VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.7% → 99.7%

Time: 7.1s

Alternatives: 21

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+24}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, t\_1\right)\\ \mathbf{elif}\;F \leq 200000:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 1, t\_1\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ (- x) (tan B))))
   (if (<= F -2e+24)
     (fma t_0 -1.0 t_1)
     (if (<= F 200000.0)
       (- (* (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F (sin B))) (/ x (tan B)))
       (fma t_0 1.0 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 <= -2e+24) {
		tmp = fma(t_0, -1.0, t_1);
	} else if (F <= 200000.0) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) * (F / sin(B))) - (x / tan(B));
	} else {
		tmp = fma(t_0, 1.0, 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 <= -2e+24)
		tmp = fma(t_0, -1.0, t_1);
	elseif (F <= 200000.0)
		tmp = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) * Float64(F / sin(B))) - Float64(x / tan(B)));
	else
		tmp = fma(t_0, 1.0, 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, -2e+24], N[(t$95$0 * -1.0 + t$95$1), $MachinePrecision], If[LessEqual[F, 200000.0], 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] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 1.0 + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2e24

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 85.0%

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

   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 55.1%

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

   if -2e24 < F < 2e5

   1. Initial program 76.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.7

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

   3. Applied rewrites 76.9%

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

   if 2e5 < F

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 85.0%

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

   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 56.6%

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

Alternative 2: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -0.42:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, t\_1\right)\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-6}:\\ \;\;\;\;{\left(\mathsf{fma}\left(2, x, 2\right)\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 1, t\_1\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ (- x) (tan B))))
   (if (<= F -0.42)
     (fma t_0 -1.0 t_1)
     (if (<= F 8e-6)
       (- (* (pow (fma 2.0 x 2.0) -0.5) (/ F (sin B))) (/ x (tan B)))
       (fma t_0 1.0 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 <= -0.42) {
		tmp = fma(t_0, -1.0, t_1);
	} else if (F <= 8e-6) {
		tmp = (pow(fma(2.0, x, 2.0), -0.5) * (F / sin(B))) - (x / tan(B));
	} else {
		tmp = fma(t_0, 1.0, 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 <= -0.42)
		tmp = fma(t_0, -1.0, t_1);
	elseif (F <= 8e-6)
		tmp = Float64(Float64((fma(2.0, x, 2.0) ^ -0.5) * Float64(F / sin(B))) - Float64(x / tan(B)));
	else
		tmp = fma(t_0, 1.0, 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, -0.42], N[(t$95$0 * -1.0 + t$95$1), $MachinePrecision], If[LessEqual[F, 8e-6], N[(N[(N[Power[N[(2.0 * x + 2.0), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * 1.0 + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.419999999999999984

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 85.0%

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

   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 55.1%

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

   if -0.419999999999999984 < F < 7.99999999999999964e-6

   1. Initial program 76.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.7

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

   3. Applied rewrites 76.9%

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

   4. Taylor expanded in F around 0

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

   6. Applied rewrites 55.3%

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

   if 7.99999999999999964e-6 < F

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 85.0%

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

   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 56.6%

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

Alternative 3: 99.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -0.42:\\ \;\;\;\;\mathsf{fma}\left(t\_0, -1, t\_1\right)\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B \cdot \sqrt{2 + 2 \cdot x}}, F, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, 1, t\_1\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ (- x) (tan B))))
   (if (<= F -0.42)
     (fma t_0 -1.0 t_1)
     (if (<= F 8e-6)
       (fma (/ 1.0 (* (sin B) (sqrt (+ 2.0 (* 2.0 x))))) F t_1)
       (fma t_0 1.0 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 <= -0.42) {
		tmp = fma(t_0, -1.0, t_1);
	} else if (F <= 8e-6) {
		tmp = fma((1.0 / (sin(B) * sqrt((2.0 + (2.0 * x))))), F, t_1);
	} else {
		tmp = fma(t_0, 1.0, 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 <= -0.42)
		tmp = fma(t_0, -1.0, t_1);
	elseif (F <= 8e-6)
		tmp = fma(Float64(1.0 / Float64(sin(B) * sqrt(Float64(2.0 + Float64(2.0 * x))))), F, t_1);
	else
		tmp = fma(t_0, 1.0, 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, -0.42], N[(t$95$0 * -1.0 + t$95$1), $MachinePrecision], If[LessEqual[F, 8e-6], N[(N[(1.0 / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F + t$95$1), $MachinePrecision], N[(t$95$0 * 1.0 + t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.419999999999999984

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 85.0%

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

   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 55.1%

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

   if -0.419999999999999984 < F < 7.99999999999999964e-6

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. add-to-fraction N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 85.0%

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in F around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 56.6

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

   8. Applied rewrites 56.6%

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

   if 7.99999999999999964e-6 < F

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 85.0%

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

   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 56.6%

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

Alternative 4: 91.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}\\ t_1 := \frac{1}{\sin B}\\ t_2 := \frac{-x}{\tan B}\\ \mathbf{if}\;F \leq -5.2 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, -1, t\_2\right)\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-155}:\\ \;\;\;\;t\_0 \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, F, -1 \cdot x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, 1, t\_2\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (fma 2.0 x (fma F F 2.0)) -0.5))
        (t_1 (/ 1.0 (sin B)))
        (t_2 (/ (- x) (tan B))))
   (if (<= F -5.2e+22)
     (fma t_1 -1.0 t_2)
     (if (<= F 3.4e-155)
       (- (* t_0 (/ F B)) (/ x (tan B)))
       (if (<= F 8e-6)
         (/ (fma t_0 F (* -1.0 x)) (sin B))
         (fma t_1 1.0 t_2))))))

