, x, fma(F, F, 2.0))))
	tmp = 0.0
	if (B <= 1.24e+14)
		tmp = Float64(Float64(fma(t_0, fma(Float64(Float64(B * B) * F), 0.16666666666666666, F), Float64(fma(fma(Float64(Float64(-F) * -0.019444444444444445), t_0, Float64(x * 0.022222222222222223)), Float64(B * B), Float64(0.3333333333333333 * x)) * Float64(B * B))) - x) / B);
	else
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / B) * Float64(1.0 / F)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B, 1.24e+14], N[(N[(N[(t$95$0 * N[(N[(N[(B * B), $MachinePrecision] * F), $MachinePrecision] * 0.16666666666666666 + F), $MachinePrecision] + N[(N[(N[(N[((-F) * -0.019444444444444445), $MachinePrecision] * t$95$0 + N[(x * 0.022222222222222223), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + N[(0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 1.24e14

   1. Initial program 75.8%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 63.5%

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

   if 1.24e14 < B

   1. Initial program 85.6%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 52.4%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 47.4%

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

4. Final simplification 60.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.24 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}, \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right), \mathsf{fma}\left(\mathsf{fma}\left(\left(-F\right) \cdot -0.019444444444444445, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}, x \cdot 0.022222222222222223\right), B \cdot B, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B} \cdot \frac{1}{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}\\ \mathbf{if}\;B \leq 1.24 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right), \mathsf{fma}\left(\mathsf{fma}\left(\left(-F\right) \cdot -0.019444444444444445, t\_0, x \cdot 0.022222222222222223\right), B \cdot B, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0))))))
   (if (<= B 1.24e+14)
     (/
      (-
       (fma
        t_0
        (fma (* (* B B) F) 0.16666666666666666 F)
        (*
         (fma
          (fma (* (- F) -0.019444444444444445) t_0 (* x 0.022222222222222223))
          (* B B)
          (* 0.3333333333333333 x))
         (* B B)))
       x)
      B)
     (+ (/ (- x) (tan B)) (/ 1.0 B)))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0))));
	double tmp;
	if (B <= 1.24e+14) {
		tmp = (fma(t_0, fma(((B * B) * F), 0.16666666666666666, F), (fma(fma((-F * -0.019444444444444445), t_0, (x * 0.022222222222222223)), (B * B), (0.3333333333333333 * x)) * (B * B))) - x) / B;
	} else {
		tmp = (-x / tan(B)) + (1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = sqrt(Float64(1.0 / fma(2.0, x, fma(F, F, 2.0))))
	tmp = 0.0
	if (B <= 1.24e+14)
		tmp = Float64(Float64(fma(t_0, fma(Float64(Float64(B * B) * F), 0.16666666666666666, F), Float64(fma(fma(Float64(Float64(-F) * -0.019444444444444445), t_0, Float64(x * 0.022222222222222223)), Float64(B * B), Float64(0.3333333333333333 * x)) * Float64(B * B))) - x) / B);
	else
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[B, 1.24e+14], N[(N[(N[(t$95$0 * N[(N[(N[(B * B), $MachinePrecision] * F), $MachinePrecision] * 0.16666666666666666 + F), $MachinePrecision] + N[(N[(N[(N[((-F) * -0.019444444444444445), $MachinePrecision] * t$95$0 + N[(x * 0.022222222222222223), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision] + N[(0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 1.24e14

   1. Initial program 75.8%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 63.5%

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

   if 1.24e14 < B

   1. Initial program 85.6%

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

   3. Taylor expanded in F around inf

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

   5. Applied rewrites 44.3%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 42.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. distribute-neg-frac N/A

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

      6. *-rgt-identity N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 42.6

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

   10. Applied rewrites 42.6%

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

4. Final simplification 59.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.24 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}, \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right), \mathsf{fma}\left(\mathsf{fma}\left(\left(-F\right) \cdot -0.019444444444444445, \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}, x \cdot 0.022222222222222223\right), B \cdot B, 0.3333333333333333 \cdot x\right) \cdot \left(B \cdot B\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 58.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{+90}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 58:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2e+90)
   (/ (- -1.0 x) B)
   (if (<= F 58.0)
     (/ (- (* (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0)))) F) x) B)
     (+ (- (/ x B)) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2e+90) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 58.0) {
		tmp = ((sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B;
	} else {
		tmp = -(x / B) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2e+90)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 58.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B);
	else
		tmp = Float64(Float64(-Float64(x / B)) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2e+90], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 58.0], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[((-N[(x / B), $MachinePrecision]) + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.99999999999999993e90

   1. Initial program 49.8%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 29.9%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 47.6%

