VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.8% → 99.6%

Time: 8.3s

Alternatives: 29

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.8% 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: 99.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{1}{\tan B}, \frac{-1}{\sin B}\right)\\ \mathbf{elif}\;F \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5e+25)
   (fma (- x) (/ 1.0 (tan B)) (/ -1.0 (sin B)))
   (if (<= F 2e+15)
     (+
      (- (/ (* x 1.0) (tan B)))
      (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
     (+ (/ (- x) (tan B)) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5e+25) {
		tmp = fma(-x, (1.0 / tan(B)), (-1.0 / sin(B)));
	} else if (F <= 2e+15) {
		tmp = -((x * 1.0) / tan(B)) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = (-x / tan(B)) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5e+25)
		tmp = fma(Float64(-x), Float64(1.0 / tan(B)), Float64(-1.0 / sin(B)));
	elseif (F <= 2e+15)
		tmp = Float64(Float64(-Float64(Float64(x * 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)))));
	else
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5e+25], N[((-x) * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2e+15], N[((-N[(N[(x * 1.0), $MachinePrecision] / N[Tan[B], $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], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.00000000000000024e25

   1. Initial program 55.6%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. lift-/.f64 99.7

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

   6. Applied rewrites 99.7%

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

   if -5.00000000000000024e25 < F < 2e15

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 99.5

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

   3. Applied rewrites 99.5%

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

   if 2e15 < F

   1. Initial program 56.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 57.0

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

   3. Applied rewrites 57.0%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 99.8

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

   6. Applied rewrites 99.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-tan.f64 99.8

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

   8. Applied rewrites 99.8%

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

Alternative 2: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -5.5 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(-x, t\_0, \frac{-1}{\sin B}\right)\\ \mathbf{elif}\;F \leq 2.85 \cdot 10^{-6}:\\ \;\;\;\;\left(-x \cdot t\_0\right) + \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (tan B))))
   (if (<= F -5.5e+25)
     (fma (- x) t_0 (/ -1.0 (sin B)))
     (if (<= F 2.85e-6)
       (+ (- (* x t_0)) (* (/ F (sin B)) (pow (fma F F 2.0) -0.5)))
       (+ (/ (- x) (tan B)) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / tan(B);
	double tmp;
	if (F <= -5.5e+25) {
		tmp = fma(-x, t_0, (-1.0 / sin(B)));
	} else if (F <= 2.85e-6) {
		tmp = -(x * t_0) + ((F / sin(B)) * pow(fma(F, F, 2.0), -0.5));
	} else {
		tmp = (-x / tan(B)) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / tan(B))
	tmp = 0.0
	if (F <= -5.5e+25)
		tmp = fma(Float64(-x), t_0, Float64(-1.0 / sin(B)));
	elseif (F <= 2.85e-6)
		tmp = Float64(Float64(-Float64(x * t_0)) + Float64(Float64(F / sin(B)) * (fma(F, F, 2.0) ^ -0.5)));
	else
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5.5e+25], N[((-x) * t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.85e-6], N[((-N[(x * t$95$0), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(F * F + 2.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.50000000000000018e25

   1. Initial program 55.6%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. lift-/.f64 99.7

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

   6. Applied rewrites 99.7%

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

   if -5.50000000000000018e25 < F < 2.8499999999999998e-6

   1. Initial program 99.4%

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

   2. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval 99.4

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

   4. Applied rewrites 99.4%

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

   if 2.8499999999999998e-6 < F

   1. Initial program 58.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 58.9

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

   3. Applied rewrites 58.9%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 98.3

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

   6. Applied rewrites 98.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-tan.f64 98.3

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

   8. Applied rewrites 98.3%

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

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -5.5 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(-x, t\_0, \frac{-1}{\sin B}\right)\\ \mathbf{elif}\;F \leq 2.85 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-x, t\_0, {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (tan B))))
   (if (<= F -5.5e+25)
     (fma (- x) t_0 (/ -1.0 (sin B)))
     (if (<= F 2.85e-6)
       (fma (- x) t_0 (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F (sin B))))
       (+ (/ (- x) (tan B)) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / tan(B);
	double tmp;
	if (F <= -5.5e+25) {
		tmp = fma(-x, t_0, (-1.0 / sin(B)));
	} else if (F <= 2.85e-6) {
		tmp = fma(-x, t_0, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / sin(B))));
	} else {
		tmp = (-x / tan(B)) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / tan(B))
	tmp = 0.0
	if (F <= -5.5e+25)
		tmp = fma(Float64(-x), t_0, Float64(-1.0 / sin(B)));
	elseif (F <= 2.85e-6)
		tmp = fma(Float64(-x), t_0, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / sin(B))));
	else
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5.5e+25], N[((-x) * t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.85e-6], N[((-x) * t$95$0 + N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.50000000000000018e25

   1. Initial program 55.6%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. lift-/.f64 99.7

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

   6. Applied rewrites 99.7%

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

   if -5.50000000000000018e25 < F < 2.8499999999999998e-6

   1. Initial program 99.4%

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

   3. Applied rewrites 99.4%

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

   if 2.8499999999999998e-6 < F

   1. Initial program 58.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 58.9

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

   3. Applied rewrites 58.9%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 98.3

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

   6. Applied rewrites 98.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-tan.f64 98.3

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

   8. Applied rewrites 98.3%

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

Alternative 4: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.45:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{1}{\tan B}, \frac{-1}{\sin B}\right)\\ \mathbf{elif}\;F \leq 2.85 \cdot 10^{-6}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5} \cdot F - \cos B \cdot x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.45)
   (fma (- x) (/ 1.0 (tan B)) (/ -1.0 (sin B)))
   (if (<= F 2.85e-6)
     (/ (- (* (pow (fma 2.0 x 2.0) -0.5) F) (* (cos B) x)) (sin B))
     (+ (/ (- x) (tan B)) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.45) {
		tmp = fma(-x, (1.0 / tan(B)), (-1.0 / sin(B)));
	} else if (F <= 2.85e-6) {
		tmp = ((pow(fma(2.0, x, 2.0), -0.5) * F) - (cos(B) * x)) / sin(B);
	} else {
		tmp = (-x / tan(B)) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.45)
		tmp = fma(Float64(-x), Float64(1.0 / tan(B)), Float64(-1.0 / sin(B)));
	elseif (F <= 2.85e-6)
		tmp = Float64(Float64(Float64((fma(2.0, x, 2.0) ^ -0.5) * F) - Float64(cos(B) * x)) / sin(B));
	else
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.45], N[((-x) * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.85e-6], N[(N[(N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.44999999999999996

   1. Initial program 58.8%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. lift-/.f64 99.3

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

   6. Applied rewrites 99.3%

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

   if -1.44999999999999996 < F < 2.8499999999999998e-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. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. sub-div N/A

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

      4. lower-/.f64 N/A

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

   4. Applied rewrites 99.3%

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

   if 2.8499999999999998e-6 < F

   1. Initial program 58.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 58.9

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

   3. Applied rewrites 58.9%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 98.3

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

   6. Applied rewrites 98.3%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-tan.f64 98.3

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

   8. Applied rewrites 98.3%

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

Alternative 5: 91.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -0.000216:\\ \;\;\;\;t\_1 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{-130}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-140}:\\ \;\;\;\;-\cos B \cdot \frac{x}{\sin B}\\ \mathbf{elif}\;F \leq 6.6 \cdot 10^{-7}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (+
          (- (/ x B))
          (*
           (/ F (sin B))
           (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0))))))
        (t_1 (/ (- x) (tan B))))
   (if (<= F -0.000216)
     (+ t_1 (/ -1.0 (sin B)))
     (if (<= F -4.5e-130)
       t_0
       (if (<= F 4.2e-140)
         (- (* (cos B) (/ x (sin B))))
         (if (<= F 6.6e-7) t_0 (+ t_1 (/ 1.0 (sin B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	double t_1 = -x / tan(B);
	double tmp;
	if (F <= -0.000216) {
		tmp = t_1 + (-1.0 / sin(B));
	} else if (F <= -4.5e-130) {
		tmp = t_0;
	} else if (F <= 4.2e-140) {
		tmp = -(cos(B) * (x / sin(B)));
	} else if (F <= 6.6e-7) {
		tmp = t_0;
	} else {
		tmp = t_1 + (1.0 / sin(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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = -(x / b) + ((f / sin(b)) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
    t_1 = -x / tan(b)
    if (f <= (-0.000216d0)) then
        tmp = t_1 + ((-1.0d0) / sin(b))
    else if (f <= (-4.5d-130)) then
        tmp = t_0
    else if (f <= 4.2d-140) then
        tmp = -(cos(b) * (x / sin(b)))
    else if (f <= 6.6d-7) then
        tmp = t_0
    else
        tmp = t_1 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -(x / B) + ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	double t_1 = -x / Math.tan(B);
	double tmp;
	if (F <= -0.000216) {
		tmp = t_1 + (-1.0 / Math.sin(B));
	} else if (F <= -4.5e-130) {
		tmp = t_0;
	} else if (F <= 4.2e-140) {
		tmp = -(Math.cos(B) * (x / Math.sin(B)));
	} else if (F <= 6.6e-7) {
		tmp = t_0;
	} else {
		tmp = t_1 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -(x / B) + ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))
	t_1 = -x / math.tan(B)
	tmp = 0
	if F <= -0.000216:
		tmp = t_1 + (-1.0 / math.sin(B))
	elif F <= -4.5e-130:
		tmp = t_0
	elif F <= 4.2e-140:
		tmp = -(math.cos(B) * (x / math.sin(B)))
	elif F <= 6.6e-7:
		tmp = t_0
	else:
		tmp = t_1 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-Float64(x / B)) + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))))
	t_1 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -0.000216)
		tmp = Float64(t_1 + Float64(-1.0 / sin(B)));
	elseif (F <= -4.5e-130)
		tmp = t_0;
	elseif (F <= 4.2e-140)
		tmp = Float64(-Float64(cos(B) * Float64(x / sin(B))));
	elseif (F <= 6.6e-7)
		tmp = t_0;
	else
		tmp = Float64(t_1 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -(x / B) + ((F / sin(B)) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
	t_1 = -x / tan(B);
	tmp = 0.0;
	if (F <= -0.000216)
		tmp = t_1 + (-1.0 / sin(B));
	elseif (F <= -4.5e-130)
		tmp = t_0;
	elseif (F <= 4.2e-140)
		tmp = -(cos(B) * (x / sin(B)));
	elseif (F <= 6.6e-7)
		tmp = t_0;
	else
		tmp = t_1 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-N[(x / B), $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]}, Block[{t$95$1 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.000216], N[(t$95$1 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -4.5e-130], t$95$0, If[LessEqual[F, 4.2e-140], (-N[(N[Cos[B], $MachinePrecision] * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 6.6e-7], t$95$0, N[(t$95$1 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.1599999999999999e-4