C

double code(double F, double B, double x) {
	double t_0 = pow(fma(2.0, x, fma(F, F, 2.0)), -0.5);
	double t_1 = 1.0 / sin(B);
	double t_2 = -x / tan(B);
	double tmp;
	if (F <= -5.2e+22) {
		tmp = fma(t_1, -1.0, t_2);
	} else if (F <= 3.4e-155) {
		tmp = (t_0 * (F / B)) - (x / tan(B));
	} else if (F <= 8e-6) {
		tmp = fma(t_0, F, (-1.0 * x)) / sin(B);
	} else {
		tmp = fma(t_1, 1.0, t_2);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(2.0, x, fma(F, F, 2.0)) ^ -0.5
	t_1 = Float64(1.0 / sin(B))
	t_2 = Float64(Float64(-x) / tan(B))
	tmp = 0.0
	if (F <= -5.2e+22)
		tmp = fma(t_1, -1.0, t_2);
	elseif (F <= 3.4e-155)
		tmp = Float64(Float64(t_0 * Float64(F / B)) - Float64(x / tan(B)));
	elseif (F <= 8e-6)
		tmp = Float64(fma(t_0, F, Float64(-1.0 * x)) / sin(B));
	else
		tmp = fma(t_1, 1.0, t_2);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5.2e+22], N[(t$95$1 * -1.0 + t$95$2), $MachinePrecision], If[LessEqual[F, 3.4e-155], N[(N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8e-6], N[(N[(t$95$0 * F + N[(-1.0 * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(t$95$1 * 1.0 + t$95$2), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -5.2e22

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 85.0%

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

   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 55.1%

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

   if -5.2e22 < F < 3.4e-155

   1. Initial program 76.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.7

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

   3. Applied rewrites 76.9%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 62.5%

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

   if 3.4e-155 < F < 7.99999999999999964e-6

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. add-to-fraction N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 60.3

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

   6. Applied rewrites 60.3%

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

   if 7.99999999999999964e-6 < F

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 85.0%

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

   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 56.6%

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

Alternative 5: 86.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}\\ \mathbf{if}\;F \leq -5.2 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B}, -1, \frac{-x}{\tan B}\right)\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-155}:\\ \;\;\;\;t\_0 \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{+146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, F, -1 \cdot x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\mathsf{fma}\left(-0.5, \frac{2 + 2 \cdot x}{{F}^{3}}, \frac{1}{F}\right)}{\sin B}, -\frac{x}{B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (fma 2.0 x (fma F F 2.0)) -0.5)))
   (if (<= F -5.2e+22)
     (fma (/ 1.0 (sin B)) -1.0 (/ (- x) (tan B)))
     (if (<= F 3.4e-155)
       (- (* t_0 (/ F B)) (/ x (tan B)))
       (if (<= F 2.7e+146)
         (/ (fma t_0 F (* -1.0 x)) (sin B))
         (fma
          F
          (/ (fma -0.5 (/ (+ 2.0 (* 2.0 x)) (pow F 3.0)) (/ 1.0 F)) (sin B))
          (- (/ x B))))))))

C

double code(double F, double B, double x) {
	double t_0 = pow(fma(2.0, x, fma(F, F, 2.0)), -0.5);
	double tmp;
	if (F <= -5.2e+22) {
		tmp = fma((1.0 / sin(B)), -1.0, (-x / tan(B)));
	} else if (F <= 3.4e-155) {
		tmp = (t_0 * (F / B)) - (x / tan(B));
	} else if (F <= 2.7e+146) {
		tmp = fma(t_0, F, (-1.0 * x)) / sin(B);
	} else {
		tmp = fma(F, (fma(-0.5, ((2.0 + (2.0 * x)) / pow(F, 3.0)), (1.0 / F)) / sin(B)), -(x / B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(2.0, x, fma(F, F, 2.0)) ^ -0.5
	tmp = 0.0
	if (F <= -5.2e+22)
		tmp = fma(Float64(1.0 / sin(B)), -1.0, Float64(Float64(-x) / tan(B)));
	elseif (F <= 3.4e-155)
		tmp = Float64(Float64(t_0 * Float64(F / B)) - Float64(x / tan(B)));
	elseif (F <= 2.7e+146)
		tmp = Float64(fma(t_0, F, Float64(-1.0 * x)) / sin(B));
	else
		tmp = fma(F, Float64(fma(-0.5, Float64(Float64(2.0 + Float64(2.0 * x)) / (F ^ 3.0)), Float64(1.0 / F)) / sin(B)), Float64(-Float64(x / B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[F, -5.2e+22], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] * -1.0 + N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.4e-155], N[(N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.7e+146], N[(N[(t$95$0 * F + N[(-1.0 * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(F * N[(N[(-0.5 * N[(N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision] / N[Power[F, 3.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / F), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + (-N[(x / B), $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -5.2e22

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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 85.0%

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

   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 55.1%

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

   if -5.2e22 < F < 3.4e-155

   1. Initial program 76.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.7

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

   3. Applied rewrites 76.9%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 62.5%

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

   if 3.4e-155 < F < 2.69999999999999989e146

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. add-to-fraction N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 60.3

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

   6. Applied rewrites 60.3%

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

   if 2.69999999999999989e146 < F

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 58.3%

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

   7. Taylor expanded in F around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-/.f64 33.3

          \[\leadsto \mathsf{fma}\left(F, \frac{\mathsf{fma}\left(-0.5, \frac{2 + 2 \cdot x}{{F}^{3}}, \frac{1}{F}\right)}{\sin B}, -\frac{x}{B}\right) \]

   9. Applied rewrites 33.3%

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

Alternative 6: 81.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}\\ \mathbf{if}\;F \leq -5.2 \cdot 10^{+22}:\\ \;\;\;\;\frac{\left(-F\right) \cdot \mathsf{fma}\left(\cos B, x, 1\right)}{\sin B \cdot F}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-155}:\\ \;\;\;\;t\_0 \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{+146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, F, -1 \cdot x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\mathsf{fma}\left(-0.5, \frac{2 + 2 \cdot x}{{F}^{3}}, \frac{1}{F}\right)}{\sin B}, -\frac{x}{B}\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (fma 2.0 x (fma F F 2.0)) -0.5)))
   (if (<= F -5.2e+22)
     (/ (* (- F) (fma (cos B) x 1.0)) (* (sin B) F))
     (if (<= F 3.4e-155)
       (- (* t_0 (/ F B)) (/ x (tan B)))
       (if (<= F 2.7e+146)
         (/ (fma t_0 F (* -1.0 x)) (sin B))
         (fma
          F
          (/ (fma -0.5 (/ (+ 2.0 (* 2.0 x)) (pow F 3.0)) (/ 1.0 F)) (sin B))
          (- (/ x B))))))))

