VandenBroeck and Keller, Equation (23)

Percentage Accurate: 75.9% → 99.6%

Time: 6.8s

Alternatives: 25

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 75.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+38}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{F \cdot \sin B}, t\_0\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -2e+38)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F 1e+116)
       (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B)) t_0)
       (fma F (/ 1.0 (* F (sin B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -2e+38) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1e+116) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), t_0);
	} else {
		tmp = fma(F, (1.0 / (F * sin(B))), t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -2e+38)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1e+116)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), t_0);
	else
		tmp = fma(F, Float64(1.0 / Float64(F * sin(B))), t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2e+38], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e+116], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(F * N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.99999999999999995e38

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 55.3%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-lft-identity N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 55.3

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

   8. Applied rewrites 55.3%

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

   if -1.99999999999999995e38 < F < 1.00000000000000002e116

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   if 1.00000000000000002e116 < F

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.3

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

   6. Applied rewrites 52.3%

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

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.5 \cdot 10^{+32}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.25 \cdot 10^{+26}:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{F \cdot \sin B}, \frac{-x}{\tan B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.5e+32)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.25e+26)
       (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F (sin B))) t_0)
       (fma F (/ 1.0 (* F (sin B))) (/ (- x) (tan B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.5e+32) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.25e+26) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / sin(B))) - t_0;
	} else {
		tmp = fma(F, (1.0 / (F * sin(B))), (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.5e+32)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.25e+26)
		tmp = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / sin(B))) - t_0);
	else
		tmp = fma(F, Float64(1.0 / Float64(F * sin(B))), Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.5e+32], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.25e+26], N[(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] - t$95$0), $MachinePrecision], N[(F * N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5e32

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 55.3%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-lft-identity N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 55.3

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

   8. Applied rewrites 55.3%

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

   if -1.5e32 < F < 1.25e26

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 75.9

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

   3. Applied rewrites 76.1%

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

   if 1.25e26 < F

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.3

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

   6. Applied rewrites 52.3%

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

Alternative 3: 99.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.5 \cdot 10^{+32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{+126}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - \cos B \cdot x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{F \cdot \sin B}, \frac{-x}{\tan B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.5e+32)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 3.5e+126)
     (/ (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F) (* (cos B) x)) (sin B))
     (fma F (/ 1.0 (* F (sin B))) (/ (- x) (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.5e+32) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 3.5e+126) {
		tmp = ((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F) - (cos(B) * x)) / sin(B);
	} else {
		tmp = fma(F, (1.0 / (F * sin(B))), (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.5e+32)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 3.5e+126)
		tmp = Float64(Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F) - Float64(cos(B) * x)) / sin(B));
	else
		tmp = fma(F, Float64(1.0 / Float64(F * sin(B))), Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.5e+32], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.5e+126], N[(N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(F * N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.5e32

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 55.3%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-lft-identity N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 55.3

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

   8. Applied rewrites 55.3%

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

   if -1.5e32 < F < 3.5000000000000003e126

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   5. Applied rewrites 84.2%

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

   if 3.5000000000000003e126 < F

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.3

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

   6. Applied rewrites 52.3%

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

Alternative 4: 97.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -2.05 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5}}{\sin B}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{F \cdot \sin B}, t\_0\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -2.05e-12)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F 2.5e-28)
       (fma F (/ (pow (fma 2.0 x 2.0) -0.5) (sin B)) t_0)
       (fma F (/ 1.0 (* F (sin B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -2.05e-12) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 2.5e-28) {
		tmp = fma(F, (pow(fma(2.0, x, 2.0), -0.5) / sin(B)), t_0);
	} else {
		tmp = fma(F, (1.0 / (F * sin(B))), t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -2.05e-12)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 2.5e-28)
		tmp = fma(F, Float64((fma(2.0, x, 2.0) ^ -0.5) / sin(B)), t_0);
	else
		tmp = fma(F, Float64(1.0 / Float64(F * sin(B))), t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.05e-12], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.5e-28], N[(F * N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], N[(F * N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.04999999999999995e-12

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 55.3%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-lft-identity N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 55.3

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

   8. Applied rewrites 55.3%

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

   if -2.04999999999999995e-12 < F < 2.5000000000000001e-28

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around 0

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

   6. Applied rewrites 56.1%

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

   if 2.5000000000000001e-28 < F

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.3

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

   6. Applied rewrites 52.3%

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

Alternative 5: 97.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -2.05 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-28}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, 2\right)\right)}^{-0.5} \cdot F - \cos B \cdot x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{F \cdot \sin B}, \frac{-x}{\tan B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.05e-12)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 2.5e-28)
     (/ (- (* (pow (fma x 2.0 2.0) -0.5) F) (* (cos B) x)) (sin B))
     (fma F (/ 1.0 (* F (sin B))) (/ (- x) (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.05e-12) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 2.5e-28) {
		tmp = ((pow(fma(x, 2.0, 2.0), -0.5) * F) - (cos(B) * x)) / sin(B);
	} else {
		tmp = fma(F, (1.0 / (F * sin(B))), (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.05e-12)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 2.5e-28)
		tmp = Float64(Float64(Float64((fma(x, 2.0, 2.0) ^ -0.5) * F) - Float64(cos(B) * x)) / sin(B));
	else
		tmp = fma(F, Float64(1.0 / Float64(F * sin(B))), Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.05e-12], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.5e-28], N[(N[(N[(N[Power[N[(x * 2.0 + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(F * N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.04999999999999995e-12

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 55.3%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-lft-identity N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 55.3

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

   8. Applied rewrites 55.3%

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

   if -2.04999999999999995e-12 < F < 2.5000000000000001e-28

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 55.8%

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

   if 2.5000000000000001e-28 < F

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.3

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

   6. Applied rewrites 52.3%

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

Alternative 6: 90.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -6 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B}, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{1}{F \cdot \sin B}, t\_0\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -6e-12)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F 2.3e+24)
       (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) B) t_0)
       (fma F (/ 1.0 (* F (sin B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -6e-12) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 2.3e+24) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / B), t_0);
	} else {
		tmp = fma(F, (1.0 / (F * sin(B))), t_0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -6e-12)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 2.3e+24)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / B), t_0);
	else
		tmp = fma(F, Float64(1.0 / Float64(F * sin(B))), t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -6e-12], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.3e+24], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / B), $MachinePrecision] + t$95$0), $MachinePrecision], N[(F * N[(1.0 / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.0000000000000003e-12