   1. Initial program 59.1%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 98.8

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

   4. Applied rewrites 98.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-/l* N/A

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

      6. distribute-neg-frac N/A

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

      7. *-rgt-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-tan.f64 98.9

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

   6. Applied rewrites 98.9%

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

   if -2.1599999999999999e-4 < F < -4.5e-130 or 4.20000000000000035e-140 < F < 6.6000000000000003e-7

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 76.6

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

   4. Applied rewrites 76.6%

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

   if -4.5e-130 < F < 4.20000000000000035e-140

   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. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 82.2

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

   4. Applied rewrites 82.2%

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

      1. associate-/l* 82.2

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-sin.f64 82.2

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

   6. Applied rewrites 82.2%

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

   if 6.6000000000000003e-7 < F

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 99.5

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

   3. Applied rewrites 99.5%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 35.9

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

   6. Applied rewrites 35.9%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-tan.f64 35.9

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

   8. Applied rewrites 35.9%

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

Alternative 6: 91.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\mathsf{fma}\left(-x, t\_0, \frac{-1}{\sin B}\right)\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-38}:\\ \;\;\;\;\left(-x \cdot t\_0\right) + \frac{{\left(\mathsf{fma}\left(F, F, x + x\right) + 2\right)}^{-0.5} \cdot F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (tan B))))
   (if (<= F -1.4)
     (fma (- x) t_0 (/ -1.0 (sin B)))
     (if (<= F 1.15e-38)
       (+ (- (* x t_0)) (/ (* (pow (+ (fma F F (+ x x)) 2.0) -0.5) F) B))
       (+ (/ (- x) (tan B)) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = fma(-x, t_0, (-1.0 / sin(B)));
	} else if (F <= 1.15e-38) {
		tmp = -(x * t_0) + ((pow((fma(F, F, (x + x)) + 2.0), -0.5) * F) / B);
	} else {
		tmp = (-x / tan(B)) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / tan(B))
	tmp = 0.0
	if (F <= -1.4)
		tmp = fma(Float64(-x), t_0, Float64(-1.0 / sin(B)));
	elseif (F <= 1.15e-38)
		tmp = Float64(Float64(-Float64(x * t_0)) + Float64(Float64((Float64(fma(F, F, Float64(x + x)) + 2.0) ^ -0.5) * F) / B));
	else
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4], N[((-x) * t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.15e-38], N[((-N[(x * t$95$0), $MachinePrecision]) + N[(N[(N[Power[N[(N[(F * F + N[(x + x), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 58.8%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-+.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-tan.f64 N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-tan.f64 N/A

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

      10. lift-/.f64 99.3

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

   6. Applied rewrites 99.3%

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

   if -1.3999999999999999 < F < 1.15000000000000001e-38

   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. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 84.0%

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

   if 1.15000000000000001e-38 < F

   1. Initial program 61.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 62.0

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

   3. Applied rewrites 62.0%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 94.0

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

   6. Applied rewrites 94.0%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-tan.f64 94.0

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

   8. Applied rewrites 94.0%

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

Alternative 7: 90.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -0.000216:\\ \;\;\;\;t\_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-38}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{{\left(\mathsf{fma}\left(F, F, x + x\right) + 2\right)}^{-0.5} \cdot F}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -0.000216)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 1.15e-38)
       (+
        (- (* x (/ 1.0 (tan B))))
        (/ (* (pow (+ (fma F F (+ x x)) 2.0) -0.5) F) B))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -0.000216) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 1.15e-38) {
		tmp = -(x * (1.0 / tan(B))) + ((pow((fma(F, F, (x + x)) + 2.0), -0.5) * F) / B);
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -0.000216)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 1.15e-38)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64((Float64(fma(F, F, Float64(x + x)) + 2.0) ^ -0.5) * F) / B));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.000216], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.15e-38], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(N[Power[N[(N[(F * F + N[(x + x), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.1599999999999999e-4

   1. Initial program 59.1%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 98.8

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

   4. Applied rewrites 98.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-/l* N/A

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

      6. distribute-neg-frac N/A

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

      7. *-rgt-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-tan.f64 98.9

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

   6. Applied rewrites 98.9%

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

   if -2.1599999999999999e-4 < F < 1.15000000000000001e-38

   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. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 84.1%

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

   if 1.15000000000000001e-38 < F

   1. Initial program 61.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 62.0

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

   3. Applied rewrites 62.0%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 94.0

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

   6. Applied rewrites 94.0%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. distribute-neg-frac N/A

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

      6. *-rgt-identity N/A

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

      7. lower-/.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lift-tan.f64 94.0

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

   8. Applied rewrites 94.0%

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

Alternative 8: 83.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{if}\;F \leq -0.000216:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{-130}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-140}:\\ \;\;\;\;-\cos B \cdot \frac{x}{\sin B}\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{+139}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{1}{\left(1 + \left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666\right) \cdot \left(B \cdot B\right)\right) \cdot B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (+
          (- (/ x B))
          (*
           (/ F (sin B))
           (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))))
   (if (<= F -0.000216)
     (+ (/ (- x) (tan B)) (/ -1.0 (sin B)))
     (if (<= F -4.5e-130)
       t_0
       (if (<= F 4.2e-140)
         (- (* (cos B) (/ x (sin B))))
         (if (<= F 4.3e+139)
           t_0
           (+
            (- (/ (* x 1.0) (tan B)))
            (/
             1.0
             (*
              (+
               1.0
               (*
                (- (* 0.008333333333333333 (* B B)) 0.16666666666666666)
                (* B B)))
              B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	double tmp;
	if (F <= -0.000216) {
		tmp = (-x / tan(B)) + (-1.0 / sin(B));
	} else if (F <= -4.5e-130) {
		tmp = t_0;
	} else if (F <= 4.2e-140) {
		tmp = -(cos(B) * (x / sin(B)));
	} else if (F <= 4.3e+139) {
		tmp = t_0;
	} else {
		tmp = -((x * 1.0) / tan(B)) + (1.0 / ((1.0 + (((0.008333333333333333 * (B * B)) - 0.16666666666666666) * (B * B))) * 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) :: t_0
    real(8) :: tmp
    t_0 = -(x / b) + ((f / sin(b)) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
    if (f <= (-0.000216d0)) then
        tmp = (-x / tan(b)) + ((-1.0d0) / sin(b))
    else if (f <= (-4.5d-130)) then
        tmp = t_0
    else if (f <= 4.2d-140) then
        tmp = -(cos(b) * (x / sin(b)))
    else if (f <= 4.3d+139) then
        tmp = t_0
    else
        tmp = -((x * 1.0d0) / tan(b)) + (1.0d0 / ((1.0d0 + (((0.008333333333333333d0 * (b * b)) - 0.16666666666666666d0) * (b * b))) * b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -(x / B) + ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	double tmp;
	if (F <= -0.000216) {
		tmp = (-x / Math.tan(B)) + (-1.0 / Math.sin(B));
	} else if (F <= -4.5e-130) {
		tmp = t_0;
	} else if (F <= 4.2e-140) {
		tmp = -(Math.cos(B) * (x / Math.sin(B)));
	} else if (F <= 4.3e+139) {
		tmp = t_0;
	} else {
		tmp = -((x * 1.0) / Math.tan(B)) + (1.0 / ((1.0 + (((0.008333333333333333 * (B * B)) - 0.16666666666666666) * (B * B))) * B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -(x / B) + ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))
	tmp = 0
	if F <= -0.000216:
		tmp = (-x / math.tan(B)) + (-1.0 / math.sin(B))
	elif F <= -4.5e-130:
		tmp = t_0
	elif F <= 4.2e-140:
		tmp = -(math.cos(B) * (x / math.sin(B)))
	elif F <= 4.3e+139:
		tmp = t_0
	else:
		tmp = -((x * 1.0) / math.tan(B)) + (1.0 / ((1.0 + (((0.008333333333333333 * (B * B)) - 0.16666666666666666) * (B * B))) * B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-Float64(x / B)) + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))))
	tmp = 0.0
	if (F <= -0.000216)
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(-1.0 / sin(B)));
	elseif (F <= -4.5e-130)
		tmp = t_0;
	elseif (F <= 4.2e-140)
		tmp = Float64(-Float64(cos(B) * Float64(x / sin(B))));
	elseif (F <= 4.3e+139)
		tmp = t_0;
	else
		tmp = Float64(Float64(-Float64(Float64(x * 1.0) / tan(B))) + Float64(1.0 / Float64(Float64(1.0 + Float64(Float64(Float64(0.008333333333333333 * Float64(B * B)) - 0.16666666666666666) * Float64(B * B))) * B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -(x / B) + ((F / sin(B)) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
	tmp = 0.0;
	if (F <= -0.000216)
		tmp = (-x / tan(B)) + (-1.0 / sin(B));
	elseif (F <= -4.5e-130)
		tmp = t_0;
	elseif (F <= 4.2e-140)
		tmp = -(cos(B) * (x / sin(B)));
	elseif (F <= 4.3e+139)
		tmp = t_0;
	else
		tmp = -((x * 1.0) / tan(B)) + (1.0 / ((1.0 + (((0.008333333333333333 * (B * B)) - 0.16666666666666666) * (B * B))) * B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-N[(x / B), $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]}, If[LessEqual[F, -0.000216], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -4.5e-130], t$95$0, If[LessEqual[F, 4.2e-140], (-N[(N[Cos[B], $MachinePrecision] * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 4.3e+139], t$95$0, N[((-N[(N[(x * 1.0), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]) + N[(1.0 / N[(N[(1.0 + N[(N[(N[(0.008333333333333333 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.1599999999999999e-4