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

   if -1.99999999999999993e90 < F < 58

   1. Initial program 98.2%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 55.8%

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

   if 58 < F

   1. Initial program 60.0%

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

   3. Taylor expanded in F around inf

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 81.0%

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

Alternative 10: 55.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 1.24 \cdot 10^{+14}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}, \mathsf{fma}\left(\left(B \cdot B\right) \cdot F, 0.16666666666666666, F\right), 0.3333333333333333 \cdot \left(\left(B \cdot B\right) \cdot x\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 1.24e+14)
   (/
    (-
     (fma
      (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0))))
      (fma (* (* B B) F) 0.16666666666666666 F)
      (* 0.3333333333333333 (* (* B B) x)))
     x)
    B)
   (+ (/ (- x) (tan B)) (/ 1.0 B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 1.24e+14) {
		tmp = (fma(sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0)))), fma(((B * B) * F), 0.16666666666666666, F), (0.3333333333333333 * ((B * B) * x))) - x) / B;
	} else {
		tmp = (-x / tan(B)) + (1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 1.24e+14)
		tmp = Float64(Float64(fma(sqrt(Float64(1.0 / fma(2.0, x, fma(F, F, 2.0)))), fma(Float64(Float64(B * B) * F), 0.16666666666666666, F), Float64(0.3333333333333333 * Float64(Float64(B * B) * x))) - x) / B);
	else
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 1.24e+14], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(B * B), $MachinePrecision] * F), $MachinePrecision] * 0.16666666666666666 + F), $MachinePrecision] + N[(0.3333333333333333 * N[(N[(B * B), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.24e14

   1. Initial program 75.8%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 63.5%

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

   if 1.24e14 < B

   1. Initial program 85.6%

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

   3. Taylor expanded in F around inf

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

   5. Applied rewrites 44.3%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 42.4%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*r/ N/A

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

      5. distribute-neg-frac N/A

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

      6. *-rgt-identity N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-/.f64 42.6

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

   10. Applied rewrites 42.6%

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

Alternative 11: 51.1% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+105}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\mathsf{fma}\left(\left(B \cdot B\right) \cdot x, -0.3333333333333333, x\right)}{B}\right) + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5e+105)
   (/ (- -1.0 x) B)
   (if (<= F 6.8e+78)
     (/ (- (* (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0)))) F) x) B)
     (+ (- (/ (fma (* (* B B) x) -0.3333333333333333 x) B)) (/ 1.0 B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5e+105) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 6.8e+78) {
		tmp = ((sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B;
	} else {
		tmp = -(fma(((B * B) * x), -0.3333333333333333, x) / B) + (1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5e+105)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 6.8e+78)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B);
	else
		tmp = Float64(Float64(-Float64(fma(Float64(Float64(B * B) * x), -0.3333333333333333, x) / B)) + Float64(1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5e+105], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 6.8e+78], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[((-N[(N[(N[(N[(B * B), $MachinePrecision] * x), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision] / B), $MachinePrecision]) + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.00000000000000046e105

   1. Initial program 47.0%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 29.6%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 48.4%

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

   if -5.00000000000000046e105 < F < 6.80000000000000014e78

   1. Initial program 98.3%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 57.1%

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

   if 6.80000000000000014e78 < F

   1. Initial program 48.5%

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

   3. Taylor expanded in F around inf

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 83.5%

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

   9. Taylor expanded in B around 0

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

   11. Applied rewrites 63.9%

       \[\leadsto \left(-\color{blue}{\frac{\mathsf{fma}\left(\left(B \cdot B\right) \cdot x, -0.3333333333333333, x\right)}{B}}\right) + \frac{1}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 50.8% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{-0.5}{F}, \frac{\mathsf{fma}\left(2, x, 2\right)}{F}, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.4)
   (/ (- -1.0 x) B)
   (if (<= F 5.8e-6)
     (/ (- (* (sqrt (/ 1.0 (fma 2.0 x 2.0))) F) x) B)
     (/ (- (fma (/ -0.5 F) (/ (fma 2.0 x 2.0) F) 1.0) x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 5.8e-6) {
		tmp = ((sqrt((1.0 / fma(2.0, x, 2.0))) * F) - x) / B;
	} else {
		tmp = (fma((-0.5 / F), (fma(2.0, x, 2.0) / F), 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 5.8e-6)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, 2.0))) * F) - x) / B);
	else
		tmp = Float64(Float64(fma(Float64(-0.5 / F), Float64(fma(2.0, x, 2.0) / F), 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.4], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 5.8e-6], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(-0.5 / F), $MachinePrecision] * N[(N[(2.0 * x + 2.0), $MachinePrecision] / F), $MachinePrecision] + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 56.1%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 36.7%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 51.3%

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

   if -1.3999999999999999 < F < 5.8000000000000004e-6

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 55.0%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 55.0%

