VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.9% → 99.7%

Time: 7.6s

Alternatives: 24

Speedup: 1.6×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -1 \cdot 10^{+14}:\\ \;\;\;\;-\frac{1 + t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{F}{\sin B} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -1e+14)
     (- (/ (+ 1.0 t_0) (sin B)))
     (if (<= F 100000000.0)
       (+
        (/ (- x) (tan B))
        (* (/ F (sin B)) (pow (fma 2.0 x (fma F F 2.0)) -0.5)))
       (/ (- 1.0 t_0) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -1e+14) {
		tmp = -((1.0 + t_0) / sin(B));
	} else if (F <= 100000000.0) {
		tmp = (-x / tan(B)) + ((F / sin(B)) * pow(fma(2.0, x, fma(F, F, 2.0)), -0.5));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -1e+14)
		tmp = Float64(-Float64(Float64(1.0 + t_0) / sin(B)));
	elseif (F <= 100000000.0)
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(Float64(F / sin(B)) * (fma(2.0, x, fma(F, F, 2.0)) ^ -0.5)));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -1e+14], (-N[(N[(1.0 + t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 100000000.0], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1e14

   1. Initial program 57.7%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if -1e14 < F < 1e8

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.4%

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

   5. Applied rewrites 99.6%

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

   if 1e8 < F

   1. Initial program 56.0%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

Alternative 2: 71.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -x \cdot \frac{1}{\tan B}\\ \mathbf{if}\;t\_0 + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \leq 5 \cdot 10^{+300}:\\ \;\;\;\;t\_0 + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{2}{\left(F \cdot F\right) \cdot B}, \frac{1 - x}{B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* x (/ 1.0 (tan B))))))
   (if (<=
        (+
         t_0
         (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
        5e+300)
     (+ t_0 (* (/ F B) (/ 1.0 (sqrt (+ 2.0 (fma 2.0 x (* F F)))))))
     (fma -0.5 (/ 2.0 (* (* F F) B)) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double t_0 = -(x * (1.0 / tan(B)));
	double tmp;
	if ((t_0 + ((F / sin(B)) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)))) <= 5e+300) {
		tmp = t_0 + ((F / B) * (1.0 / sqrt((2.0 + fma(2.0, x, (F * F))))));
	} else {
		tmp = fma(-0.5, (2.0 / ((F * F) * B)), ((1.0 - x) / B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(-Float64(x * Float64(1.0 / tan(B))))
	tmp = 0.0
	if (Float64(t_0 + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0))))) <= 5e+300)
		tmp = Float64(t_0 + Float64(Float64(F / B) * Float64(1.0 / sqrt(Float64(2.0 + fma(2.0, x, Float64(F * F)))))));
	else
		tmp = fma(-0.5, Float64(2.0 / Float64(Float64(F * F) * B)), Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = (-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision])}, If[LessEqual[N[(t$95$0 + 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], 5e+300], N[(t$95$0 + N[(N[(F / B), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(2.0 + N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(2.0 / N[(N[(F * F), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64)))))) < 5.00000000000000026e300

   1. Initial program 87.4%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 51.4

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

   4. Applied rewrites 51.4%

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

   5. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 70.1

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

   7. Applied rewrites 70.1%

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

   if 5.00000000000000026e300 < (+.f64 (neg.f64 (*.f64 x (/.f64 #s(literal 1 binary64) (tan.f64 B)))) (*.f64 (/.f64 F (sin.f64 B)) (pow.f64 (+.f64 (+.f64 (*.f64 F F) #s(literal 2 binary64)) (*.f64 #s(literal 2 binary64) x)) (neg.f64 (/.f64 #s(literal 1 binary64) #s(literal 2 binary64))))))

   1. Initial program 20.8%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 64.9%

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

   5. Taylor expanded in F around inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{B \cdot {F}^{2}} + \frac{1 - x}{B} \]