   1. Initial program 59.1%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 98.8

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

   4. Applied rewrites 98.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-/l* N/A

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

      6. distribute-neg-frac N/A

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

      7. *-rgt-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-tan.f64 98.9

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

   6. Applied rewrites 98.9%

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

   if -2.1599999999999999e-4 < F < -4.5e-130 or 4.20000000000000035e-140 < F < 4.2999999999999998e139

   1. Initial program 95.9%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 74.1

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

   4. Applied rewrites 74.1%

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

   if -4.5e-130 < F < 4.20000000000000035e-140

   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. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 82.2

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

   4. Applied rewrites 82.2%

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

      1. associate-/l* 82.2

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-sin.f64 82.2

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

   6. Applied rewrites 82.2%

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

   if 4.2999999999999998e139 < F

   1. Initial program 95.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 96.0

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

   3. Applied rewrites 96.0%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 58.1

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

   6. Applied rewrites 58.1%

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

   7. Taylor expanded in B around 0

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

      1. *-commutative N/A

         \[\leadsto \left(-\frac{x \cdot 1}{\tan B}\right) + \frac{1}{\left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right) \cdot B} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-\frac{x \cdot 1}{\tan B}\right) + \frac{1}{\left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right) \cdot B} \]

      3. lower-+.f64 N/A

         \[\leadsto \left(-\frac{x \cdot 1}{\tan B}\right) + \frac{1}{\left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right) \cdot B} \]

      4. *-commutative N/A

         \[\leadsto \left(-\frac{x \cdot 1}{\tan B}\right) + \frac{1}{\left(1 + \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right) \cdot {B}^{2}\right) \cdot B} \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-\frac{x \cdot 1}{\tan B}\right) + \frac{1}{\left(1 + \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right) \cdot {B}^{2}\right) \cdot B} \]

      6. lower--.f64 N/A

         \[\leadsto \left(-\frac{x \cdot 1}{\tan B}\right) + \frac{1}{\left(1 + \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right) \cdot {B}^{2}\right) \cdot B} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(-\frac{x \cdot 1}{\tan B}\right) + \frac{1}{\left(1 + \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right) \cdot {B}^{2}\right) \cdot B} \]

      8. unpow2 N/A

         \[\leadsto \left(-\frac{x \cdot 1}{\tan B}\right) + \frac{1}{\left(1 + \left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}\right) \cdot {B}^{2}\right) \cdot B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{x \cdot 1}{\tan B}\right) + \frac{1}{\left(1 + \left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}\right) \cdot {B}^{2}\right) \cdot B} \]

      10. unpow2 N/A

          \[\leadsto \left(-\frac{x \cdot 1}{\tan B}\right) + \frac{1}{\left(1 + \left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}\right) \cdot \left(B \cdot B\right)\right) \cdot B} \]

      11. lower-*.f64 52.0

          \[\leadsto \left(-\frac{x \cdot 1}{\tan B}\right) + \frac{1}{\left(1 + \left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666\right) \cdot \left(B \cdot B\right)\right) \cdot B} \]

   9. Applied rewrites 52.0%

      \[\leadsto \left(-\frac{x \cdot 1}{\tan B}\right) + \frac{1}{\left(1 + \left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666\right) \cdot \left(B \cdot B\right)\right) \cdot \color{blue}{B}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 75.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-8}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B}\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-18}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;-\cos B \cdot \frac{x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -6.8e-8)
   (+ (- (* x (/ 1.0 (tan B)))) (/ -1.0 B))
   (if (<= x 5.7e-18)
     (+
      (- (/ x B))
      (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
     (- (* (cos B) (/ x (sin B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -6.8e-8) {
		tmp = -(x * (1.0 / tan(B))) + (-1.0 / B);
	} else if (x <= 5.7e-18) {
		tmp = -(x / B) + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = -(cos(B) * (x / sin(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 (x <= (-6.8d-8)) then
        tmp = -(x * (1.0d0 / tan(b))) + ((-1.0d0) / b)
    else if (x <= 5.7d-18) then
        tmp = -(x / b) + ((f / sin(b)) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
    else
        tmp = -(cos(b) * (x / sin(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (x <= -6.8e-8) {
		tmp = -(x * (1.0 / Math.tan(B))) + (-1.0 / B);
	} else if (x <= 5.7e-18) {
		tmp = -(x / B) + ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = -(Math.cos(B) * (x / Math.sin(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if x <= -6.8e-8:
		tmp = -(x * (1.0 / math.tan(B))) + (-1.0 / B)
	elif x <= 5.7e-18:
		tmp = -(x / B) + ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))
	else:
		tmp = -(math.cos(B) * (x / math.sin(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -6.8e-8)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(-1.0 / B));
	elseif (x <= 5.7e-18)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))));
	else
		tmp = Float64(-Float64(cos(B) * Float64(x / sin(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (x <= -6.8e-8)
		tmp = -(x * (1.0 / tan(B))) + (-1.0 / B);
	elseif (x <= 5.7e-18)
		tmp = -(x / B) + ((F / sin(B)) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
	else
		tmp = -(cos(B) * (x / sin(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -6.8e-8], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.7e-18], N[((-N[(x / B), $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], (-N[(N[Cos[B], $MachinePrecision] * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if x < -6.8e-8

   1. Initial program 70.8%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 94.6

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

   4. Applied rewrites 94.6%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 94.4%

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

   if -6.8e-8 < x < 5.69999999999999971e-18

   1. Initial program 72.1%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 60.6

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

   4. Applied rewrites 60.6%

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

   if 5.69999999999999971e-18 < x

   1. Initial program 87.7%

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

   2. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 96.1

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

   4. Applied rewrites 96.1%

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

      1. associate-/l* 96.1

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-sin.f64 96.1

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

   6. Applied rewrites 96.1%

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

Alternative 10: 75.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-8}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B}\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-18}:\\ \;\;\;\;\left(-x \cdot \frac{1}{B}\right) + \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;-\cos B \cdot \frac{x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -6.8e-8)
   (+ (- (* x (/ 1.0 (tan B)))) (/ -1.0 B))
   (if (<= x 5.7e-18)
     (+ (- (* x (/ 1.0 B))) (* (/ F (sin B)) (pow (fma F F 2.0) -0.5)))
     (- (* (cos B) (/ x (sin B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -6.8e-8) {
		tmp = -(x * (1.0 / tan(B))) + (-1.0 / B);
	} else if (x <= 5.7e-18) {
		tmp = -(x * (1.0 / B)) + ((F / sin(B)) * pow(fma(F, F, 2.0), -0.5));
	} else {
		tmp = -(cos(B) * (x / sin(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -6.8e-8)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(-1.0 / B));
	elseif (x <= 5.7e-18)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / B))) + Float64(Float64(F / sin(B)) * (fma(F, F, 2.0) ^ -0.5)));
	else
		tmp = Float64(-Float64(cos(B) * Float64(x / sin(B))));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -6.8e-8], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.7e-18], N[((-N[(x * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(F * F + 2.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[Cos[B], $MachinePrecision] * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if x < -6.8e-8

   1. Initial program 70.8%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 94.6

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

   4. Applied rewrites 94.6%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 94.4%

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

   if -6.8e-8 < x < 5.69999999999999971e-18

   1. Initial program 72.1%

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

   2. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval 72.1

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

   4. Applied rewrites 72.1%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 60.6%

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

   if 5.69999999999999971e-18 < x

   1. Initial program 87.7%

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

   2. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 96.1

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

   4. Applied rewrites 96.1%

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

      1. associate-/l* 96.1

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. lift-sin.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lift-cos.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-sin.f64 96.1