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

   if 5.8000000000000004e-6 < F

   1. Initial program 60.6%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 65.0%

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

Alternative 13: 51.0% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.65 \cdot 10^{+161}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\mathsf{fma}\left(\left(B \cdot B\right) \cdot x, -0.3333333333333333, x\right)}{B}\right) + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.65e+161)
   (/ (- -1.0 x) B)
   (if (<= F 7.8e+79)
     (/ (- (/ F (sqrt (fma x 2.0 (fma F F 2.0)))) x) B)
     (+ (- (/ (fma (* (* B B) x) -0.3333333333333333 x) B)) (/ 1.0 B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.65e+161) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 7.8e+79) {
		tmp = ((F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B;
	} else {
		tmp = -(fma(((B * B) * x), -0.3333333333333333, x) / B) + (1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.65e+161)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 7.8e+79)
		tmp = Float64(Float64(Float64(F / sqrt(fma(x, 2.0, fma(F, F, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(-Float64(fma(Float64(Float64(B * B) * x), -0.3333333333333333, x) / B)) + Float64(1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.65e+161], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 7.8e+79], N[(N[(N[(F / N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[((-N[(N[(N[(N[(B * B), $MachinePrecision] * x), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision] / B), $MachinePrecision]) + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.64999999999999999e161

   1. Initial program 33.6%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 27.8%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 52.2%

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

   if -1.64999999999999999e161 < F < 7.7999999999999994e79

   1. Initial program 97.8%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 55.5%

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

   7. Applied rewrites 55.5%

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

   if 7.7999999999999994e79 < F

   1. Initial program 48.5%

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

   3. Taylor expanded in F around inf

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 83.5%

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

   9. Taylor expanded in B around 0

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

   11. Applied rewrites 63.9%

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

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.65 \cdot 10^{+161}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{+79}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{\mathsf{fma}\left(\left(B \cdot B\right) \cdot x, -0.3333333333333333, x\right)}{B}\right) + \frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 50.9% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 5.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, 2\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.4)
   (/ (- -1.0 x) B)
   (if (<= F 5.8e-6)
     (/ (- (* (sqrt (/ 1.0 (fma 2.0 x 2.0))) F) x) B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 5.8e-6) {
		tmp = ((sqrt((1.0 / fma(2.0, x, 2.0))) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 5.8e-6)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, 2.0))) * F) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.4], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 5.8e-6], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 56.1%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 36.7%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 51.3%

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

   if -1.3999999999999999 < F < 5.8000000000000004e-6

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 55.0%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 55.0%

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

   if 5.8000000000000004e-6 < F

   1. Initial program 60.6%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 57.6%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 65.0%

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

Alternative 15: 43.4% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.45 \cdot 10^{-35}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{-68}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.45e-35)
   (/ (- -1.0 x) B)
   (if (<= F 1.45e-68) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.45e-35) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.45e-68) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.45d-35)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.45d-68) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.45e-35) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.45e-68) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.45e-35:
		tmp = (-1.0 - x) / B
	elif F <= 1.45e-68:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.45e-35)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.45e-68)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.45e-35)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.45e-68)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.45e-35], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.45e-68], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4500000000000001e-35

   1. Initial program 60.2%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 41.1%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 48.5%

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

   if -1.4500000000000001e-35 < F < 1.45e-68

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 55.2%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 42.3%

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

   if 1.45e-68 < F

   1. Initial program 63.8%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 54.4%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 60.1%

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

Alternative 16: 36.2% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.45 \cdot 10^{-35}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.45e-35) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.45e-35) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.45d-35)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.45e-35) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.45e-35:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.45e-35)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.45e-35)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.45e-35], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.4500000000000001e-35

   1. Initial program 60.2%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 41.1%

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

   6. Taylor expanded in F around -inf

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

   8. Applied rewrites 48.5%

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

   if -1.4500000000000001e-35 < F

   1. Initial program 85.2%

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

   3. Taylor expanded in B around 0

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

   5. Applied rewrites 54.9%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 37.8%

      \[\leadsto \frac{-x}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 29.2% accurate, 26.3× speedup

Math

\[\begin{array}{l} \\ \frac{-x}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ (- x) B))

C

double code(double F, double B, double x) {
	return -x / B;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -x / b
end function

Java

public static double code(double F, double B, double x) {
	return -x / B;
}

Python

def code(F, B, x):
	return -x / B

Julia

function code(F, B, x)
	return Float64(Float64(-x) / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -x / B;
end

Wolfram

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

Derivation

1. Initial program 78.0%

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

3. Taylor expanded in B around 0

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

5. Applied rewrites 50.9%

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

6. Taylor expanded in F around 0

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

8. Applied rewrites 33.2%

   \[\leadsto \frac{-x}{B} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025018 
(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))))))