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 55.3%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-lft-identity N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 55.3

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

   8. Applied rewrites 55.3%

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

   if -6.0000000000000003e-12 < F < 2.2999999999999999e24

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 69.4%

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

   if 2.2999999999999999e24 < F

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sin.f64 52.3

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

   6. Applied rewrites 52.3%

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

Alternative 7: 90.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -6 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 9 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{F} \cdot F - \cos B \cdot x}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6e-12)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 9e-25)
     (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) B) (/ (- x) (tan B)))
     (/ (- (* (/ 1.0 F) F) (* (cos B) x)) (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6e-12) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 9e-25) {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / B), (-x / tan(B)));
	} else {
		tmp = (((1.0 / F) * F) - (cos(B) * x)) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6e-12)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 9e-25)
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / B), Float64(Float64(-x) / tan(B)));
	else
		tmp = Float64(Float64(Float64(Float64(1.0 / F) * F) - Float64(cos(B) * x)) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6e-12], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9e-25], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / B), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(1.0 / F), $MachinePrecision] * F), $MachinePrecision] - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.0000000000000003e-12

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 55.3%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-lft-identity N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 55.3

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

   8. Applied rewrites 55.3%

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

   if -6.0000000000000003e-12 < F < 9.0000000000000002e-25

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 69.4%

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

   if 9.0000000000000002e-25 < F

   1. Initial program 75.9%

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

   2. Taylor expanded in F around inf

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

      1. lower-/.f64 47.6

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

   4. Applied rewrites 47.6%

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

      1. metadata-eval 47.6

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

      2. metadata-eval 47.6

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-neg.f64 N/A

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

      6. sub-flip-reverse N/A

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

   6. Applied rewrites 56.0%

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

Alternative 8: 79.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -6 \cdot 10^{-12}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B}, \frac{-x}{\tan B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6e-12)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) B) (/ (- x) (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6e-12) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / B), (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6e-12)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	else
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / B), Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6e-12], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / B), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -6.0000000000000003e-12

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 55.3%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. lower--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. lower-/.f64 N/A

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

      7. *-lft-identity N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac2 N/A

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

      10. lift-neg.f64 N/A

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

      11. frac-2neg N/A

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

      12. lower-/.f64 55.3

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

   8. Applied rewrites 55.3%

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

   if -6.0000000000000003e-12 < F

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 69.4%

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

Alternative 9: 75.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -6.8 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, -1, -1 \cdot \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq -5.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{B}, \frac{-x}{\tan B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.8e+148)
   (fma (/ 1.0 (sin B)) -1.0 (* -1.0 (/ x B)))
   (if (<= F -5.5e-160)
     (/ (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F) x) (sin B))
     (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) B) (/ (- x) (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.8e+148) {
		tmp = fma((1.0 / sin(B)), -1.0, (-1.0 * (x / B)));
	} else if (F <= -5.5e-160) {
		tmp = ((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F) - x) / sin(B);
	} else {
		tmp = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / B), (-x / tan(B)));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.8e+148)
		tmp = fma(Float64(1.0 / sin(B)), -1.0, Float64(-1.0 * Float64(x / B)));
	elseif (F <= -5.5e-160)
		tmp = Float64(Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F) - x) / sin(B));
	else
		tmp = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / B), Float64(Float64(-x) / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.8e+148], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * -1.0 + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -5.5e-160], N[(N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / B), $MachinePrecision] + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.8000000000000006e148

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 55.3%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.1

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

   9. Applied rewrites 36.1%

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

   if -6.8000000000000006e148 < F < -5.5e-160

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 59.4%

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

   if -5.5e-160 < F

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 69.4%

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

Alternative 10: 74.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;x \leq -8 \cdot 10^{-117}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot F - x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= x -8e-117)
     t_0
     (if (<= x 8.6e-14)
       (/ (- (* (pow (fma x 2.0 (fma F F 2.0)) -0.5) F) x) (sin B))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (x <= -8e-117) {
		tmp = t_0;
	} else if (x <= 8.6e-14) {
		tmp = ((pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) * F) - x) / sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (x <= -8e-117)
		tmp = t_0;
	elseif (x <= 8.6e-14)
		tmp = Float64(Float64(Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) * F) - x) / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8e-117], t$95$0, If[LessEqual[x, 8.6e-14], N[(N[(N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -8.00000000000000024e-117 or 8.59999999999999996e-14 < x

   1. Initial program 75.9%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

   6. Applied rewrites 55.4%

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

   if -8.00000000000000024e-117 < x < 8.59999999999999996e-14

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   5. Applied rewrites 84.2%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 59.4%

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

Alternative 11: 70.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -2.05 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, -1 \cdot \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq -1.7 \cdot 10^{-54}:\\ \;\;\;\;\frac{{2}^{-0.5}}{\sin B} \cdot F\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{+132}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ (- x) (tan B))))
   (if (<= F -2.05e-12)
     (fma t_0 -1.0 (* -1.0 (/ x B)))
     (if (<= F -1.7e-54)
       (* (/ (pow 2.0 -0.5) (sin B)) F)
       (if (<= F 2.35e+21) t_1 (if (<= F 1.05e+132) t_0 t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = -x / tan(B);
	double tmp;
	if (F <= -2.05e-12) {
		tmp = fma(t_0, -1.0, (-1.0 * (x / B)));
	} else if (F <= -1.7e-54) {
		tmp = (pow(2.0, -0.5) / sin(B)) * F;
	} else if (F <= 2.35e+21) {
		tmp = t_1;
	} else if (F <= 1.05e+132) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -2.05e-12)
		tmp = fma(t_0, -1.0, Float64(-1.0 * Float64(x / B)));
	elseif (F <= -1.7e-54)
		tmp = Float64(Float64((2.0 ^ -0.5) / sin(B)) * F);
	elseif (F <= 2.35e+21)
		tmp = t_1;
	elseif (F <= 1.05e+132)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.05e-12], N[(t$95$0 * -1.0 + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.7e-54], N[(N[(N[Power[2.0, -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision], If[LessEqual[F, 2.35e+21], t$95$1, If[LessEqual[F, 1.05e+132], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.04999999999999995e-12