      3. lower-fma.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 54.1

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

   7. Applied rewrites 54.1%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 76.1

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

   10. Applied rewrites 76.1%

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

Alternative 3: 91.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-\frac{x}{B}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}\\ t_1 := \cos B \cdot x\\ \mathbf{if}\;F \leq -1.65 \cdot 10^{-10}:\\ \;\;\;\;-\frac{1 + t\_1}{\sin B}\\ \mathbf{elif}\;F \leq -4.8 \cdot 10^{-127}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-158}:\\ \;\;\;\;\left(-\frac{x \cdot 1}{\tan B}\right) + \frac{F}{B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}\\ \mathbf{elif}\;F \leq 40000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (+
          (- (/ x B))
          (/ (* F (pow (fma 2.0 x (fma F F 2.0)) -0.5)) (sin B))))
        (t_1 (* (cos B) x)))
   (if (<= F -1.65e-10)
     (- (/ (+ 1.0 t_1) (sin B)))
     (if (<= F -4.8e-127)
       t_0
       (if (<= F 1.75e-158)
         (+
          (- (/ (* x 1.0) (tan B)))
          (* (/ F B) (pow (+ (+ (* F F) 2.0) (* 2.0 x)) (- (/ 1.0 2.0)))))
         (if (<= F 40000000.0) t_0 (/ (- 1.0 t_1) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B) + ((F * pow(fma(2.0, x, fma(F, F, 2.0)), -0.5)) / sin(B));
	double t_1 = cos(B) * x;
	double tmp;
	if (F <= -1.65e-10) {
		tmp = -((1.0 + t_1) / sin(B));
	} else if (F <= -4.8e-127) {
		tmp = t_0;
	} else if (F <= 1.75e-158) {
		tmp = -((x * 1.0) / tan(B)) + ((F / B) * pow((((F * F) + 2.0) + (2.0 * x)), -(1.0 / 2.0)));
	} else if (F <= 40000000.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 - t_1) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-Float64(x / B)) + Float64(Float64(F * (fma(2.0, x, fma(F, F, 2.0)) ^ -0.5)) / sin(B)))
	t_1 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -1.65e-10)
		tmp = Float64(-Float64(Float64(1.0 + t_1) / sin(B)));
	elseif (F <= -4.8e-127)
		tmp = t_0;
	elseif (F <= 1.75e-158)
		tmp = Float64(Float64(-Float64(Float64(x * 1.0) / tan(B))) + Float64(Float64(F / B) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(2.0 * x)) ^ Float64(-Float64(1.0 / 2.0)))));
	elseif (F <= 40000000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 - t_1) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-N[(x / B), $MachinePrecision]) + N[(N[(F * N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -1.65e-10], (-N[(N[(1.0 + t$95$1), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, -4.8e-127], t$95$0, If[LessEqual[F, 1.75e-158], N[((-N[(N[(x * 1.0), $MachinePrecision] / N[Tan[B], $MachinePrecision]), $MachinePrecision]) + 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], If[LessEqual[F, 40000000.0], t$95$0, N[(N[(1.0 - t$95$1), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.65e-10

   1. Initial program 60.2%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 97.5

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

   4. Applied rewrites 97.5%

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

   if -1.65e-10 < F < -4.79999999999999964e-127 or 1.75000000000000006e-158 < F < 4e7

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in B around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 74.1

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

   6. Applied rewrites 74.1%

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

   if -4.79999999999999964e-127 < F < 1.75000000000000006e-158

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in B around 0

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

   6. Applied rewrites 89.8%

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

   if 4e7 < F

   1. Initial program 56.1%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

Alternative 4: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -55000000000000:\\ \;\;\;\;-\frac{1 + t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 120000000:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -55000000000000.0)
     (- (/ (+ 1.0 t_0) (sin B)))
     (if (<= F 120000000.0)
       (+
        (- (* x (/ 1.0 (tan B))))
        (* (/ F (sin B)) (/ 1.0 (sqrt (+ 2.0 (fma 2.0 x (* F F)))))))
       (/ (- 1.0 t_0) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -55000000000000.0) {
		tmp = -((1.0 + t_0) / sin(B));
	} else if (F <= 120000000.0) {
		tmp = -(x * (1.0 / tan(B))) + ((F / sin(B)) * (1.0 / sqrt((2.0 + fma(2.0, x, (F * F))))));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -55000000000000.0)
		tmp = Float64(-Float64(Float64(1.0 + t_0) / sin(B)));
	elseif (F <= 120000000.0)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / sin(B)) * Float64(1.0 / sqrt(Float64(2.0 + fma(2.0, x, Float64(F * F)))))));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -55000000000000.0], (-N[(N[(1.0 + t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 120000000.0], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(2.0 + N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.5e13