C

double code(double F, double B, double x) {
	double t_0 = pow(fma(2.0, x, fma(F, F, 2.0)), -0.5);
	double tmp;
	if (F <= -5.2e+22) {
		tmp = (-F * fma(cos(B), x, 1.0)) / (sin(B) * F);
	} else if (F <= 3.4e-155) {
		tmp = (t_0 * (F / B)) - (x / tan(B));
	} else if (F <= 2.7e+146) {
		tmp = fma(t_0, F, (-1.0 * x)) / sin(B);
	} else {
		tmp = fma(F, (fma(-0.5, ((2.0 + (2.0 * x)) / pow(F, 3.0)), (1.0 / F)) / sin(B)), -(x / B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(2.0, x, fma(F, F, 2.0)) ^ -0.5
	tmp = 0.0
	if (F <= -5.2e+22)
		tmp = Float64(Float64(Float64(-F) * fma(cos(B), x, 1.0)) / Float64(sin(B) * F));
	elseif (F <= 3.4e-155)
		tmp = Float64(Float64(t_0 * Float64(F / B)) - Float64(x / tan(B)));
	elseif (F <= 2.7e+146)
		tmp = Float64(fma(t_0, F, Float64(-1.0 * x)) / sin(B));
	else
		tmp = fma(F, Float64(fma(-0.5, Float64(Float64(2.0 + Float64(2.0 * x)) / (F ^ 3.0)), Float64(1.0 / F)) / sin(B)), Float64(-Float64(x / B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, If[LessEqual[F, -5.2e+22], N[(N[((-F) * N[(N[Cos[B], $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.4e-155], N[(N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.7e+146], N[(N[(t$95$0 * F + N[(-1.0 * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(F * N[(N[(-0.5 * N[(N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision] / N[Power[F, 3.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / F), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + (-N[(x / B), $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -5.2e22

   1. Initial program 76.7%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.1

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

   4. Applied rewrites 49.1%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

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

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. div-add-rev N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

   6. Applied rewrites 45.5%

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

   if -5.2e22 < F < 3.4e-155

   1. Initial program 76.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.7

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

   3. Applied rewrites 76.9%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 62.5%

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

   if 3.4e-155 < F < 2.69999999999999989e146

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. add-to-fraction N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 60.3

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

   6. Applied rewrites 60.3%

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

   if 2.69999999999999989e146 < F

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 58.3%

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

   7. Taylor expanded in F around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-/.f64 33.3

          \[\leadsto \mathsf{fma}\left(F, \frac{\mathsf{fma}\left(-0.5, \frac{2 + 2 \cdot x}{{F}^{3}}, \frac{1}{F}\right)}{\sin B}, -\frac{x}{B}\right) \]

   9. Applied rewrites 33.3%

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

Alternative 7: 79.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}\\ t_1 := -\frac{x}{B}\\ \mathbf{if}\;F \leq -1.1 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{-1}{F}}{\sin B}, t\_1\right)\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-155}:\\ \;\;\;\;t\_0 \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{+146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, F, -1 \cdot x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\mathsf{fma}\left(-0.5, \frac{2 + 2 \cdot x}{{F}^{3}}, \frac{1}{F}\right)}{\sin B}, t\_1\right)\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (fma 2.0 x (fma F F 2.0)) -0.5)) (t_1 (- (/ x B))))
   (if (<= F -1.1e+27)
     (fma F (/ (/ -1.0 F) (sin B)) t_1)
     (if (<= F 3.4e-155)
       (- (* t_0 (/ F B)) (/ x (tan B)))
       (if (<= F 2.7e+146)
         (/ (fma t_0 F (* -1.0 x)) (sin B))
         (fma
          F
          (/ (fma -0.5 (/ (+ 2.0 (* 2.0 x)) (pow F 3.0)) (/ 1.0 F)) (sin B))
          t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = pow(fma(2.0, x, fma(F, F, 2.0)), -0.5);
	double t_1 = -(x / B);
	double tmp;
	if (F <= -1.1e+27) {
		tmp = fma(F, ((-1.0 / F) / sin(B)), t_1);
	} else if (F <= 3.4e-155) {
		tmp = (t_0 * (F / B)) - (x / tan(B));
	} else if (F <= 2.7e+146) {
		tmp = fma(t_0, F, (-1.0 * x)) / sin(B);
	} else {
		tmp = fma(F, (fma(-0.5, ((2.0 + (2.0 * x)) / pow(F, 3.0)), (1.0 / F)) / sin(B)), t_1);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(2.0, x, fma(F, F, 2.0)) ^ -0.5
	t_1 = Float64(-Float64(x / B))
	tmp = 0.0
	if (F <= -1.1e+27)
		tmp = fma(F, Float64(Float64(-1.0 / F) / sin(B)), t_1);
	elseif (F <= 3.4e-155)
		tmp = Float64(Float64(t_0 * Float64(F / B)) - Float64(x / tan(B)));
	elseif (F <= 2.7e+146)
		tmp = Float64(fma(t_0, F, Float64(-1.0 * x)) / sin(B));
	else
		tmp = fma(F, Float64(fma(-0.5, Float64(Float64(2.0 + Float64(2.0 * x)) / (F ^ 3.0)), Float64(1.0 / F)) / sin(B)), t_1);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$1 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[F, -1.1e+27], N[(F * N[(N[(-1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[F, 3.4e-155], N[(N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.7e+146], N[(N[(t$95$0 * F + N[(-1.0 * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(F * N[(N[(-0.5 * N[(N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision] / N[Power[F, 3.0], $MachinePrecision]), $MachinePrecision] + N[(1.0 / F), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.0999999999999999e27