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

   6. Applied rewrites 96.1%

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

Alternative 11: 75.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-8}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B}\\ \mathbf{elif}\;x \leq 5.7 \cdot 10^{-18}:\\ \;\;\;\;\left(-x \cdot \frac{1}{B}\right) + \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -6.8e-8)
   (+ (- (* x (/ 1.0 (tan B)))) (/ -1.0 B))
   (if (<= x 5.7e-18)
     (+ (- (* x (/ 1.0 B))) (* (/ F (sin B)) (pow (fma F F 2.0) -0.5)))
     (- (/ (* (cos B) x) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -6.8e-8) {
		tmp = -(x * (1.0 / tan(B))) + (-1.0 / B);
	} else if (x <= 5.7e-18) {
		tmp = -(x * (1.0 / B)) + ((F / sin(B)) * pow(fma(F, F, 2.0), -0.5));
	} else {
		tmp = -((cos(B) * x) / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -6.8e-8)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(-1.0 / B));
	elseif (x <= 5.7e-18)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / B))) + Float64(Float64(F / sin(B)) * (fma(F, F, 2.0) ^ -0.5)));
	else
		tmp = Float64(-Float64(Float64(cos(B) * x) / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -6.8e-8], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.7e-18], N[((-N[(x * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(F * F + 2.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[(N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if x < -6.8e-8

   1. Initial program 70.8%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 94.6

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

   4. Applied rewrites 94.6%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 94.4%

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

   if -6.8e-8 < x < 5.69999999999999971e-18

   1. Initial program 72.1%

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

   2. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval 72.1

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

   4. Applied rewrites 72.1%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 60.6%

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

   if 5.69999999999999971e-18 < x

   1. Initial program 87.7%

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

   2. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 96.1

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

   4. Applied rewrites 96.1%

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

Alternative 12: 75.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-8}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B}\\ \mathbf{elif}\;x \leq 0.015:\\ \;\;\;\;\left(-x \cdot \frac{1}{B}\right) + \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -6.8e-8)
   (+ (- (* x (/ 1.0 (tan B)))) (/ -1.0 B))
   (if (<= x 0.015)
     (+ (- (* x (/ 1.0 B))) (* (/ F (sin B)) (pow (fma F F 2.0) -0.5)))
     (+ (- (/ (* x 1.0) (tan B))) (/ 1.0 B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -6.8e-8) {
		tmp = -(x * (1.0 / tan(B))) + (-1.0 / B);
	} else if (x <= 0.015) {
		tmp = -(x * (1.0 / B)) + ((F / sin(B)) * pow(fma(F, F, 2.0), -0.5));
	} else {
		tmp = -((x * 1.0) / tan(B)) + (1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -6.8e-8)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(-1.0 / B));
	elseif (x <= 0.015)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / B))) + Float64(Float64(F / sin(B)) * (fma(F, F, 2.0) ^ -0.5)));
	else
		tmp = Float64(Float64(-Float64(Float64(x * 1.0) / tan(B))) + Float64(1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -6.8e-8], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.015], N[((-N[(x * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(F * F + 2.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[((-N[(N[(x * 1.0), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]) + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.8e-8

   1. Initial program 70.8%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 94.6

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

   4. Applied rewrites 94.6%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 94.4%

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

   if -6.8e-8 < x < 0.014999999999999999

   1. Initial program 72.0%

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

   2. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval 71.9

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

   4. Applied rewrites 71.9%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 60.0%

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

   if 0.014999999999999999 < x

   1. Initial program 88.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 88.8

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

   3. Applied rewrites 88.8%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 97.7

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

   6. Applied rewrites 97.7%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 98.4

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

   9. Applied rewrites 98.4%

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

Alternative 13: 69.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{-109}:\\ \;\;\;\;t\_0 + \frac{-1}{\mathsf{fma}\left(B \cdot B, -0.16666666666666666, 1\right) \cdot B}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5} \cdot F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{-1}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666, B \cdot B, 1\right) \cdot B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* x (/ 1.0 (tan B))))))
   (if (<= x -3.1e-109)
     (+ t_0 (/ -1.0 (* (fma (* B B) -0.16666666666666666 1.0) B)))
     (if (<= x 7.5e-119)
       (/ (* (pow (fma F F 2.0) -0.5) F) (sin B))
       (+
        t_0
        (/
         -1.0
         (*
          (fma
           (- (* 0.008333333333333333 (* B B)) 0.16666666666666666)
           (* B B)
           1.0)
          B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B)));
	double tmp;
	if (x <= -3.1e-109) {
		tmp = t_0 + (-1.0 / (fma((B * B), -0.16666666666666666, 1.0) * B));
	} else if (x <= 7.5e-119) {
		tmp = (pow(fma(F, F, 2.0), -0.5) * F) / sin(B);
	} else {
		tmp = t_0 + (-1.0 / (fma(((0.008333333333333333 * (B * B)) - 0.16666666666666666), (B * B), 1.0) * B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (x <= -3.1e-109)
		tmp = Float64(t_0 + Float64(-1.0 / Float64(fma(Float64(B * B), -0.16666666666666666, 1.0) * B)));
	elseif (x <= 7.5e-119)
		tmp = Float64(Float64((fma(F, F, 2.0) ^ -0.5) * F) / sin(B));
	else
		tmp = Float64(t_0 + Float64(-1.0 / Float64(fma(Float64(Float64(0.008333333333333333 * Float64(B * B)) - 0.16666666666666666), Float64(B * B), 1.0) * B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[x, -3.1e-109], N[(t$95$0 + N[(-1.0 / N[(N[(N[(B * B), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.5e-119], N[(N[(N[Power[N[(F * F + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(-1.0 / N[(N[(N[(N[(0.008333333333333333 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.1e-109

   1. Initial program 71.4%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 63.8

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

   4. Applied rewrites 63.8%

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

   5. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(B \cdot B, \frac{-1}{6}, 1\right) \cdot B} \]

      7. lower-*.f64 65.7

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

   7. Applied rewrites 65.7%

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

   if -3.1e-109 < x < 7.50000000000000044e-119

   1. Initial program 72.3%

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

   2. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lift-sin.f64 60.2

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

   4. Applied rewrites 60.2%

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

   if 7.50000000000000044e-119 < x

   1. Initial program 83.6%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 77.4

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

   4. Applied rewrites 77.4%

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

   5. Taylor expanded in B around 0

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

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right) \cdot B} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right) \cdot B} \]

      3. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\left({B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right) + 1\right) \cdot B} \]

      4. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\left(\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right) \cdot {B}^{2} + 1\right) \cdot B} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      6. lower--.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}, {B}^{2}, 1\right) \cdot B} \]

      10. unpow2 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}, B \cdot B, 1\right) \cdot B} \]

      11. lower-*.f64 79.7

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666, B \cdot B, 1\right) \cdot B} \]

   7. Applied rewrites 79.7%

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

Alternative 14: 69.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(B \cdot B, -0.16666666666666666, 1\right) \cdot B}\\ \mathbf{if}\;x \leq -3.1 \cdot 10^{-109}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-119}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5} \cdot F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (+
          (- (* x (/ 1.0 (tan B))))
          (/ -1.0 (* (fma (* B B) -0.16666666666666666 1.0) B)))))
   (if (<= x -3.1e-109)
     t_0
     (if (<= x 7.5e-119) (/ (* (pow (fma F F 2.0) -0.5) F) (sin B)) t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B))) + (-1.0 / (fma((B * B), -0.16666666666666666, 1.0) * B));
	double tmp;
	if (x <= -3.1e-109) {
		tmp = t_0;
	} else if (x <= 7.5e-119) {
		tmp = (pow(fma(F, F, 2.0), -0.5) * F) / sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(-1.0 / Float64(fma(Float64(B * B), -0.16666666666666666, 1.0) * B)))
	tmp = 0.0
	if (x <= -3.1e-109)
		tmp = t_0;
	elseif (x <= 7.5e-119)
		tmp = Float64(Float64((fma(F, F, 2.0) ^ -0.5) * F) / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / N[(N[(N[(B * B), $MachinePrecision] * -0.16666666666666666 + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.1e-109], t$95$0, If[LessEqual[x, 7.5e-119], N[(N[(N[Power[N[(F * F + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.1e-109 or 7.50000000000000044e-119 < x

   1. Initial program 79.4%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 72.7

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

   4. Applied rewrites 72.7%

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

   5. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\mathsf{fma}\left(B \cdot B, \frac{-1}{6}, 1\right) \cdot B} \]

      7. lower-*.f64 74.8

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

   7. Applied rewrites 74.8%

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

   if -3.1e-109 < x < 7.50000000000000044e-119

   1. Initial program 72.3%

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

   2. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lift-sin.f64 60.2

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

   4. Applied rewrites 60.2%

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

Alternative 15: 69.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B}\\ \mathbf{if}\;x \leq -1.85 \cdot 10^{-73}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-119}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5} \cdot F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (+ (- (* x (/ 1.0 (tan B)))) (/ -1.0 B))))
   (if (<= x -1.85e-73)
     t_0
     (if (<= x 8.2e-119) (/ (* (pow (fma F F 2.0) -0.5) F) (sin B)) t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B))) + (-1.0 / B);
	double tmp;
	if (x <= -1.85e-73) {
		tmp = t_0;
	} else if (x <= 8.2e-119) {
		tmp = (pow(fma(F, F, 2.0), -0.5) * F) / sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(-1.0 / B))
	tmp = 0.0
	if (x <= -1.85e-73)
		tmp = t_0;
	elseif (x <= 8.2e-119)
		tmp = Float64(Float64((fma(F, F, 2.0) ^ -0.5) * F) / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.85e-73], t$95$0, If[LessEqual[x, 8.2e-119], N[(N[(N[Power[N[(F * F + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.85e-73 or 8.20000000000000041e-119 < x