   1. Initial program 57.7%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if -5.5e13 < F < 1.2e8

   1. Initial program 99.4%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 36.3

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

   4. Applied rewrites 36.3%

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

   5. Taylor expanded in B around inf

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

      1. lower-*.f64 N/A

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

      2. lift-sin.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sqrt-div N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. pow2 N/A

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

      9. lower-+.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. lift-fma.f64 99.4

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

   7. Applied rewrites 99.4%

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

   if 1.2e8 < F

   1. Initial program 56.0%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

Alternative 5: 91.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-\frac{x}{B}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}\\ t_1 := \cos B \cdot x\\ \mathbf{if}\;F \leq -1.65 \cdot 10^{-10}:\\ \;\;\;\;-\frac{1 + t\_1}{\sin B}\\ \mathbf{elif}\;F \leq -4.4 \cdot 10^{-134}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-158}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}}\\ \mathbf{elif}\;F \leq 40000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (+
          (- (/ x B))
          (/ (* F (pow (fma 2.0 x (fma F F 2.0)) -0.5)) (sin B))))
        (t_1 (* (cos B) x)))
   (if (<= F -1.65e-10)
     (- (/ (+ 1.0 t_1) (sin B)))
     (if (<= F -4.4e-134)
       t_0
       (if (<= F 1.7e-158)
         (+
          (- (* x (/ 1.0 (tan B))))
          (* (/ F B) (/ 1.0 (sqrt (+ 2.0 (fma 2.0 x (* F F)))))))
         (if (<= F 40000000.0) t_0 (/ (- 1.0 t_1) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = -(x / B) + ((F * pow(fma(2.0, x, fma(F, F, 2.0)), -0.5)) / sin(B));
	double t_1 = cos(B) * x;
	double tmp;
	if (F <= -1.65e-10) {
		tmp = -((1.0 + t_1) / sin(B));
	} else if (F <= -4.4e-134) {
		tmp = t_0;
	} else if (F <= 1.7e-158) {
		tmp = -(x * (1.0 / tan(B))) + ((F / B) * (1.0 / sqrt((2.0 + fma(2.0, x, (F * F))))));
	} else if (F <= 40000000.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 - t_1) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-Float64(x / B)) + Float64(Float64(F * (fma(2.0, x, fma(F, F, 2.0)) ^ -0.5)) / sin(B)))
	t_1 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -1.65e-10)
		tmp = Float64(-Float64(Float64(1.0 + t_1) / sin(B)));
	elseif (F <= -4.4e-134)
		tmp = t_0;
	elseif (F <= 1.7e-158)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / B) * Float64(1.0 / sqrt(Float64(2.0 + fma(2.0, x, Float64(F * F)))))));
	elseif (F <= 40000000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 - t_1) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[((-N[(x / B), $MachinePrecision]) + N[(N[(F * N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -1.65e-10], (-N[(N[(1.0 + t$95$1), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, -4.4e-134], t$95$0, If[LessEqual[F, 1.7e-158], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / B), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(2.0 + N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 40000000.0], t$95$0, N[(N[(1.0 - t$95$1), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.65e-10

   1. Initial program 60.2%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 97.5

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

   4. Applied rewrites 97.5%

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

   if -1.65e-10 < F < -4.3999999999999999e-134 or 1.7e-158 < F < 4e7

   1. Initial program 99.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-sin.f64 N/A

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

      4. lift-pow.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-neg.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. associate-*l/ N/A

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

      12. lower-/.f64 N/A

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in B around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 74.1