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 58.3%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 34.4

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

   9. Applied rewrites 34.4%

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

   if -1.0999999999999999e27 < F < 3.4e-155

   1. Initial program 76.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.7

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

   3. Applied rewrites 76.9%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 62.5%

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

   if 3.4e-155 < F < 2.69999999999999989e146

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. add-to-fraction N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 60.3

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

   6. Applied rewrites 60.3%

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

   if 2.69999999999999989e146 < F

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 58.3%

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

   7. Taylor expanded in F around inf

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-/.f64 33.3

          \[\leadsto \mathsf{fma}\left(F, \frac{\mathsf{fma}\left(-0.5, \frac{2 + 2 \cdot x}{{F}^{3}}, \frac{1}{F}\right)}{\sin B}, -\frac{x}{B}\right) \]

   9. Applied rewrites 33.3%

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

Alternative 8: 79.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}\\ t_1 := -\frac{x}{B}\\ \mathbf{if}\;F \leq -1.1 \cdot 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{-1}{F}}{\sin B}, t\_1\right)\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-155}:\\ \;\;\;\;t\_0 \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{+146}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t\_0, F, -1 \cdot x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (fma 2.0 x (fma F F 2.0)) -0.5)) (t_1 (- (/ x B))))
   (if (<= F -1.1e+27)
     (fma F (/ (/ -1.0 F) (sin B)) t_1)
     (if (<= F 3.4e-155)
       (- (* t_0 (/ F B)) (/ x (tan B)))
       (if (<= F 1.8e+146)
         (/ (fma t_0 F (* -1.0 x)) (sin B))
         (+ t_1 (/ 1.0 (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = pow(fma(2.0, x, fma(F, F, 2.0)), -0.5);
	double t_1 = -(x / B);
	double tmp;
	if (F <= -1.1e+27) {
		tmp = fma(F, ((-1.0 / F) / sin(B)), t_1);
	} else if (F <= 3.4e-155) {
		tmp = (t_0 * (F / B)) - (x / tan(B));
	} else if (F <= 1.8e+146) {
		tmp = fma(t_0, F, (-1.0 * x)) / sin(B);
	} else {
		tmp = t_1 + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(2.0, x, fma(F, F, 2.0)) ^ -0.5
	t_1 = Float64(-Float64(x / B))
	tmp = 0.0
	if (F <= -1.1e+27)
		tmp = fma(F, Float64(Float64(-1.0 / F) / sin(B)), t_1);
	elseif (F <= 3.4e-155)
		tmp = Float64(Float64(t_0 * Float64(F / B)) - Float64(x / tan(B)));
	elseif (F <= 1.8e+146)
		tmp = Float64(fma(t_0, F, Float64(-1.0 * x)) / sin(B));
	else
		tmp = Float64(t_1 + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$1 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[F, -1.1e+27], N[(F * N[(N[(-1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[F, 3.4e-155], N[(N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.8e+146], N[(N[(t$95$0 * F + N[(-1.0 * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.0999999999999999e27

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 58.3%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 34.4

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

   9. Applied rewrites 34.4%

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

   if -1.0999999999999999e27 < F < 3.4e-155

   1. Initial program 76.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 76.7

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

   3. Applied rewrites 76.9%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 62.5%

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

   if 3.4e-155 < F < 1.7999999999999999e146

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. add-to-fraction N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 60.3

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

   6. Applied rewrites 60.3%

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

   if 1.7999999999999999e146 < F

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

   6. Applied rewrites 58.3%

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

   7. Taylor expanded in F around inf

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

      1. lower-sin.f64 37.1

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

   9. Applied rewrites 37.1%

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

Alternative 9: 78.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := \frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, F, -1 \cdot x\right)}{\sin B}\\ t_1 := -\frac{x}{B}\\ \mathbf{if}\;F \leq -1.85 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{-1}{F}}{\sin B}, t\_1\right)\\ \mathbf{elif}\;F \leq -4.5 \cdot 10^{-126}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{-192}:\\ \;\;\;\;-1 \cdot \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{+146}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (/ (fma (pow (fma 2.0 x (fma F F 2.0)) -0.5) F (* -1.0 x)) (sin B)))
        (t_1 (- (/ x B))))
   (if (<= F -1.85e+30)
     (fma F (/ (/ -1.0 F) (sin B)) t_1)
     (if (<= F -4.5e-126)
       t_0
       (if (<= F 5.2e-192)
         (* -1.0 (/ (* x (cos B)) (sin B)))
         (if (<= F 1.8e+146) t_0 (+ t_1 (/ 1.0 (sin B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = fma(pow(fma(2.0, x, fma(F, F, 2.0)), -0.5), F, (-1.0 * x)) / sin(B);
	double t_1 = -(x / B);
	double tmp;
	if (F <= -1.85e+30) {
		tmp = fma(F, ((-1.0 / F) / sin(B)), t_1);
	} else if (F <= -4.5e-126) {
		tmp = t_0;
	} else if (F <= 5.2e-192) {
		tmp = -1.0 * ((x * cos(B)) / sin(B));
	} else if (F <= 1.8e+146) {
		tmp = t_0;
	} else {
		tmp = t_1 + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(fma((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5), F, Float64(-1.0 * x)) / sin(B))
	t_1 = Float64(-Float64(x / B))
	tmp = 0.0
	if (F <= -1.85e+30)
		tmp = fma(F, Float64(Float64(-1.0 / F) / sin(B)), t_1);
	elseif (F <= -4.5e-126)
		tmp = t_0;
	elseif (F <= 5.2e-192)
		tmp = Float64(-1.0 * Float64(Float64(x * cos(B)) / sin(B)));
	elseif (F <= 1.8e+146)
		tmp = t_0;
	else
		tmp = Float64(t_1 + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F + N[(-1.0 * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[F, -1.85e+30], N[(F * N[(N[(-1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], If[LessEqual[F, -4.5e-126], t$95$0, If[LessEqual[F, 5.2e-192], N[(-1.0 * N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.8e+146], t$95$0, N[(t$95$1 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.85000000000000008e30