   1. Initial program 79.8%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 75.1

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

   4. Applied rewrites 75.1%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 76.9%

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

   if -1.85e-73 < x < 8.20000000000000041e-119

   1. Initial program 72.3%

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

   2. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-pow.f64 N/A

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

      7. +-commutative N/A

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

      8. pow2 N/A

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

      9. lower-fma.f64 N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lift-sin.f64 58.7

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

   4. Applied rewrites 58.7%

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

Alternative 16: 63.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -x \cdot \frac{1}{B}\\ t_1 := \left(-\frac{x \cdot 1}{\tan B}\right) + \frac{1}{B}\\ \mathbf{if}\;F \leq -2.55 \cdot 10^{+119}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -0.000216:\\ \;\;\;\;t\_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-107}:\\ \;\;\;\;t\_0 + \frac{F}{B \cdot \left(1 + \left(B \cdot B\right) \cdot \left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666\right)\right)} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{+139}:\\ \;\;\;\;t\_0 + \frac{F}{\sin B} \cdot \frac{1}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* x (/ 1.0 B))))
        (t_1 (+ (- (/ (* x 1.0) (tan B))) (/ 1.0 B))))
   (if (<= F -2.55e+119)
     (+ (- (* x (/ 1.0 (tan B)))) (/ -1.0 B))
     (if (<= F -0.000216)
       (+ t_0 (/ -1.0 (sin B)))
       (if (<= F 7.5e-107)
         (+
          t_0
          (*
           (/
            F
            (*
             B
             (+
              1.0
              (*
               (* B B)
               (- (* 0.008333333333333333 (* B B)) 0.16666666666666666)))))
           (pow (fma F F 2.0) -0.5)))
         (if (<= F 1.15e+36)
           t_1
           (if (<= F 4.3e+139) (+ t_0 (* (/ F (sin B)) (/ 1.0 F))) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / B));
	double t_1 = -((x * 1.0) / tan(B)) + (1.0 / B);
	double tmp;
	if (F <= -2.55e+119) {
		tmp = -(x * (1.0 / tan(B))) + (-1.0 / B);
	} else if (F <= -0.000216) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 7.5e-107) {
		tmp = t_0 + ((F / (B * (1.0 + ((B * B) * ((0.008333333333333333 * (B * B)) - 0.16666666666666666))))) * pow(fma(F, F, 2.0), -0.5));
	} else if (F <= 1.15e+36) {
		tmp = t_1;
	} else if (F <= 4.3e+139) {
		tmp = t_0 + ((F / sin(B)) * (1.0 / F));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x * Float64(1.0 / B)))
	t_1 = Float64(Float64(-Float64(Float64(x * 1.0) / tan(B))) + Float64(1.0 / B))
	tmp = 0.0
	if (F <= -2.55e+119)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(-1.0 / B));
	elseif (F <= -0.000216)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 7.5e-107)
		tmp = Float64(t_0 + Float64(Float64(F / Float64(B * Float64(1.0 + Float64(Float64(B * B) * Float64(Float64(0.008333333333333333 * Float64(B * B)) - 0.16666666666666666))))) * (fma(F, F, 2.0) ^ -0.5)));
	elseif (F <= 1.15e+36)
		tmp = t_1;
	elseif (F <= 4.3e+139)
		tmp = Float64(t_0 + Float64(Float64(F / sin(B)) * Float64(1.0 / F)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x * N[(1.0 / B), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$1 = N[((-N[(N[(x * 1.0), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]) + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.55e+119], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -0.000216], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.5e-107], N[(t$95$0 + N[(N[(F / N[(B * N[(1.0 + N[(N[(B * B), $MachinePrecision] * N[(N[(0.008333333333333333 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(F * F + 2.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.15e+36], t$95$1, If[LessEqual[F, 4.3e+139], N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -2.54999999999999992e119

   1. Initial program 39.4%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 74.5%

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

   if -2.54999999999999992e119 < F < -2.1599999999999999e-4

   1. Initial program 93.3%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 97.3

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

   4. Applied rewrites 97.3%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 77.7%

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

   if -2.1599999999999999e-4 < F < 7.50000000000000047e-107

   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. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 64.7%

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

   8. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{F}{B \cdot \color{blue}{\left(1 + {B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right)}} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}} \]

      2. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{F}{B \cdot \left(1 + \color{blue}{{B}^{2} \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)}\right)} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{F}{B \cdot \left(1 + {B}^{2} \cdot \color{blue}{\left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)}\right)} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}} \]

      4. pow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{F}{B \cdot \left(1 + \left(B \cdot B\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {B}^{2}} - \frac{1}{6}\right)\right)} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{F}{B \cdot \left(1 + \left(B \cdot B\right) \cdot \left(\color{blue}{\frac{1}{120} \cdot {B}^{2}} - \frac{1}{6}\right)\right)} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}} \]

      6. lower--.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{F}{B \cdot \left(1 + \left(B \cdot B\right) \cdot \left(\frac{1}{120} \cdot {B}^{2} - \color{blue}{\frac{1}{6}}\right)\right)} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{F}{B \cdot \left(1 + \left(B \cdot B\right) \cdot \left(\frac{1}{120} \cdot {B}^{2} - \frac{1}{6}\right)\right)} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}} \]

      8. pow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{F}{B \cdot \left(1 + \left(B \cdot B\right) \cdot \left(\frac{1}{120} \cdot \left(B \cdot B\right) - \frac{1}{6}\right)\right)} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}} \]

      9. lift-*.f64 52.0

         \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{F}{B \cdot \left(1 + \left(B \cdot B\right) \cdot \left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666\right)\right)} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5} \]

   10. Applied rewrites 52.0%

       \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{F}{\color{blue}{B \cdot \left(1 + \left(B \cdot B\right) \cdot \left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666\right)\right)}} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5} \]

   if 7.50000000000000047e-107 < F < 1.14999999999999998e36 or 4.2999999999999998e139 < F

   1. Initial program 61.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 61.0

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

   3. Applied rewrites 61.0%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 81.3

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

   6. Applied rewrites 81.3%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 64.6

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

   9. Applied rewrites 64.6%

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

   if 1.14999999999999998e36 < F < 4.2999999999999998e139

   1. Initial program 87.0%

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

   2. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval 87.0

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

   4. Applied rewrites 87.0%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 67.3%

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

   8. Taylor expanded in F around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. sqr-pow N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. pow-prod-up N/A

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

      14. unpow2 N/A

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

      15. lower-/.f64 67.3

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

   10. Applied rewrites 67.3%

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

Alternative 17: 63.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -x \cdot \frac{1}{B}\\ t_1 := \left(-\frac{x \cdot 1}{\tan B}\right) + \frac{1}{B}\\ \mathbf{if}\;F \leq -2.55 \cdot 10^{+119}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B}\\ \mathbf{elif}\;F \leq -0.000216:\\ \;\;\;\;t\_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-107}:\\ \;\;\;\;t\_0 + \frac{F}{B \cdot \left(1 - 0.16666666666666666 \cdot \left(B \cdot B\right)\right)} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{+36}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{+139}:\\ \;\;\;\;t\_0 + \frac{F}{\sin B} \cdot \frac{1}{F}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* x (/ 1.0 B))))
        (t_1 (+ (- (/ (* x 1.0) (tan B))) (/ 1.0 B))))
   (if (<= F -2.55e+119)
     (+ (- (* x (/ 1.0 (tan B)))) (/ -1.0 B))
     (if (<= F -0.000216)
       (+ t_0 (/ -1.0 (sin B)))
       (if (<= F 7.5e-107)
         (+
          t_0
          (*
           (/ F (* B (- 1.0 (* 0.16666666666666666 (* B B)))))
           (pow (fma F F 2.0) -0.5)))
         (if (<= F 1.15e+36)
           t_1
           (if (<= F 4.3e+139) (+ t_0 (* (/ F (sin B)) (/ 1.0 F))) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / B));
	double t_1 = -((x * 1.0) / tan(B)) + (1.0 / B);
	double tmp;
	if (F <= -2.55e+119) {
		tmp = -(x * (1.0 / tan(B))) + (-1.0 / B);
	} else if (F <= -0.000216) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 7.5e-107) {
		tmp = t_0 + ((F / (B * (1.0 - (0.16666666666666666 * (B * B))))) * pow(fma(F, F, 2.0), -0.5));
	} else if (F <= 1.15e+36) {
		tmp = t_1;
	} else if (F <= 4.3e+139) {
		tmp = t_0 + ((F / sin(B)) * (1.0 / F));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x * Float64(1.0 / B)))
	t_1 = Float64(Float64(-Float64(Float64(x * 1.0) / tan(B))) + Float64(1.0 / B))
	tmp = 0.0
	if (F <= -2.55e+119)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(-1.0 / B));
	elseif (F <= -0.000216)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 7.5e-107)
		tmp = Float64(t_0 + Float64(Float64(F / Float64(B * Float64(1.0 - Float64(0.16666666666666666 * Float64(B * B))))) * (fma(F, F, 2.0) ^ -0.5)));
	elseif (F <= 1.15e+36)
		tmp = t_1;
	elseif (F <= 4.3e+139)
		tmp = Float64(t_0 + Float64(Float64(F / sin(B)) * Float64(1.0 / F)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x * N[(1.0 / B), $MachinePrecision]), $MachinePrecision])}, Block[{t$95$1 = N[((-N[(N[(x * 1.0), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]) + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.55e+119], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -0.000216], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.5e-107], N[(t$95$0 + N[(N[(F / N[(B * N[(1.0 - N[(0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[(F * F + 2.0), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.15e+36], t$95$1, If[LessEqual[F, 4.3e+139], N[(t$95$0 + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -2.54999999999999992e119