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 55.3%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.1

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

   9. Applied rewrites 36.1%

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

   if -2.04999999999999995e-12 < F < -1.69999999999999994e-54

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{\sin B}} \]
   5. 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. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-sin.f64 30.3

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

   6. Applied rewrites 30.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 30.3

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-fma.f64 30.3

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

   8. Applied rewrites 30.3%

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

   9. Taylor expanded in F around 0

      \[\leadsto \frac{{2}^{\frac{-1}{2}}}{\sin B} \cdot F \]
   10. Step-by-step derivation

   11. Applied rewrites 19.6%

       \[\leadsto \frac{{2}^{-0.5}}{\sin B} \cdot F \]

   if -1.69999999999999994e-54 < F < 2.35e21 or 1.04999999999999997e132 < F

   1. Initial program 75.9%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

   6. Applied rewrites 55.4%

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

   if 2.35e21 < F < 1.04999999999999997e132

   1. Initial program 75.9%

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

   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. lower-sin.f64 17.4

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

   4. Applied rewrites 17.4%

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

Alternative 12: 68.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;x \leq -8 \cdot 10^{-120}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-38}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}{\sin B} \cdot F\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= x -8e-120)
     t_0
     (if (<= x 1.35e-38) (* (/ (pow (fma F F 2.0) -0.5) (sin B)) F) t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (x <= -8e-120) {
		tmp = t_0;
	} else if (x <= 1.35e-38) {
		tmp = (pow(fma(F, F, 2.0), -0.5) / sin(B)) * F;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (x <= -8e-120)
		tmp = t_0;
	elseif (x <= 1.35e-38)
		tmp = Float64(Float64((fma(F, F, 2.0) ^ -0.5) / sin(B)) * F);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -8e-120], t$95$0, If[LessEqual[x, 1.35e-38], N[(N[(N[Power[N[(F * F + 2.0), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -7.99999999999999983e-120 or 1.35000000000000003e-38 < x

   1. Initial program 75.9%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

   6. Applied rewrites 55.4%

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

   if -7.99999999999999983e-120 < x < 1.35000000000000003e-38

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{\sin B}} \]
   5. 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. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-sin.f64 30.3

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

   6. Applied rewrites 30.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 30.3

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

      7. lift-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-pow.f64 N/A

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

      10. pow2 N/A

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

      11. lift-fma.f64 30.3

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

   8. Applied rewrites 30.3%

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

Alternative 13: 64.9% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ t_1 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -2.2 \cdot 10^{-55}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, -1, -1 \cdot \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{+132}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))) (t_1 (/ 1.0 (sin B))))
   (if (<= F -2.2e-55)
     (fma t_1 -1.0 (* -1.0 (/ x B)))
     (if (<= F 2.35e+21) t_0 (if (<= F 1.05e+132) t_1 t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double t_1 = 1.0 / sin(B);
	double tmp;
	if (F <= -2.2e-55) {
		tmp = fma(t_1, -1.0, (-1.0 * (x / B)));
	} else if (F <= 2.35e+21) {
		tmp = t_0;
	} else if (F <= 1.05e+132) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	t_1 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -2.2e-55)
		tmp = fma(t_1, -1.0, Float64(-1.0 * Float64(x / B)));
	elseif (F <= 2.35e+21)
		tmp = t_0;
	elseif (F <= 1.05e+132)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.2e-55], N[(t$95$1 * -1.0 + N[(-1.0 * N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.35e+21], t$95$0, If[LessEqual[F, 1.05e+132], t$95$1, t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.2e-55

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. mult-flip N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in F around -inf

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

   6. Applied rewrites 55.3%

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

   7. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 36.1

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

   9. Applied rewrites 36.1%

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

   if -2.2e-55 < F < 2.35e21 or 1.04999999999999997e132 < F

   1. Initial program 75.9%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

   6. Applied rewrites 55.4%

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

   if 2.35e21 < F < 1.04999999999999997e132

   1. Initial program 75.9%

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

   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. lower-sin.f64 17.4

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

   4. Applied rewrites 17.4%

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

Alternative 14: 63.7% accurate, 1.9× speedup

Math

\[\mathsf{copysign}\left(1, B\right) \cdot \begin{array}{l} \mathbf{if}\;\left|B\right| \leq 9 \cdot 10^{-18}:\\ \;\;\;\;\frac{F \cdot {\left(2 + \mathsf{fma}\left(2, x, {F}^{2}\right)\right)}^{-0.5} - x}{\left|B\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan \left(\left|B\right|\right)}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (*
  (copysign 1.0 B)
  (if (<= (fabs B) 9e-18)
    (/ (- (* F (pow (+ 2.0 (fma 2.0 x (pow F 2.0))) -0.5)) x) (fabs B))
    (/ (- x) (tan (fabs B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (fabs(B) <= 9e-18) {
		tmp = ((F * pow((2.0 + fma(2.0, x, pow(F, 2.0))), -0.5)) - x) / fabs(B);
	} else {
		tmp = -x / tan(fabs(B));
	}
	return copysign(1.0, B) * tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (abs(B) <= 9e-18)
		tmp = Float64(Float64(Float64(F * (Float64(2.0 + fma(2.0, x, (F ^ 2.0))) ^ -0.5)) - x) / abs(B));
	else
		tmp = Float64(Float64(-x) / tan(abs(B)));
	end
	return Float64(copysign(1.0, B) * tmp)
end