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 58.3%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 34.4

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

   9. Applied rewrites 34.4%

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

   if -1.85000000000000008e30 < F < -4.50000000000000025e-126 or 5.2000000000000003e-192 < F < 1.7999999999999999e146

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. add-to-fraction N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 60.3

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

   6. Applied rewrites 60.3%

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

   if -4.50000000000000025e-126 < F < 5.2000000000000003e-192

   1. Initial program 76.7%

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

   2. Taylor expanded in F around 0

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

      1. 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 55.8

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

   4. Applied rewrites 55.8%

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

   if 1.7999999999999999e146 < F

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

   6. Applied rewrites 58.3%

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

   7. Taylor expanded in F around inf

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

      1. lower-sin.f64 37.1

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

   9. Applied rewrites 37.1%

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

Alternative 10: 73.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;F \leq -1.85 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{-1}{F}}{\sin B}, t\_0\right)\\ \mathbf{elif}\;F \leq 1.8 \cdot 10^{+146}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, F, -1 \cdot x\right)}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x B))))
   (if (<= F -1.85e+30)
     (fma F (/ (/ -1.0 F) (sin B)) t_0)
     (if (<= F 1.8e+146)
       (/ (fma (pow (fma 2.0 x (fma F F 2.0)) -0.5) F (* -1.0 x)) (sin B))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (F <= -1.85e+30) {
		tmp = fma(F, ((-1.0 / F) / sin(B)), t_0);
	} else if (F <= 1.8e+146) {
		tmp = fma(pow(fma(2.0, x, fma(F, F, 2.0)), -0.5), F, (-1.0 * x)) / sin(B);
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	tmp = 0.0
	if (F <= -1.85e+30)
		tmp = fma(F, Float64(Float64(-1.0 / F) / sin(B)), t_0);
	elseif (F <= 1.8e+146)
		tmp = Float64(fma((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5), F, Float64(-1.0 * x)) / sin(B));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[F, -1.85e+30], N[(F * N[(N[(-1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, 1.8e+146], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * F + N[(-1.0 * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.85000000000000008e30

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 58.3%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 34.4

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

   9. Applied rewrites 34.4%

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

   if -1.85000000000000008e30 < F < 1.7999999999999999e146

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. add-to-fraction N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 85.0%

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

   4. Taylor expanded in B around 0

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

      1. lower-*.f64 60.3

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

   6. Applied rewrites 60.3%

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

   if 1.7999999999999999e146 < F

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

   6. Applied rewrites 58.3%

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

   7. Taylor expanded in F around inf

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

      1. lower-sin.f64 37.1

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

   9. Applied rewrites 37.1%

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

Alternative 11: 72.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{-1}{F}}{\sin B}, t\_0\right)\\ \mathbf{elif}\;F \leq 200000:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}, F, t\_0\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x B))))
   (if (<= F -2e+101)
     (fma F (/ (/ -1.0 F) (sin B)) t_0)
     (if (<= F 200000.0)
       (fma (/ 1.0 (* (sin B) (sqrt (fma 2.0 x (fma F F 2.0))))) F t_0)
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (F <= -2e+101) {
		tmp = fma(F, ((-1.0 / F) / sin(B)), t_0);
	} else if (F <= 200000.0) {
		tmp = fma((1.0 / (sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))), F, t_0);
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	tmp = 0.0
	if (F <= -2e+101)
		tmp = fma(F, Float64(Float64(-1.0 / F) / sin(B)), t_0);
	elseif (F <= 200000.0)
		tmp = fma(Float64(1.0 / Float64(sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))), F, t_0);
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[F, -2e+101], N[(F * N[(N[(-1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, 200000.0], N[(N[(1.0 / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F + t$95$0), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2e101

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 58.3%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 34.4

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

   9. Applied rewrites 34.4%

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

   if -2e101 < F < 2e5

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 58.3%

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

      1. metadata-eval N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. pow-flip N/A

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

      6. lift-pow.f64 N/A

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

      7. associate-/r* N/A

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

      8. *-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-sin.f64 N/A

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

      11. lift-/.f64 58.3

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

      12. lower-fma.f64 N/A

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

      13. *-commutative N/A

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

   8. Applied rewrites 58.3%

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

   if 2e5 < F

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

   6. Applied rewrites 58.3%

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

   7. Taylor expanded in F around inf

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

      1. lower-sin.f64 37.1

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

   9. Applied rewrites 37.1%

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

Alternative 12: 72.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+101}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{-1}{F}}{\sin B}, t\_0\right)\\ \mathbf{elif}\;F \leq 200000:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x B))))
   (if (<= F -2e+101)
     (fma F (/ (/ -1.0 F) (sin B)) t_0)
     (if (<= F 200000.0)
       (- (/ F (* (sin B) (sqrt (fma 2.0 x (fma F F 2.0))))) (/ x B))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (F <= -2e+101) {
		tmp = fma(F, ((-1.0 / F) / sin(B)), t_0);
	} else if (F <= 200000.0) {
		tmp = (F / (sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - (x / B);
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	tmp = 0.0
	if (F <= -2e+101)
		tmp = fma(F, Float64(Float64(-1.0 / F) / sin(B)), t_0);
	elseif (F <= 200000.0)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(2.0, x, fma(F, F, 2.0))))) - Float64(x / B));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[F, -2e+101], N[(F * N[(N[(-1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, 200000.0], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2e101

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 58.3%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 34.4

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

   9. Applied rewrites 34.4%

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

   if -2e101 < F < 2e5

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 58.3%

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

      1. metadata-eval N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-neg.f64 N/A