   1. Initial program 39.4%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 74.5%

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

   if -2.54999999999999992e119 < F < -2.1599999999999999e-4

   1. Initial program 93.3%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 97.3

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

   4. Applied rewrites 97.3%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 77.7%

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

   if -2.1599999999999999e-4 < F < 7.50000000000000047e-107

   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. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 64.7%

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

   8. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

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

         \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{F}{B \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {B}^{2}}\right)} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}} \]

      3. lower--.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{F}{B \cdot \left(1 - \color{blue}{\left(\mathsf{neg}\left(\frac{-1}{6}\right)\right) \cdot {B}^{2}}\right)} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}} \]

      4. metadata-eval N/A

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

      5. lower-*.f64 N/A

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

      6. pow2 N/A

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

      7. lift-*.f64 52.0

         \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{F}{B \cdot \left(1 - 0.16666666666666666 \cdot \left(B \cdot \color{blue}{B}\right)\right)} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5} \]

   10. Applied rewrites 52.0%

       \[\leadsto \left(-x \cdot \frac{1}{B}\right) + \frac{F}{\color{blue}{B \cdot \left(1 - 0.16666666666666666 \cdot \left(B \cdot B\right)\right)}} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5} \]

   if 7.50000000000000047e-107 < F < 1.14999999999999998e36 or 4.2999999999999998e139 < F

   1. Initial program 61.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 61.0

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

   3. Applied rewrites 61.0%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 81.3

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

   6. Applied rewrites 81.3%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 64.6

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

   9. Applied rewrites 64.6%

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

   if 1.14999999999999998e36 < F < 4.2999999999999998e139

   1. Initial program 87.0%

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

   2. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-pow.f64 N/A

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

      4. +-commutative N/A

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

      5. pow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval 87.0

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

   4. Applied rewrites 87.0%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 67.3%

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

   8. Taylor expanded in F around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. sqr-pow N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. pow-prod-up N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. metadata-eval N/A

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

      13. pow-prod-up N/A

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

      14. unpow2 N/A

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

      15. lower-/.f64 67.3

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

   10. Applied rewrites 67.3%

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

Alternative 18: 58.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.002:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(F, F, x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.002)
   (/ (- (* (pow (+ (fma F F (+ x x)) 2.0) -0.5) F) x) B)
   (+ (- (/ (* x 1.0) (tan B))) (/ 1.0 B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.002) {
		tmp = ((pow((fma(F, F, (x + x)) + 2.0), -0.5) * F) - x) / B;
	} else {
		tmp = -((x * 1.0) / tan(B)) + (1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.002)
		tmp = Float64(Float64(Float64((Float64(fma(F, F, Float64(x + x)) + 2.0) ^ -0.5) * F) - x) / B);
	else
		tmp = Float64(Float64(-Float64(Float64(x * 1.0) / tan(B))) + Float64(1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.002], N[(N[(N[(N[Power[N[(N[(F * F + N[(x + x), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[((-N[(N[(x * 1.0), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]) + N[(1.0 / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2e-3

   1. Initial program 74.0%

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

   2. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

   4. Applied rewrites 57.7%

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

   if 2e-3 < B

   1. Initial program 85.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 85.1

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

   3. Applied rewrites 85.1%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 57.3

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

   6. Applied rewrites 57.3%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 52.3

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

   9. Applied rewrites 52.3%

      \[\leadsto \left(-\frac{x \cdot 1}{\tan B}\right) + \frac{1}{\color{blue}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 56.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.0024:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(F, F, x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.0024)
   (/ (- (* (pow (+ (fma F F (+ x x)) 2.0) -0.5) F) x) B)
   (+ (- (* x (/ 1.0 (tan B)))) (/ -1.0 B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.0024) {
		tmp = ((pow((fma(F, F, (x + x)) + 2.0), -0.5) * F) - x) / B;
	} else {
		tmp = -(x * (1.0 / tan(B))) + (-1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.0024)
		tmp = Float64(Float64(Float64((Float64(fma(F, F, Float64(x + x)) + 2.0) ^ -0.5) * F) - x) / B);
	else
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(-1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.0024], N[(N[(N[(N[Power[N[(N[(F * F + N[(x + x), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.00239999999999999979

   1. Initial program 74.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 57.7%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B}} \]

   if 0.00239999999999999979 < B

   1. Initial program 85.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\color{blue}{\sin B}} \]

      2. lift-sin.f64 56.9

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B} \]

   4. Applied rewrites 56.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 52.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 56.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\left(-x \cdot \frac{1}{B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 7.7 \cdot 10^{+40}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(F, F, x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B}\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{+159}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.4)
   (+ (- (* x (/ 1.0 B))) (/ -1.0 (sin B)))
   (if (<= F 7.7e+40)
     (/ (- (* (pow (+ (fma F F (+ x x)) 2.0) -0.5) F) x) B)
     (if (<= F 5.6e+159) (/ 1.0 (sin B)) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4) {
		tmp = -(x * (1.0 / B)) + (-1.0 / sin(B));
	} else if (F <= 7.7e+40) {
		tmp = ((pow((fma(F, F, (x + x)) + 2.0), -0.5) * F) - x) / B;
	} else if (F <= 5.6e+159) {
		tmp = 1.0 / sin(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(-Float64(x * Float64(1.0 / B))) + Float64(-1.0 / sin(B)));
	elseif (F <= 7.7e+40)
		tmp = Float64(Float64(Float64((Float64(fma(F, F, Float64(x + x)) + 2.0) ^ -0.5) * F) - x) / B);
	elseif (F <= 5.6e+159)
		tmp = Float64(1.0 / sin(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[(x * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.7e+40], N[(N[(N[(N[Power[N[(N[(F * F + N[(x + x), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 5.6e+159], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.3999999999999999

   1. Initial program 58.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\color{blue}{\sin B}} \]

      2. lift-sin.f64 99.2

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\sin B} \]

   4. Applied rewrites 99.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{-1}{\sin B} \]
   6. Step-by-step derivation

   7. Applied rewrites 76.8%

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B}}\right) + \frac{-1}{\sin B} \]

   if -1.3999999999999999 < F < 7.69999999999999964e40

   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. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 51.5%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B}} \]

   if 7.69999999999999964e40 < F < 5.6000000000000002e159

   1. Initial program 84.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lift-sin.f64 54.8

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]

   if 5.6000000000000002e159 < F

   1. Initial program 32.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\color{blue}{x \cdot \frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. lift-tan.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{x \cdot 1}}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      7. lift-tan.f64 32.2

         \[\leadsto \left(-\frac{x \cdot 1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied rewrites 32.2%

      \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   6. Applied rewrites 25.1%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B}} \]

   7. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 51.1%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 21: 52.7% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{+120}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -210000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1300000000000:\\ \;\;\;\;\frac{{\left(\left(x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B}\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{+159}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4e+120)
   (/ (- -1.0 x) B)
   (if (<= F -210000.0)
     (/ -1.0 (sin B))
     (if (<= F 1300000000000.0)
       (/ (- (* (pow (+ (+ x x) 2.0) -0.5) F) x) B)
       (if (<= F 5.6e+159) (/ 1.0 (sin B)) (/ (- 1.0 x) B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4e+120) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -210000.0) {
		tmp = -1.0 / sin(B);
	} else if (F <= 1300000000000.0) {
		tmp = ((pow(((x + x) + 2.0), -0.5) * F) - x) / B;
	} else if (F <= 5.6e+159) {
		tmp = 1.0 / sin(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 <= (-4d+120)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= (-210000.0d0)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 1300000000000.0d0) then
        tmp = (((((x + x) + 2.0d0) ** (-0.5d0)) * f) - x) / b
    else if (f <= 5.6d+159) then
        tmp = 1.0d0 / sin(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 <= -4e+120) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -210000.0) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 1300000000000.0) {
		tmp = ((Math.pow(((x + x) + 2.0), -0.5) * F) - x) / B;
	} else if (F <= 5.6e+159) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4e+120:
		tmp = (-1.0 - x) / B
	elif F <= -210000.0:
		tmp = -1.0 / math.sin(B)
	elif F <= 1300000000000.0:
		tmp = ((math.pow(((x + x) + 2.0), -0.5) * F) - x) / B
	elif F <= 5.6e+159:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4e+120)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -210000.0)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 1300000000000.0)
		tmp = Float64(Float64(Float64((Float64(Float64(x + x) + 2.0) ^ -0.5) * F) - x) / B);
	elseif (F <= 5.6e+159)
		tmp = Float64(1.0 / sin(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 <= -4e+120)
		tmp = (-1.0 - x) / B;
	elseif (F <= -210000.0)
		tmp = -1.0 / sin(B);
	elseif (F <= 1300000000000.0)
		tmp = (((((x + x) + 2.0) ^ -0.5) * F) - x) / B;
	elseif (F <= 5.6e+159)
		tmp = 1.0 / sin(B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4e+120], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -210000.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1300000000000.0], N[(N[(N[(N[Power[N[(N[(x + x), $MachinePrecision] + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 5.6e+159], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -3.9999999999999999e120