Wolfram

code[F_, B_, x_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[B]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[B], $MachinePrecision], 9e-18], N[(N[(N[(F * N[Power[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[Abs[B], $MachinePrecision]), $MachinePrecision], N[((-x) / N[Tan[N[Abs[B], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if B < 8.99999999999999987e-18

   1. Initial program 75.9%

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

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + \left(2 \cdot x + {F}^{2}\right)\right)}^{\frac{-1}{2}} - x}{B}} \]
   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 43.4%

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

   if 8.99999999999999987e-18 < B

   1. Initial program 75.9%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

   6. Applied rewrites 55.4%

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

Alternative 15: 57.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -1.7 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{+132}:\\ \;\;\;\;\frac{1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- x) (tan B))))
   (if (<= F -1.7e-54)
     (/ -1.0 (sin B))
     (if (<= F 2.35e+21) t_0 (if (<= F 1.05e+132) (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -x / tan(B);
	double tmp;
	if (F <= -1.7e-54) {
		tmp = -1.0 / sin(B);
	} else if (F <= 2.35e+21) {
		tmp = t_0;
	} else if (F <= 1.05e+132) {
		tmp = 1.0 / sin(B);
	} else {
		tmp = t_0;
	}
	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 / tan(b)
    if (f <= (-1.7d-54)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 2.35d+21) then
        tmp = t_0
    else if (f <= 1.05d+132) then
        tmp = 1.0d0 / sin(b)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -x / Math.tan(B);
	double tmp;
	if (F <= -1.7e-54) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 2.35e+21) {
		tmp = t_0;
	} else if (F <= 1.05e+132) {
		tmp = 1.0 / Math.sin(B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -x / math.tan(B)
	tmp = 0
	if F <= -1.7e-54:
		tmp = -1.0 / math.sin(B)
	elif F <= 2.35e+21:
		tmp = t_0
	elif F <= 1.05e+132:
		tmp = 1.0 / math.sin(B)
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -1.7e-54)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 2.35e+21)
		tmp = t_0;
	elseif (F <= 1.05e+132)
		tmp = Float64(1.0 / sin(B));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -x / tan(B);
	tmp = 0.0;
	if (F <= -1.7e-54)
		tmp = -1.0 / sin(B);
	elseif (F <= 2.35e+21)
		tmp = t_0;
	elseif (F <= 1.05e+132)
		tmp = 1.0 / sin(B);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.7e-54], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.35e+21], t$95$0, If[LessEqual[F, 1.05e+132], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.69999999999999994e-54

   1. Initial program 75.9%

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

   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. lower-sin.f64 17.2

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

   4. Applied rewrites 17.2%

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

   if -1.69999999999999994e-54 < F < 2.35e21 or 1.04999999999999997e132 < F

   1. Initial program 75.9%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

   6. Applied rewrites 55.4%

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

   if 2.35e21 < F < 1.04999999999999997e132

   1. Initial program 75.9%

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

   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. lower-sin.f64 17.4

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

   4. Applied rewrites 17.4%

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

Alternative 16: 45.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.15 \cdot 10^{-54}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{+21}:\\ \;\;\;\;-1 \cdot \frac{x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.15e-54)
   (/ -1.0 (sin B))
   (if (<= F 2.35e+21) (* -1.0 (/ x (sin B))) (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.15e-54) {
		tmp = -1.0 / sin(B);
	} else if (F <= 2.35e+21) {
		tmp = -1.0 * (x / sin(B));
	} else {
		tmp = 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) :: tmp
    if (f <= (-1.15d-54)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 2.35d+21) then
        tmp = (-1.0d0) * (x / sin(b))
    else
        tmp = 1.0d0 / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.15e-54) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 2.35e+21) {
		tmp = -1.0 * (x / Math.sin(B));
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.15e-54:
		tmp = -1.0 / math.sin(B)
	elif F <= 2.35e+21:
		tmp = -1.0 * (x / math.sin(B))
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.15e-54)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 2.35e+21)
		tmp = Float64(-1.0 * Float64(x / sin(B)));
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.15e-54)
		tmp = -1.0 / sin(B);
	elseif (F <= 2.35e+21)
		tmp = -1.0 * (x / sin(B));
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.15e-54], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.35e+21], N[(-1.0 * N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.1499999999999999e-54

   1. Initial program 75.9%

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

   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. lower-sin.f64 17.2

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

   4. Applied rewrites 17.2%

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

   if -1.1499999999999999e-54 < F < 2.35e21

   1. Initial program 75.9%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 31.6%

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

   if 2.35e21 < F

   1. Initial program 75.9%

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

   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. lower-sin.f64 17.4

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

   4. Applied rewrites 17.4%

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

Alternative 17: 44.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{+21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, 0.3333333333333333 \cdot \left({B}^{2} \cdot x\right)\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.8e-76)
   (/ -1.0 (sin B))
   (if (<= F 2.35e+21)
     (/ (fma -1.0 x (* 0.3333333333333333 (* (pow B 2.0) x))) B)
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.8e-76) {
		tmp = -1.0 / sin(B);
	} else if (F <= 2.35e+21) {
		tmp = fma(-1.0, x, (0.3333333333333333 * (pow(B, 2.0) * x))) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.8e-76)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 2.35e+21)
		tmp = Float64(fma(-1.0, x, Float64(0.3333333333333333 * Float64((B ^ 2.0) * x))) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.8e-76], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.35e+21], N[(N[(-1.0 * x + N[(0.3333333333333333 * N[(N[Power[B, 2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.8e-76

   1. Initial program 75.9%

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

   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. lower-sin.f64 17.2

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

   4. Applied rewrites 17.2%

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

   if -1.8e-76 < F < 2.35e21

   1. Initial program 75.9%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

   6. Applied rewrites 55.4%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{1}{3} \cdot \left({B}^{2} \cdot x\right)\right)}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{1}{3} \cdot \left({B}^{2} \cdot x\right)\right)}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{1}{3} \cdot \left({B}^{2} \cdot x\right)\right)}{B} \]