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

      4. sub-flip-reverse N/A

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

      5. lower--.f64 N/A

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

   8. Applied rewrites 58.3%

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

   if 2e5 < F

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

   6. Applied rewrites 58.3%

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

   7. Taylor expanded in F around inf

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

      1. lower-sin.f64 37.1

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

   9. Applied rewrites 37.1%

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

Alternative 13: 65.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;F \leq -5.2 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{\frac{-1}{F}}{\sin B}, t\_0\right)\\ \mathbf{elif}\;F \leq 190:\\ \;\;\;\;t\_0 + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x B))))
   (if (<= F -5.2e+22)
     (fma F (/ (/ -1.0 F) (sin B)) t_0)
     (if (<= F 190.0)
       (+ t_0 (* (/ F B) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (F <= -5.2e+22) {
		tmp = fma(F, ((-1.0 / F) / sin(B)), t_0);
	} else if (F <= 190.0) {
		tmp = t_0 + ((F / B) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	tmp = 0.0
	if (F <= -5.2e+22)
		tmp = fma(F, Float64(Float64(-1.0 / F) / sin(B)), t_0);
	elseif (F <= 190.0)
		tmp = Float64(t_0 + Float64(Float64(F / B) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[F, -5.2e+22], N[(F * N[(N[(-1.0 / F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision], If[LessEqual[F, 190.0], N[(t$95$0 + N[(N[(F / B), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.2e22

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

   6. Applied rewrites 58.3%

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

   7. Taylor expanded in F around -inf

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

      1. lower-/.f64 34.4

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

   9. Applied rewrites 34.4%

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

   if -5.2e22 < F < 190

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 36.4%

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

   if 190 < F

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

   6. Applied rewrites 58.3%

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

   7. Taylor expanded in F around inf

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

      1. lower-sin.f64 37.1

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

   9. Applied rewrites 37.1%

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

Alternative 14: 59.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} t_0 := -\frac{x}{B}\\ \mathbf{if}\;F \leq -4.9 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 190:\\ \;\;\;\;t\_0 + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_0 + \frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ x B))))
   (if (<= F -4.9e+27)
     (/ -1.0 (sin B))
     (if (<= F 190.0)
       (+ t_0 (* (/ F B) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (F <= -4.9e+27) {
		tmp = -1.0 / sin(B);
	} else if (F <= 190.0) {
		tmp = t_0 + ((F / B) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = -(x / b)
    if (f <= (-4.9d+27)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 190.0d0) then
        tmp = t_0 + ((f / b) * ((((f * f) + 2.0d0) + (2.0d0 * x)) ** -(1.0d0 / 2.0d0)))
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -(x / B);
	double tmp;
	if (F <= -4.9e+27) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 190.0) {
		tmp = t_0 + ((F / B) * Math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -(x / B)
	tmp = 0
	if F <= -4.9e+27:
		tmp = -1.0 / math.sin(B)
	elif F <= 190.0:
		tmp = t_0 + ((F / B) * math.pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x / B))
	tmp = 0.0
	if (F <= -4.9e+27)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 190.0)
		tmp = Float64(t_0 + Float64(Float64(F / B) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -(x / B);
	tmp = 0.0;
	if (F <= -4.9e+27)
		tmp = -1.0 / sin(B);
	elseif (F <= 190.0)
		tmp = t_0 + ((F / B) * ((((F * F) + 2.0) + (2.0 * x)) ^ -(1.0 / 2.0)));
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x / B), $MachinePrecision])}, If[LessEqual[F, -4.9e+27], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 190.0], N[(t$95$0 + N[(N[(F / B), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], (-N[(1.0 / 2.0), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.90000000000000015e27

   1. Initial program 76.7%

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

   2. Taylor expanded in F around -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 16.6

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

   4. Applied rewrites 16.6%

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

   if -4.90000000000000015e27 < F < 190

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 36.4%

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

   if 190 < F

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

   6. Applied rewrites 58.3%

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

   7. Taylor expanded in F around inf

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

      1. lower-sin.f64 37.1

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

   9. Applied rewrites 37.1%

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

Alternative 15: 59.1% accurate, 2.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -4.9 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 190:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \mathsf{fma}\left(2, x, {F}^{2}\right)}}\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.9e+27)
   (/ -1.0 (sin B))
   (if (<= F 190.0)
     (/ (fma -1.0 x (/ F (sqrt (+ 2.0 (fma 2.0 x (pow F 2.0)))))) B)
     (+ (- (/ x B)) (/ 1.0 (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.9e+27) {
		tmp = -1.0 / sin(B);
	} else if (F <= 190.0) {
		tmp = fma(-1.0, x, (F / sqrt((2.0 + fma(2.0, x, pow(F, 2.0)))))) / B;
	} else {
		tmp = -(x / B) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.9e+27)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 190.0)
		tmp = Float64(fma(-1.0, x, Float64(F / sqrt(Float64(2.0 + fma(2.0, x, (F ^ 2.0)))))) / B);
	else
		tmp = Float64(Float64(-Float64(x / B)) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.9e+27], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 190.0], N[(N[(-1.0 * x + N[(F / N[Sqrt[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[((-N[(x / B), $MachinePrecision]) + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.90000000000000015e27

   1. Initial program 76.7%

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

   2. Taylor expanded in F around -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 16.6

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

   4. Applied rewrites 16.6%

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

   if -4.90000000000000015e27 < F < 190

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. add-to-fraction N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 85.0%

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-pow.f64 44.5

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

   8. Applied rewrites 44.5%

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

   if 190 < F

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lift-+.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. +-commutative N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lift-/.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. *-commutative N/A