   1. Initial program 39.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\color{blue}{x \cdot \frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. lift-tan.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{x \cdot 1}}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      7. lift-tan.f64 39.2

         \[\leadsto \left(-\frac{x \cdot 1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied rewrites 39.2%

      \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   6. Applied rewrites 30.8%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B}} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 52.1%

      \[\leadsto \frac{-1 - x}{B} \]

   if -3.9999999999999999e120 < F < -2.1e5

   1. Initial program 92.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lift-sin.f64 59.7

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 59.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -2.1e5 < F < 1.3e12

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\color{blue}{x \cdot \frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. lift-tan.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{x \cdot 1}}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      7. lift-tan.f64 99.6

         \[\leadsto \left(-\frac{x \cdot 1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied rewrites 99.6%

      \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   6. Applied rewrites 51.8%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B}} \]

   7. Taylor expanded in F around 0

      \[\leadsto \frac{{\left(2 \cdot x + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]
   8. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto \frac{{\left(\left(x + x\right) + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      2. lift-+.f64 51.1

         \[\leadsto \frac{{\left(\left(x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B} \]

   9. Applied rewrites 51.1%

      \[\leadsto \frac{{\left(\left(x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B} \]

   if 1.3e12 < F < 5.6000000000000002e159

   1. Initial program 86.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lift-sin.f64 56.1

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 56.1%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]

   if 5.6000000000000002e159 < F

   1. Initial program 32.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\color{blue}{x \cdot \frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. lift-tan.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{x \cdot 1}}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      7. lift-tan.f64 32.2

         \[\leadsto \left(-\frac{x \cdot 1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied rewrites 32.2%

      \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   6. Applied rewrites 25.1%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B}} \]

   7. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 51.1%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 22: 51.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ \mathbf{if}\;B \leq 0.92:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(F, F, x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B}\\ \mathbf{elif}\;B \leq 6.1 \cdot 10^{+29}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;B \leq 5 \cdot 10^{+105}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))))
   (if (<= B 0.92)
     (/ (- (* (pow (+ (fma F F (+ x x)) 2.0) -0.5) F) x) B)
     (if (<= B 6.1e+29) t_0 (if (<= B 5e+105) (/ -1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double tmp;
	if (B <= 0.92) {
		tmp = ((pow((fma(F, F, (x + x)) + 2.0), -0.5) * F) - x) / B;
	} else if (B <= 6.1e+29) {
		tmp = t_0;
	} else if (B <= 5e+105) {
		tmp = -1.0 / sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (B <= 0.92)
		tmp = Float64(Float64(Float64((Float64(fma(F, F, Float64(x + x)) + 2.0) ^ -0.5) * F) - x) / B);
	elseif (B <= 6.1e+29)
		tmp = t_0;
	elseif (B <= 5e+105)
		tmp = Float64(-1.0 / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 0.92], N[(N[(N[(N[Power[N[(N[(F * F + N[(x + x), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[B, 6.1e+29], t$95$0, If[LessEqual[B, 5e+105], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if B < 0.92000000000000004

   1. Initial program 74.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   3. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   4. Applied rewrites 57.6%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B}} \]

   if 0.92000000000000004 < B < 6.0999999999999998e29 or 5.00000000000000046e105 < B

   1. Initial program 84.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} \]

      2. lift-sin.f64 17.8

         \[\leadsto \frac{1}{\sin B} \]

   4. Applied rewrites 17.8%

      \[\leadsto \color{blue}{\frac{1}{\sin B}} \]

   if 6.0999999999999998e29 < B < 5.00000000000000046e105

   1. Initial program 85.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lift-sin.f64 17.2

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 17.2%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 50.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{+120}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -210000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-38}:\\ \;\;\;\;\frac{{\left(\left(x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4e+120)
   (/ (- -1.0 x) B)
   (if (<= F -210000.0)
     (/ -1.0 (sin B))
     (if (<= F 1.15e-38)
       (/ (- (* (pow (+ (+ x x) 2.0) -0.5) F) x) B)
       (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4e+120) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -210000.0) {
		tmp = -1.0 / sin(B);
	} else if (F <= 1.15e-38) {
		tmp = ((pow(((x + x) + 2.0), -0.5) * F) - 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 <= (-4d+120)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= (-210000.0d0)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 1.15d-38) then
        tmp = (((((x + x) + 2.0d0) ** (-0.5d0)) * f) - 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 <= -4e+120) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -210000.0) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 1.15e-38) {
		tmp = ((Math.pow(((x + x) + 2.0), -0.5) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4e+120:
		tmp = (-1.0 - x) / B
	elif F <= -210000.0:
		tmp = -1.0 / math.sin(B)
	elif F <= 1.15e-38:
		tmp = ((math.pow(((x + x) + 2.0), -0.5) * F) - x) / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4e+120)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -210000.0)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 1.15e-38)
		tmp = Float64(Float64(Float64((Float64(Float64(x + x) + 2.0) ^ -0.5) * F) - 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 <= -4e+120)
		tmp = (-1.0 - x) / B;
	elseif (F <= -210000.0)
		tmp = -1.0 / sin(B);
	elseif (F <= 1.15e-38)
		tmp = (((((x + x) + 2.0) ^ -0.5) * F) - x) / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4e+120], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -210000.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.15e-38], N[(N[(N[(N[Power[N[(N[(x + x), $MachinePrecision] + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.9999999999999999e120

   1. Initial program 39.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\color{blue}{x \cdot \frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. lift-tan.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{x \cdot 1}}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      7. lift-tan.f64 39.2

         \[\leadsto \left(-\frac{x \cdot 1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied rewrites 39.2%

      \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   6. Applied rewrites 30.8%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B}} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 52.1%

      \[\leadsto \frac{-1 - x}{B} \]

   if -3.9999999999999999e120 < F < -2.1e5

   1. Initial program 92.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lift-sin.f64 59.7

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 59.7%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   if -2.1e5 < F < 1.15000000000000001e-38

   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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\color{blue}{x \cdot \frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. lift-tan.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{x \cdot 1}}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      7. lift-tan.f64 99.6

         \[\leadsto \left(-\frac{x \cdot 1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied rewrites 99.6%

      \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   6. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B}} \]

   7. Taylor expanded in F around 0

      \[\leadsto \frac{{\left(2 \cdot x + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]
   8. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto \frac{{\left(\left(x + x\right) + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      2. lift-+.f64 51.9

         \[\leadsto \frac{{\left(\left(x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B} \]

   9. Applied rewrites 51.9%

      \[\leadsto \frac{{\left(\left(x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B} \]

   if 1.15000000000000001e-38 < F

   1. Initial program 61.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\color{blue}{x \cdot \frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. lift-tan.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{x \cdot 1}}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      7. lift-tan.f64 62.0

         \[\leadsto \left(-\frac{x \cdot 1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied rewrites 62.0%

      \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   6. Applied rewrites 37.5%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B}} \]

   7. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 47.8%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 24: 47.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -210000:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{-38}:\\ \;\;\;\;\frac{{\left(\left(x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -210000.0)
   (/ (- -1.0 x) B)
   (if (<= F 1.15e-38)
     (/ (- (* (pow (+ (+ x x) 2.0) -0.5) F) x) B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -210000.0) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.15e-38) {
		tmp = ((pow(((x + x) + 2.0), -0.5) * F) - 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 <= (-210000.0d0)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.15d-38) then
        tmp = (((((x + x) + 2.0d0) ** (-0.5d0)) * f) - 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 <= -210000.0) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.15e-38) {
		tmp = ((Math.pow(((x + x) + 2.0), -0.5) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -210000.0:
		tmp = (-1.0 - x) / B
	elif F <= 1.15e-38:
		tmp = ((math.pow(((x + x) + 2.0), -0.5) * F) - x) / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -210000.0)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.15e-38)
		tmp = Float64(Float64(Float64((Float64(Float64(x + x) + 2.0) ^ -0.5) * F) - 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 <= -210000.0)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.15e-38)
		tmp = (((((x + x) + 2.0) ^ -0.5) * F) - x) / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -210000.0], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.15e-38], N[(N[(N[(N[Power[N[(N[(x + x), $MachinePrecision] + 2.0), $MachinePrecision], -0.5], $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 < -2.1e5

   1. Initial program 58.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\color{blue}{x \cdot \frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. lift-tan.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{x \cdot 1}}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      7. lift-tan.f64 58.2

         \[\leadsto \left(-\frac{x \cdot 1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied rewrites 58.2%