      5. lower-pow.f64 29.3

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

   9. Applied rewrites 29.3%

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

   if 2.35e21 < F

   1. Initial program 75.9%

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

   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. lower-sin.f64 17.4

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

   4. Applied rewrites 17.4%

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

Alternative 18: 36.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -1.8 \cdot 10^{-76}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{+21}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, 0.3333333333333333 \cdot \left({B}^{2} \cdot x\right)\right)}{B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{+132}:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.8e-76)
   (/ -1.0 (sin B))
   (if (<= F 2.35e+21)
     (/ (fma -1.0 x (* 0.3333333333333333 (* (pow B 2.0) x))) B)
     (if (<= F 1.1e+132) (/ 1.0 B) (- (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.8e-76) {
		tmp = -1.0 / sin(B);
	} else if (F <= 2.35e+21) {
		tmp = fma(-1.0, x, (0.3333333333333333 * (pow(B, 2.0) * x))) / B;
	} else if (F <= 1.1e+132) {
		tmp = 1.0 / B;
	} else {
		tmp = -(x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.8e-76)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 2.35e+21)
		tmp = Float64(fma(-1.0, x, Float64(0.3333333333333333 * Float64((B ^ 2.0) * x))) / B);
	elseif (F <= 1.1e+132)
		tmp = Float64(1.0 / B);
	else
		tmp = Float64(-Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.8e-76], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.35e+21], N[(N[(-1.0 * x + N[(0.3333333333333333 * N[(N[Power[B, 2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.1e+132], N[(1.0 / B), $MachinePrecision], (-N[(x / B), $MachinePrecision])]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.8e-76

   1. Initial program 75.9%

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

   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. lower-sin.f64 17.2

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

   4. Applied rewrites 17.2%

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

   if -1.8e-76 < F < 2.35e21

   1. Initial program 75.9%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

   6. Applied rewrites 55.4%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{1}{3} \cdot \left({B}^{2} \cdot x\right)\right)}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{1}{3} \cdot \left({B}^{2} \cdot x\right)\right)}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{1}{3} \cdot \left({B}^{2} \cdot x\right)\right)}{B} \]

      5. lower-pow.f64 29.3

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

   9. Applied rewrites 29.3%

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

   if 2.35e21 < F < 1.09999999999999994e132

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{\sin B}} \]
   5. 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. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-sin.f64 30.3

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

   6. Applied rewrites 30.3%

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

   7. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 16.2

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{-0.5}}{B} \]

   9. Applied rewrites 16.2%

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

   10. Taylor expanded in F around inf

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

       1. lower-/.f64 10.3

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

   12. Applied rewrites 10.3%

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

   if 1.09999999999999994e132 < F

   1. Initial program 75.9%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 29.4

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

   7. Applied rewrites 29.4%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 29.4

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

   9. Applied rewrites 29.4%

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

Alternative 19: 36.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-123}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, 0.3333333333333333 \cdot \left({B}^{2} \cdot x\right)\right)}{B}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-154}:\\ \;\;\;\;\left({\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5} \cdot F\right) \cdot \frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -1e-123)
   (/ (fma -1.0 x (* 0.3333333333333333 (* (pow B 2.0) x))) B)
   (if (<= x 4.6e-154)
     (* (* (pow (fma F F 2.0) -0.5) F) (/ 1.0 B))
     (- (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -1e-123) {
		tmp = fma(-1.0, x, (0.3333333333333333 * (pow(B, 2.0) * x))) / B;
	} else if (x <= 4.6e-154) {
		tmp = (pow(fma(F, F, 2.0), -0.5) * F) * (1.0 / B);
	} else {
		tmp = -(x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -1e-123)
		tmp = Float64(fma(-1.0, x, Float64(0.3333333333333333 * Float64((B ^ 2.0) * x))) / B);
	elseif (x <= 4.6e-154)
		tmp = Float64(Float64((fma(F, F, 2.0) ^ -0.5) * F) * Float64(1.0 / B));
	else
		tmp = Float64(-Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -1e-123], N[(N[(-1.0 * x + N[(0.3333333333333333 * N[(N[Power[B, 2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[x, 4.6e-154], N[(N[(N[Power[N[(F * F + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * F), $MachinePrecision] * N[(1.0 / B), $MachinePrecision]), $MachinePrecision], (-N[(x / B), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if x < -1.0000000000000001e-123

   1. Initial program 75.9%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

   6. Applied rewrites 55.4%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{1}{3} \cdot \left({B}^{2} \cdot x\right)\right)}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{1}{3} \cdot \left({B}^{2} \cdot x\right)\right)}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{1}{3} \cdot \left({B}^{2} \cdot x\right)\right)}{B} \]

      5. lower-pow.f64 29.3

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

   9. Applied rewrites 29.3%

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

   if -1.0000000000000001e-123 < x < 4.5999999999999999e-154

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{\sin B}} \]
   5. 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. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-sin.f64 30.3

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

   6. Applied rewrites 30.3%

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

   7. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 16.2

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{-0.5}}{B} \]