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

      16. lift-*.f64 N/A

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

   6. Applied rewrites 58.3%

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

   7. Taylor expanded in F around inf

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

      1. lower-sin.f64 37.1

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

   9. Applied rewrites 37.1%

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

Alternative 16: 52.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -4.9 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.15 \cdot 10^{+116}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-1, x, \frac{F}{\sqrt{2 + \mathsf{fma}\left(2, x, {F}^{2}\right)}}\right)}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.9e+27)
   (/ -1.0 (sin B))
   (if (<= F 1.15e+116)
     (/ (fma -1.0 x (/ F (sqrt (+ 2.0 (fma 2.0 x (pow F 2.0)))))) B)
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.9e+27) {
		tmp = -1.0 / sin(B);
	} else if (F <= 1.15e+116) {
		tmp = fma(-1.0, x, (F / sqrt((2.0 + fma(2.0, x, pow(F, 2.0)))))) / B;
	} else {
		tmp = 1.0 / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.9e+27)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 1.15e+116)
		tmp = Float64(fma(-1.0, x, Float64(F / sqrt(Float64(2.0 + fma(2.0, x, (F ^ 2.0)))))) / B);
	else
		tmp = Float64(1.0 / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.9e+27], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.15e+116], N[(N[(-1.0 * x + N[(F / N[Sqrt[N[(2.0 + N[(2.0 * x + N[Power[F, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.90000000000000015e27

   1. Initial program 76.7%

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

   2. Taylor expanded in F around -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 16.6

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

   4. Applied rewrites 16.6%

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

   if -4.90000000000000015e27 < F < 1.14999999999999997e116

   1. Initial program 76.7%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. 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. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. add-to-fraction N/A

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

      6. lower-/.f64 N/A

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

   3. Applied rewrites 85.0%

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-pow.f64 44.5

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

   8. Applied rewrites 44.5%

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

   if 1.14999999999999997e116 < F

   1. Initial program 76.7%

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

   2. Taylor expanded in F around 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: 52.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -6200000000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 0.092:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{-0.5}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6200000000000.0)
   (/ -1.0 (sin B))
   (if (<= F 0.092)
     (+ (- (/ x B)) (/ (* F (pow (+ 2.0 (* 2.0 x)) -0.5)) B))
     (/ 1.0 (sin B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6200000000000.0) {
		tmp = -1.0 / sin(B);
	} else if (F <= 0.092) {
		tmp = -(x / B) + ((F * pow((2.0 + (2.0 * x)), -0.5)) / 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 <= (-6200000000000.0d0)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 0.092d0) then
        tmp = -(x / b) + ((f * ((2.0d0 + (2.0d0 * x)) ** (-0.5d0))) / 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 <= -6200000000000.0) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 0.092) {
		tmp = -(x / B) + ((F * Math.pow((2.0 + (2.0 * x)), -0.5)) / B);
	} else {
		tmp = 1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6200000000000.0:
		tmp = -1.0 / math.sin(B)
	elif F <= 0.092:
		tmp = -(x / B) + ((F * math.pow((2.0 + (2.0 * x)), -0.5)) / B)
	else:
		tmp = 1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6200000000000.0)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 0.092)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F * (Float64(2.0 + Float64(2.0 * x)) ^ -0.5)) / 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 <= -6200000000000.0)
		tmp = -1.0 / sin(B);
	elseif (F <= 0.092)
		tmp = -(x / B) + ((F * ((2.0 + (2.0 * x)) ^ -0.5)) / B);
	else
		tmp = 1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6200000000000.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.092], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F * N[Power[N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.2e12

   1. Initial program 76.7%

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

   2. Taylor expanded in F around -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 16.6

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

   4. Applied rewrites 16.6%

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

   if -6.2e12 < F < 0.091999999999999998

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-sin.f64 37.0

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

   7. Applied rewrites 37.0%

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

   8. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 29.9

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

   10. Applied rewrites 29.9%

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

   if 0.091999999999999998 < F

   1. Initial program 76.7%

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

   2. Taylor expanded in F around 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: 43.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -6200000000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{+89}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{-0.5}}{B}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{x \cdot \left(1 + \frac{1}{x}\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6200000000000.0)
   (/ -1.0 (sin B))
   (if (<= F 1.7e+89)
     (+ (- (/ x B)) (/ (* F (pow (+ 2.0 (* 2.0 x)) -0.5)) B))
     (* -1.0 (/ (* x (+ 1.0 (/ 1.0 x))) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6200000000000.0) {
		tmp = -1.0 / sin(B);
	} else if (F <= 1.7e+89) {
		tmp = -(x / B) + ((F * pow((2.0 + (2.0 * x)), -0.5)) / B);
	} else {
		tmp = -1.0 * ((x * (1.0 + (1.0 / x))) / B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-6200000000000.0d0)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 1.7d+89) then
        tmp = -(x / b) + ((f * ((2.0d0 + (2.0d0 * x)) ** (-0.5d0))) / b)
    else
        tmp = (-1.0d0) * ((x * (1.0d0 + (1.0d0 / x))) / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -6200000000000.0) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 1.7e+89) {
		tmp = -(x / B) + ((F * Math.pow((2.0 + (2.0 * x)), -0.5)) / B);
	} else {
		tmp = -1.0 * ((x * (1.0 + (1.0 / x))) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6200000000000.0:
		tmp = -1.0 / math.sin(B)
	elif F <= 1.7e+89:
		tmp = -(x / B) + ((F * math.pow((2.0 + (2.0 * x)), -0.5)) / B)
	else:
		tmp = -1.0 * ((x * (1.0 + (1.0 / x))) / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6200000000000.0)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 1.7e+89)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F * (Float64(2.0 + Float64(2.0 * x)) ^ -0.5)) / B));
	else
		tmp = Float64(-1.0 * Float64(Float64(x * Float64(1.0 + Float64(1.0 / x))) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -6200000000000.0)
		tmp = -1.0 / sin(B);
	elseif (F <= 1.7e+89)
		tmp = -(x / B) + ((F * ((2.0 + (2.0 * x)) ^ -0.5)) / B);
	else
		tmp = -1.0 * ((x * (1.0 + (1.0 / x))) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6200000000000.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.7e+89], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F * N[Power[N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(x * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.2e12