      \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   6. Applied rewrites 37.8%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B}} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 51.5%

      \[\leadsto \frac{-1 - x}{B} \]

   if -2.1e5 < F < 1.15000000000000001e-38

   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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\color{blue}{x \cdot \frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. lift-tan.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{x \cdot 1}}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      7. lift-tan.f64 99.6

         \[\leadsto \left(-\frac{x \cdot 1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied rewrites 99.6%

      \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   6. Applied rewrites 52.2%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B}} \]

   7. Taylor expanded in F around 0

      \[\leadsto \frac{{\left(2 \cdot x + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]
   8. Step-by-step derivation

      1. count-2-rev N/A

         \[\leadsto \frac{{\left(\left(x + x\right) + 2\right)}^{\frac{-1}{2}} \cdot F - x}{B} \]

      2. lift-+.f64 51.9

         \[\leadsto \frac{{\left(\left(x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B} \]

   9. Applied rewrites 51.9%

      \[\leadsto \frac{{\left(\left(x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B} \]

   if 1.15000000000000001e-38 < F

   1. Initial program 61.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\color{blue}{x \cdot \frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. lift-tan.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{x \cdot 1}}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      7. lift-tan.f64 62.0

         \[\leadsto \left(-\frac{x \cdot 1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied rewrites 62.0%

      \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   6. Applied rewrites 37.5%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B}} \]

   7. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 47.8%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 25: 43.8% accurate, 7.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{-59}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3e-54)
   (/ (- -1.0 x) B)
   (if (<= F 1.5e-59) (- (/ x B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3e-54) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.5e-59) {
		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 <= (-3d-54)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.5d-59) 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 <= -3e-54) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.5e-59) {
		tmp = -(x / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3e-54:
		tmp = (-1.0 - x) / B
	elif F <= 1.5e-59:
		tmp = -(x / B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3e-54)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.5e-59)
		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 <= -3e-54)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.5e-59)
		tmp = -(x / B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3e-54], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.5e-59], (-N[(x / B), $MachinePrecision]), N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.00000000000000009e-54

   1. Initial program 64.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\color{blue}{x \cdot \frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. lift-tan.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{x \cdot 1}}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      7. lift-tan.f64 64.0

         \[\leadsto \left(-\frac{x \cdot 1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied rewrites 64.0%

      \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   6. Applied rewrites 39.8%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B}} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 47.6%

      \[\leadsto \frac{-1 - x}{B} \]

   if -3.00000000000000009e-54 < F < 1.5e-59

   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. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{x \cdot \cos B}{\sin B} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{x \cdot \cos B}{\sin B} \]

      4. *-commutative N/A

         \[\leadsto -\frac{\cos B \cdot x}{\sin B} \]

      5. lower-*.f64 N/A

         \[\leadsto -\frac{\cos B \cdot x}{\sin B} \]

      6. lower-cos.f64 N/A

         \[\leadsto -\frac{\cos B \cdot x}{\sin B} \]

      7. lift-sin.f64 72.2

         \[\leadsto -\frac{\cos B \cdot x}{\sin B} \]

   4. Applied rewrites 72.2%

      \[\leadsto \color{blue}{-\frac{\cos B \cdot x}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto -\frac{x}{B} \]
   6. Step-by-step derivation

      1. lower-/.f64 38.1

         \[\leadsto -\frac{x}{B} \]

   7. Applied rewrites 38.1%

      \[\leadsto -\frac{x}{B} \]

   if 1.5e-59 < 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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\color{blue}{x \cdot \frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. lift-tan.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{x \cdot 1}}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      7. lift-tan.f64 63.8

         \[\leadsto \left(-\frac{x \cdot 1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied rewrites 63.8%

      \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   6. Applied rewrites 38.2%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B}} \]

   7. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 46.5%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 26: 36.6% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3e-54) (/ (- -1.0 x) B) (- (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3e-54) {
		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 <= (-3d-54)) 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 <= -3e-54) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -(x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3e-54:
		tmp = (-1.0 - x) / B
	else:
		tmp = -(x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3e-54)
		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 <= -3e-54)
		tmp = (-1.0 - x) / B;
	else
		tmp = -(x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3e-54], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], (-N[(x / B), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if F < -3.00000000000000009e-54

   1. Initial program 64.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\color{blue}{x \cdot \frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. lift-tan.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{x \cdot 1}}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      7. lift-tan.f64 64.0

         \[\leadsto \left(-\frac{x \cdot 1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied rewrites 64.0%

      \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   6. Applied rewrites 39.8%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B}} \]

   7. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 47.6%

      \[\leadsto \frac{-1 - x}{B} \]

   if -3.00000000000000009e-54 < F

   1. Initial program 82.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{x \cdot \cos B}{\sin B} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{x \cdot \cos B}{\sin B} \]

      4. *-commutative N/A

         \[\leadsto -\frac{\cos B \cdot x}{\sin B} \]

      5. lower-*.f64 N/A

         \[\leadsto -\frac{\cos B \cdot x}{\sin B} \]

      6. lower-cos.f64 N/A

         \[\leadsto -\frac{\cos B \cdot x}{\sin B} \]

      7. lift-sin.f64 60.2

         \[\leadsto -\frac{\cos B \cdot x}{\sin B} \]

   4. Applied rewrites 60.2%

      \[\leadsto \color{blue}{-\frac{\cos B \cdot x}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto -\frac{x}{B} \]
   6. Step-by-step derivation

      1. lower-/.f64 31.6

         \[\leadsto -\frac{x}{B} \]

   7. Applied rewrites 31.6%

      \[\leadsto -\frac{x}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 27: 29.7% accurate, 12.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+155}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (if (<= F -5e+155) (/ -1.0 B) (- (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5e+155) {
		tmp = -1.0 / 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 <= (-5d+155)) then
        tmp = (-1.0d0) / 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 <= -5e+155) {
		tmp = -1.0 / B;
	} else {
		tmp = -(x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5e+155:
		tmp = -1.0 / B
	else:
		tmp = -(x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5e+155)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(-Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5e+155)
		tmp = -1.0 / B;
	else
		tmp = -(x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5e+155], N[(-1.0 / B), $MachinePrecision], (-N[(x / B), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if F < -4.9999999999999999e155

   1. Initial program 30.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lift-sin.f64 51.0

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 51.0%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-1}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 27.7%

      \[\leadsto \frac{-1}{B} \]

   if -4.9999999999999999e155 < F

   1. Initial program 84.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x \cdot \cos B}{\sin B}\right) \]

      2. lower-neg.f64 N/A

         \[\leadsto -\frac{x \cdot \cos B}{\sin B} \]

      3. lower-/.f64 N/A

         \[\leadsto -\frac{x \cdot \cos B}{\sin B} \]

      4. *-commutative N/A

         \[\leadsto -\frac{\cos B \cdot x}{\sin B} \]

      5. lower-*.f64 N/A

         \[\leadsto -\frac{\cos B \cdot x}{\sin B} \]

      6. lower-cos.f64 N/A

         \[\leadsto -\frac{\cos B \cdot x}{\sin B} \]

      7. lift-sin.f64 56.7

         \[\leadsto -\frac{\cos B \cdot x}{\sin B} \]

   4. Applied rewrites 56.7%

      \[\leadsto \color{blue}{-\frac{\cos B \cdot x}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto -\frac{x}{B} \]
   6. Step-by-step derivation

      1. lower-/.f64 30.0

         \[\leadsto -\frac{x}{B} \]

   7. Applied rewrites 30.0%

      \[\leadsto -\frac{x}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 28: 17.4% accurate, 14.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 1300000000000:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 1300000000000.0) (/ -1.0 B) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 1300000000000.0) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= 1300000000000.0d0) then
        tmp = (-1.0d0) / b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 1300000000000.0) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 1300000000000.0:
		tmp = -1.0 / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 1300000000000.0)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 1300000000000.0)
		tmp = -1.0 / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 1300000000000.0], N[(-1.0 / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.3e12

   1. Initial program 84.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

      2. lift-sin.f64 22.9

         \[\leadsto \frac{-1}{\sin B} \]

   4. Applied rewrites 22.9%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-1}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 13.3%

      \[\leadsto \frac{-1}{B} \]

   if 1.3e12 < F

   1. Initial program 57.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\color{blue}{x \cdot \frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. lift-tan.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. associate-*r/ N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      6. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{x \cdot 1}}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      7. lift-tan.f64 57.2

         \[\leadsto \left(-\frac{x \cdot 1}{\color{blue}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Applied rewrites 57.2%

      \[\leadsto \left(-\color{blue}{\frac{x \cdot 1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} - x}{\color{blue}{B}} \]

   6. Applied rewrites 36.3%

      \[\leadsto \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, x + x\right) + 2\right)}^{-0.5} \cdot F - x}{B}} \]

   7. Taylor expanded in F around inf

      \[\leadsto \frac{1}{B} \]
   8. Step-by-step derivation

   9. Applied rewrites 28.3%

      \[\leadsto \frac{1}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 29: 10.4% accurate, 26.5× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 76.8%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in F around -inf

   \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{-1}{\color{blue}{\sin B}} \]

   2. lift-sin.f64 17.2

      \[\leadsto \frac{-1}{\sin B} \]

4. Applied rewrites 17.2%

   \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

5. Taylor expanded in B around 0

   \[\leadsto \frac{-1}{B} \]
6. Step-by-step derivation

7. Applied rewrites 10.4%

   \[\leadsto \frac{-1}{B} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025133 
(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))))))