   9. Applied rewrites 16.2%

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

       1. lift-/.f64 N/A

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

       2. mult-flip N/A

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

       3. lower-*.f64 N/A

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

       4. lift-*.f64 N/A

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

       5. *-commutative N/A

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

       6. lift-+.f64 N/A

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

       7. +-commutative N/A

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

       8. lift-pow.f64 N/A

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

       9. pow2 N/A

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

       10. lift-fma.f64 N/A

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

       11. lower-*.f64 N/A

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

       12. lower-/.f64 16.2

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

   11. Applied rewrites 16.2%

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

   if 4.5999999999999999e-154 < x

   1. Initial program 75.9%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 29.4

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

   7. Applied rewrites 29.4%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 29.4

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

   9. Applied rewrites 29.4%

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

Alternative 20: 36.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-123}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, 0.3333333333333333 \cdot \left({B}^{2} \cdot x\right)\right)}{B}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-154}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}{B} \cdot F\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -1e-123)
   (/ (fma -1.0 x (* 0.3333333333333333 (* (pow B 2.0) x))) B)
   (if (<= x 4.6e-154) (* (/ (pow (fma F F 2.0) -0.5) B) F) (- (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -1e-123) {
		tmp = fma(-1.0, x, (0.3333333333333333 * (pow(B, 2.0) * x))) / B;
	} else if (x <= 4.6e-154) {
		tmp = (pow(fma(F, F, 2.0), -0.5) / B) * F;
	} else {
		tmp = -(x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -1e-123)
		tmp = Float64(fma(-1.0, x, Float64(0.3333333333333333 * Float64((B ^ 2.0) * x))) / B);
	elseif (x <= 4.6e-154)
		tmp = Float64(Float64((fma(F, F, 2.0) ^ -0.5) / B) * F);
	else
		tmp = Float64(-Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -1e-123], N[(N[(-1.0 * x + N[(0.3333333333333333 * N[(N[Power[B, 2.0], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[x, 4.6e-154], N[(N[(N[Power[N[(F * F + 2.0), $MachinePrecision], -0.5], $MachinePrecision] / B), $MachinePrecision] * F), $MachinePrecision], (-N[(x / B), $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if x < -1.0000000000000001e-123

   1. Initial program 75.9%

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

   2. Taylor expanded in F around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-cos.f64 N/A

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

      5. lower-sin.f64 55.4

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

   4. Applied rewrites 55.4%

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

   6. Applied rewrites 55.4%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{1}{3} \cdot \left({B}^{2} \cdot x\right)\right)}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{1}{3} \cdot \left({B}^{2} \cdot x\right)\right)}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(-1, x, \frac{1}{3} \cdot \left({B}^{2} \cdot x\right)\right)}{B} \]

      5. lower-pow.f64 29.3

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

   9. Applied rewrites 29.3%

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

   if -1.0000000000000001e-123 < x < 4.5999999999999999e-154

   1. Initial program 75.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. associate-*l* N/A

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

      7. lower-fma.f64 N/A

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

   3. Applied rewrites 84.3%

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

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{\sin B}} \]
   5. 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. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-pow.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-sin.f64 30.3

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

   6. Applied rewrites 30.3%

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

   7. Taylor expanded in B around 0

      \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      8. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      9. metadata-eval 16.2

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{-0.5}}{B} \]

   9. Applied rewrites 16.2%

      \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{-0.5}}{\color{blue}{B}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{B} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{B} \]

       3. associate-/l* N/A

          \[\leadsto F \cdot \frac{{\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{\color{blue}{B}} \]

       4. *-commutative N/A

          \[\leadsto \frac{{\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{B} \cdot F \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{{\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{B} \cdot F \]

       6. lift-+.f64 N/A

          \[\leadsto \frac{{\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{B} \cdot F \]

       7. +-commutative N/A

          \[\leadsto \frac{{\left({F}^{2} + 2\right)}^{\frac{-1}{2}}}{B} \cdot F \]

       8. lift-pow.f64 N/A

          \[\leadsto \frac{{\left({F}^{2} + 2\right)}^{\frac{-1}{2}}}{B} \cdot F \]

       9. pow2 N/A

          \[\leadsto \frac{{\left(F \cdot F + 2\right)}^{\frac{-1}{2}}}{B} \cdot F \]

       10. lift-fma.f64 N/A

           \[\leadsto \frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}}}{B} \cdot F \]

       11. lower-/.f64 16.2

           \[\leadsto \frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}{B} \cdot F \]

   11. Applied rewrites 16.2%

       \[\leadsto \frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}{B} \cdot \color{blue}{F} \]

   if 4.5999999999999999e-154 < x

   1. Initial program 75.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.4

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 29.4

         \[\leadsto -1 \cdot \frac{x}{B} \]

   7. Applied rewrites 29.4%

      \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x}{B}\right) \]

      3. lower-neg.f64 29.4

         \[\leadsto -\frac{x}{B} \]

   9. Applied rewrites 29.4%

      \[\leadsto -\frac{x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 21: 36.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;x \leq -1 \cdot 10^{-123}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-154}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}{B} \cdot F\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x B))))
   (if (<= x -1e-123)
     t_0
     (if (<= x 4.6e-154) (* (/ (pow (fma F F 2.0) -0.5) B) F) t_0))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (x <= -1e-123) {
		tmp = t_0;
	} else if (x <= 4.6e-154) {
		tmp = (pow(fma(F, F, 2.0), -0.5) / B) * F;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	tmp = 0.0
	if (x <= -1e-123)
		tmp = t_0;
	elseif (x <= 4.6e-154)
		tmp = Float64(Float64((fma(F, F, 2.0) ^ -0.5) / B) * F);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[x, -1e-123], t$95$0, If[LessEqual[x, 4.6e-154], N[(N[(N[Power[N[(F * F + 2.0), $MachinePrecision], -0.5], $MachinePrecision] / B), $MachinePrecision] * F), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0000000000000001e-123 or 4.5999999999999999e-154 < x

   1. Initial program 75.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.4

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 29.4

         \[\leadsto -1 \cdot \frac{x}{B} \]

   7. Applied rewrites 29.4%

      \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x}{B}\right) \]

      3. lower-neg.f64 29.4

         \[\leadsto -\frac{x}{B} \]

   9. Applied rewrites 29.4%

      \[\leadsto -\frac{x}{B} \]

   if -1.0000000000000001e-123 < x < 4.5999999999999999e-154

   1. Initial program 75.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. mult-flip N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right)} \]

   3. Applied rewrites 84.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{\sin B}} \]
   5. 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. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin \color{blue}{B}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      8. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{\sin B} \]

      10. lower-sin.f64 30.3

          \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{-0.5}}{\sin B} \]

   6. Applied rewrites 30.3%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + {F}^{2}\right)}^{-0.5}}{\sin B}} \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      8. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      9. metadata-eval 16.2

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{-0.5}}{B} \]

   9. Applied rewrites 16.2%

      \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{-0.5}}{\color{blue}{B}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

          \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{B} \]

       2. lift-*.f64 N/A

          \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{B} \]

       3. associate-/l* N/A

          \[\leadsto F \cdot \frac{{\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{\color{blue}{B}} \]