   1. Initial program 76.7%

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

   2. Taylor expanded in F around -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 16.6

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

   4. Applied rewrites 16.6%

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

   if -6.2e12 < F < 1.7000000000000001e89

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-sin.f64 37.0

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

   7. Applied rewrites 37.0%

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

   8. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 29.9

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

   10. Applied rewrites 29.9%

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

   if 1.7000000000000001e89 < F

   1. Initial program 76.7%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.1

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

   4. Applied rewrites 49.1%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 28.2

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

   7. Applied rewrites 28.2%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 29.4

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

   10. Applied rewrites 29.4%

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

Alternative 19: 43.6% accurate, 2.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;F \leq -0.42:\\ \;\;\;\;-\frac{x + 1}{B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{+89}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{F \cdot {\left(2 + 2 \cdot x\right)}^{-0.5}}{B}\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \frac{x \cdot \left(1 + \frac{1}{x}\right)}{B}\\ \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.42)
   (- (/ (+ x 1.0) B))
   (if (<= F 1.7e+89)
     (+ (- (/ x B)) (/ (* F (pow (+ 2.0 (* 2.0 x)) -0.5)) B))
     (* -1.0 (/ (* x (+ 1.0 (/ 1.0 x))) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.42) {
		tmp = -((x + 1.0) / B);
	} else if (F <= 1.7e+89) {
		tmp = -(x / B) + ((F * pow((2.0 + (2.0 * x)), -0.5)) / B);
	} else {
		tmp = -1.0 * ((x * (1.0 + (1.0 / x))) / B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.42d0)) then
        tmp = -((x + 1.0d0) / b)
    else if (f <= 1.7d+89) then
        tmp = -(x / b) + ((f * ((2.0d0 + (2.0d0 * x)) ** (-0.5d0))) / b)
    else
        tmp = (-1.0d0) * ((x * (1.0d0 + (1.0d0 / x))) / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.42) {
		tmp = -((x + 1.0) / B);
	} else if (F <= 1.7e+89) {
		tmp = -(x / B) + ((F * Math.pow((2.0 + (2.0 * x)), -0.5)) / B);
	} else {
		tmp = -1.0 * ((x * (1.0 + (1.0 / x))) / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.42:
		tmp = -((x + 1.0) / B)
	elif F <= 1.7e+89:
		tmp = -(x / B) + ((F * math.pow((2.0 + (2.0 * x)), -0.5)) / B)
	else:
		tmp = -1.0 * ((x * (1.0 + (1.0 / x))) / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.42)
		tmp = Float64(-Float64(Float64(x + 1.0) / B));
	elseif (F <= 1.7e+89)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(Float64(F * (Float64(2.0 + Float64(2.0 * x)) ^ -0.5)) / B));
	else
		tmp = Float64(-1.0 * Float64(Float64(x * Float64(1.0 + Float64(1.0 / x))) / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.42)
		tmp = -((x + 1.0) / B);
	elseif (F <= 1.7e+89)
		tmp = -(x / B) + ((F * ((2.0 + (2.0 * x)) ^ -0.5)) / B);
	else
		tmp = -1.0 * ((x * (1.0 + (1.0 / x))) / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.42], (-N[(N[(x + 1.0), $MachinePrecision] / B), $MachinePrecision]), If[LessEqual[F, 1.7e+89], N[((-N[(x / B), $MachinePrecision]) + N[(N[(F * N[Power[N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(-1.0 * N[(N[(x * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.419999999999999984

   1. Initial program 76.7%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.1

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

   4. Applied rewrites 49.1%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 28.2

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

   7. Applied rewrites 28.2%

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

      1. lift-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 28.2

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

   9. Applied rewrites 29.4%

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

   if -0.419999999999999984 < F < 1.7000000000000001e89

   1. Initial program 76.7%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 50.1

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

   4. Applied rewrites 50.1%

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

   5. Taylor expanded in F around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. lower-sin.f64 37.0

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

   7. Applied rewrites 37.0%

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

   8. Taylor expanded in B around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-pow.f64 N/A

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

      6. lower-+.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval 29.9

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

   10. Applied rewrites 29.9%

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

   if 1.7000000000000001e89 < F

   1. Initial program 76.7%

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

   2. Taylor expanded in F around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sin.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-sin.f64 49.1

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

   4. Applied rewrites 49.1%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 28.2

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

   7. Applied rewrites 28.2%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-/.f64 29.4

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

   10. Applied rewrites 29.4%

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

Alternative 20: 29.4% accurate, 14.3× speedup

Math

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

FPCore

(FPCore (F B x) :precision binary64 (- (/ (+ x 1.0) B)))

C

double code(double F, double B, double x) {
	return -((x + 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 = -((x + 1.0d0) / b)
end function

Java

public static double code(double F, double B, double x) {
	return -((x + 1.0) / B);
}

Python

def code(F, B, x):
	return -((x + 1.0) / B)

Julia

function code(F, B, x)
	return Float64(-Float64(Float64(x + 1.0) / B))
end

MATLAB

function tmp = code(F, B, x)
	tmp = -((x + 1.0) / B);
end

Wolfram

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

Derivation

1. Initial program 76.7%

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

2. Taylor expanded in F around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sin.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-cos.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-sin.f64 49.1

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

4. Applied rewrites 49.1%

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

5. Taylor expanded in B around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-+.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 28.2

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

7. Applied rewrites 28.2%

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

   1. lift-*.f64 N/A

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

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B}\right) \]

   3. lower-neg.f64 28.2

      \[\leadsto -\frac{F \cdot \left(\frac{1}{F} + \frac{x}{F}\right)}{B} \]

9. Applied rewrites 29.4%

   \[\leadsto -\frac{x + 1}{B} \]
10. Add Preprocessing

Alternative 21: 10.2% 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 76.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in F around -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 16.6

      \[\leadsto \frac{-1}{\sin B} \]

4. Applied rewrites 16.6%

   \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]

5. Taylor expanded in B around 0

   \[\leadsto \frac{-1}{B} \]
6. Step-by-step derivation

7. Applied rewrites 10.2%

   \[\leadsto \frac{-1}{B} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025180 
(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))))))