       4. *-commutative N/A

          \[\leadsto \frac{{\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{B} \cdot F \]

       5. lower-*.f64 N/A

          \[\leadsto \frac{{\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{B} \cdot F \]

       6. lift-+.f64 N/A

          \[\leadsto \frac{{\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{B} \cdot F \]

       7. +-commutative N/A

          \[\leadsto \frac{{\left({F}^{2} + 2\right)}^{\frac{-1}{2}}}{B} \cdot F \]

       8. lift-pow.f64 N/A

          \[\leadsto \frac{{\left({F}^{2} + 2\right)}^{\frac{-1}{2}}}{B} \cdot F \]

       9. pow2 N/A

          \[\leadsto \frac{{\left(F \cdot F + 2\right)}^{\frac{-1}{2}}}{B} \cdot F \]

       10. lift-fma.f64 N/A

           \[\leadsto \frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{\frac{-1}{2}}}{B} \cdot F \]

       11. lower-/.f64 16.2

           \[\leadsto \frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}{B} \cdot F \]

   11. Applied rewrites 16.2%

       \[\leadsto \frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}{B} \cdot \color{blue}{F} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 29.4% accurate, 3.9× speedup

Math

\[\begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;F \leq -5.2 \cdot 10^{+148}:\\ \;\;\;\;\frac{-0.16666666666666666 \cdot {B}^{2} - 1}{B}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{+132}:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x B))))
   (if (<= F -5.2e+148)
     (/ (- (* -0.16666666666666666 (pow B 2.0)) 1.0) B)
     (if (<= F 2.35e+21) t_0 (if (<= F 1.1e+132) (/ 1.0 B) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (F <= -5.2e+148) {
		tmp = ((-0.16666666666666666 * pow(B, 2.0)) - 1.0) / B;
	} else if (F <= 2.35e+21) {
		tmp = t_0;
	} else if (F <= 1.1e+132) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	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)
    if (f <= (-5.2d+148)) then
        tmp = (((-0.16666666666666666d0) * (b ** 2.0d0)) - 1.0d0) / b
    else if (f <= 2.35d+21) then
        tmp = t_0
    else if (f <= 1.1d+132) then
        tmp = 1.0d0 / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (F <= -5.2e+148) {
		tmp = ((-0.16666666666666666 * Math.pow(B, 2.0)) - 1.0) / B;
	} else if (F <= 2.35e+21) {
		tmp = t_0;
	} else if (F <= 1.1e+132) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -(x / B)
	tmp = 0
	if F <= -5.2e+148:
		tmp = ((-0.16666666666666666 * math.pow(B, 2.0)) - 1.0) / B
	elif F <= 2.35e+21:
		tmp = t_0
	elif F <= 1.1e+132:
		tmp = 1.0 / B
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	tmp = 0.0
	if (F <= -5.2e+148)
		tmp = Float64(Float64(Float64(-0.16666666666666666 * (B ^ 2.0)) - 1.0) / B);
	elseif (F <= 2.35e+21)
		tmp = t_0;
	elseif (F <= 1.1e+132)
		tmp = Float64(1.0 / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -(x / B);
	tmp = 0.0;
	if (F <= -5.2e+148)
		tmp = ((-0.16666666666666666 * (B ^ 2.0)) - 1.0) / B;
	elseif (F <= 2.35e+21)
		tmp = t_0;
	elseif (F <= 1.1e+132)
		tmp = 1.0 / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[F, -5.2e+148], N[(N[(N[(-0.16666666666666666 * N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 2.35e+21], t$95$0, If[LessEqual[F, 1.1e+132], N[(1.0 / B), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.2e148

   1. Initial program 75.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   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. lower-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

      \[\leadsto \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]

      2. lower--.f64 N/A

         \[\leadsto \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{6} \cdot {B}^{2} - 1}{B} \]

      4. lower-pow.f64 9.7

         \[\leadsto \frac{-0.16666666666666666 \cdot {B}^{2} - 1}{B} \]

   7. Applied rewrites 9.7%

      \[\leadsto \frac{-0.16666666666666666 \cdot {B}^{2} - 1}{\color{blue}{B}} \]

   if -5.2e148 < F < 2.35e21 or 1.09999999999999994e132 < F

   1. Initial program 75.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.4

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 29.4

         \[\leadsto -1 \cdot \frac{x}{B} \]

   7. Applied rewrites 29.4%

      \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x}{B}\right) \]

      3. lower-neg.f64 29.4

         \[\leadsto -\frac{x}{B} \]

   9. Applied rewrites 29.4%

      \[\leadsto -\frac{x}{B} \]

   if 2.35e21 < F < 1.09999999999999994e132

   1. Initial program 75.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. mult-flip N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right)} \]

   3. Applied rewrites 84.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{\sin B}} \]
   5. 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. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin \color{blue}{B}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      8. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{\sin B} \]

      10. lower-sin.f64 30.3

          \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{-0.5}}{\sin B} \]

   6. Applied rewrites 30.3%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + {F}^{2}\right)}^{-0.5}}{\sin B}} \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      8. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      9. metadata-eval 16.2

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{-0.5}}{B} \]

   9. Applied rewrites 16.2%

      \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{-0.5}}{\color{blue}{B}} \]

   10. Taylor expanded in F around inf

       \[\leadsto \frac{1}{B} \]
   11. Step-by-step derivation

       1. lower-/.f64 10.3

          \[\leadsto \frac{1}{B} \]

   12. Applied rewrites 10.3%

       \[\leadsto \frac{1}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 23: 29.4% accurate, 6.9× speedup

Math

\[\begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;F \leq -5.2 \cdot 10^{+148}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{elif}\;F \leq 2.35 \cdot 10^{+21}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{+132}:\\ \;\;\;\;\frac{1}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x B))))
   (if (<= F -5.2e+148)
     (/ -1.0 B)
     (if (<= F 2.35e+21) t_0 (if (<= F 1.1e+132) (/ 1.0 B) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (F <= -5.2e+148) {
		tmp = -1.0 / B;
	} else if (F <= 2.35e+21) {
		tmp = t_0;
	} else if (F <= 1.1e+132) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	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)
    if (f <= (-5.2d+148)) then
        tmp = (-1.0d0) / b
    else if (f <= 2.35d+21) then
        tmp = t_0
    else if (f <= 1.1d+132) then
        tmp = 1.0d0 / b
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (F <= -5.2e+148) {
		tmp = -1.0 / B;
	} else if (F <= 2.35e+21) {
		tmp = t_0;
	} else if (F <= 1.1e+132) {
		tmp = 1.0 / B;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -(x / B)
	tmp = 0
	if F <= -5.2e+148:
		tmp = -1.0 / B
	elif F <= 2.35e+21:
		tmp = t_0
	elif F <= 1.1e+132:
		tmp = 1.0 / B
	else:
		tmp = t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	tmp = 0.0
	if (F <= -5.2e+148)
		tmp = Float64(-1.0 / B);
	elseif (F <= 2.35e+21)
		tmp = t_0;
	elseif (F <= 1.1e+132)
		tmp = Float64(1.0 / B);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -(x / B);
	tmp = 0.0;
	if (F <= -5.2e+148)
		tmp = -1.0 / B;
	elseif (F <= 2.35e+21)
		tmp = t_0;
	elseif (F <= 1.1e+132)
		tmp = 1.0 / B;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[F, -5.2e+148], N[(-1.0 / B), $MachinePrecision], If[LessEqual[F, 2.35e+21], t$95$0, If[LessEqual[F, 1.1e+132], N[(1.0 / B), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.2e148

   1. Initial program 75.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   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. lower-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 10.0%

      \[\leadsto \frac{-1}{B} \]

   if -5.2e148 < F < 2.35e21 or 1.09999999999999994e132 < F

   1. Initial program 75.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in F around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x \cdot \cos B}{\sin B}} \]

      2. lower-/.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\color{blue}{\sin B}} \]

      3. lower-*.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin \color{blue}{B}} \]

      4. lower-cos.f64 N/A

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

      5. lower-sin.f64 55.4

         \[\leadsto -1 \cdot \frac{x \cdot \cos B}{\sin B} \]

   4. Applied rewrites 55.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \cos B}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]
   6. Step-by-step derivation

      1. lower-/.f64 29.4

         \[\leadsto -1 \cdot \frac{x}{B} \]

   7. Applied rewrites 29.4%

      \[\leadsto -1 \cdot \frac{x}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto -1 \cdot \color{blue}{\frac{x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x}{B}\right) \]

      3. lower-neg.f64 29.4

         \[\leadsto -\frac{x}{B} \]

   9. Applied rewrites 29.4%

      \[\leadsto -\frac{x}{B} \]

   if 2.35e21 < F < 1.09999999999999994e132

   1. Initial program 75.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. mult-flip N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right)} \]

   3. Applied rewrites 84.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{\sin B}} \]
   5. 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. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin \color{blue}{B}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      8. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{\sin B} \]

      10. lower-sin.f64 30.3

          \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{-0.5}}{\sin B} \]

   6. Applied rewrites 30.3%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + {F}^{2}\right)}^{-0.5}}{\sin B}} \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      8. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      9. metadata-eval 16.2

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{-0.5}}{B} \]

   9. Applied rewrites 16.2%

      \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{-0.5}}{\color{blue}{B}} \]

   10. Taylor expanded in F around inf

       \[\leadsto \frac{1}{B} \]
   11. Step-by-step derivation

       1. lower-/.f64 10.3

          \[\leadsto \frac{1}{B} \]

   12. Applied rewrites 10.3%

       \[\leadsto \frac{1}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 24: 17.5% accurate, 14.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq 4.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (if (<= F 4.8e-93) (/ -1.0 B) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 4.8e-93) {
		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 <= 4.8d-93) 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 <= 4.8e-93) {
		tmp = -1.0 / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 4.8e-93:
		tmp = -1.0 / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 4.8e-93)
		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 <= 4.8e-93)
		tmp = -1.0 / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 4.8e-93], N[(-1.0 / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 4.8000000000000002e-93

   1. Initial program 75.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   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. lower-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 10.0%

      \[\leadsto \frac{-1}{B} \]

   if 4.8000000000000002e-93 < F

   1. Initial program 75.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      5. mult-flip N/A

         \[\leadsto \color{blue}{\left(F \cdot \frac{1}{\sin B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      6. associate-*l* N/A

         \[\leadsto \color{blue}{F \cdot \left(\frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\right)} + \left(-x \cdot \frac{1}{\tan B}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{1}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}, -x \cdot \frac{1}{\tan B}\right)} \]

   3. Applied rewrites 84.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, \frac{-x}{\tan B}\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{\sin B}} \]
   5. 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. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin \color{blue}{B}} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      8. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B} \]

      9. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{\sin B} \]

      10. lower-sin.f64 30.3

          \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{-0.5}}{\sin B} \]

   6. Applied rewrites 30.3%

      \[\leadsto \color{blue}{\frac{F \cdot {\left(2 + {F}^{2}\right)}^{-0.5}}{\sin B}} \]

   7. Taylor expanded in B around 0

      \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\frac{-1}{2}}}{\color{blue}{B}} \]
   8. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      5. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      7. lower-pow.f64 N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      8. metadata-eval N/A

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{B} \]

      9. metadata-eval 16.2

         \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{-0.5}}{B} \]

   9. Applied rewrites 16.2%

      \[\leadsto \frac{F \cdot {\left(2 + {F}^{2}\right)}^{-0.5}}{\color{blue}{B}} \]

   10. Taylor expanded in F around inf

       \[\leadsto \frac{1}{B} \]
   11. Step-by-step derivation

       1. lower-/.f64 10.3

          \[\leadsto \frac{1}{B} \]

   12. Applied rewrites 10.3%

       \[\leadsto \frac{1}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 10.0% accurate, 26.5× speedup

Math

\[\frac{-1}{B} \]

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 75.9%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

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. lower-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 10.0%

   \[\leadsto \frac{-1}{B} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025170 
(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))))))