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

   6. Applied rewrites 74.1%

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

   if -4.3999999999999999e-134 < F < 1.7e-158

   1. Initial program 99.4%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 35.1

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

   4. Applied rewrites 35.1%

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

   5. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 89.8

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

   7. Applied rewrites 89.8%

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

   if 4e7 < F

   1. Initial program 56.1%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

Alternative 6: 85.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+119}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{-1}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666, B \cdot B, 1\right) \cdot B}\\ \mathbf{elif}\;F \leq -8.5 \cdot 10^{+25}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 175:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1e+119)
   (+
    (/ (- x) (tan B))
    (/
     -1.0
     (*
      (fma
       (- (* 0.008333333333333333 (* B B)) 0.16666666666666666)
       (* B B)
       1.0)
      B)))
   (if (<= F -8.5e+25)
     (+ (- (/ x B)) (/ -1.0 (sin B)))
     (if (<= F 175.0)
       (+
        (- (* x (/ 1.0 (tan B))))
        (* (/ F B) (/ 1.0 (sqrt (+ 2.0 (fma 2.0 x (* F F)))))))
       (/ (- 1.0 (* (cos B) x)) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e+119) {
		tmp = (-x / tan(B)) + (-1.0 / (fma(((0.008333333333333333 * (B * B)) - 0.16666666666666666), (B * B), 1.0) * B));
	} else if (F <= -8.5e+25) {
		tmp = -(x / B) + (-1.0 / sin(B));
	} else if (F <= 175.0) {
		tmp = -(x * (1.0 / tan(B))) + ((F / B) * (1.0 / sqrt((2.0 + fma(2.0, x, (F * F))))));
	} else {
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1e+119)
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(-1.0 / Float64(fma(Float64(Float64(0.008333333333333333 * Float64(B * B)) - 0.16666666666666666), Float64(B * B), 1.0) * B)));
	elseif (F <= -8.5e+25)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(-1.0 / sin(B)));
	elseif (F <= 175.0)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / B) * Float64(1.0 / sqrt(Float64(2.0 + fma(2.0, x, Float64(F * F)))))));
	else
		tmp = Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1e+119], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(N[(N[(N[(0.008333333333333333 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -8.5e+25], N[((-N[(x / B), $MachinePrecision]) + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 175.0], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / B), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(2.0 + N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -9.99999999999999944e118

   1. Initial program 42.0%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-*r/ N/A

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

      6. distribute-neg-frac N/A

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

      7. *-rgt-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-tan.f64 99.8

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 73.8

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

   9. Applied rewrites 73.8%

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

   if -9.99999999999999944e118 < F < -8.5000000000000007e25

   1. Initial program 91.0%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 99.6

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

   4. Applied rewrites 99.6%

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

   5. Taylor expanded in B around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 76.6

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

   7. Applied rewrites 76.6%

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

   if -8.5000000000000007e25 < F < 175

   1. Initial program 99.4%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 37.4

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

   4. Applied rewrites 37.4%

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

   5. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 82.3

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

   7. Applied rewrites 82.3%

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

   if 175 < F

   1. Initial program 56.6%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 99.5

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

   4. Applied rewrites 99.5%

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

Alternative 7: 91.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -1450000000:\\ \;\;\;\;-\frac{1 + t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 175:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -1450000000.0)
     (- (/ (+ 1.0 t_0) (sin B)))
     (if (<= F 175.0)
       (+
        (- (* x (/ 1.0 (tan B))))
        (* (/ F B) (/ 1.0 (sqrt (+ 2.0 (fma 2.0 x (* F F)))))))
       (/ (- 1.0 t_0) (sin B))))))

C

double code(double F, double B, double x) {
	double t_0 = cos(B) * x;
	double tmp;
	if (F <= -1450000000.0) {
		tmp = -((1.0 + t_0) / sin(B));
	} else if (F <= 175.0) {
		tmp = -(x * (1.0 / tan(B))) + ((F / B) * (1.0 / sqrt((2.0 + fma(2.0, x, (F * F))))));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -1450000000.0)
		tmp = Float64(-Float64(Float64(1.0 + t_0) / sin(B)));
	elseif (F <= 175.0)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(Float64(F / B) * Float64(1.0 / sqrt(Float64(2.0 + fma(2.0, x, Float64(F * F)))))));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -1450000000.0], (-N[(N[(1.0 + t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), If[LessEqual[F, 175.0], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(N[(F / B), $MachinePrecision] * N[(1.0 / N[Sqrt[N[(2.0 + N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.45e9

   1. Initial program 58.3%

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

   2. Taylor expanded in F around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. div-add-rev N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lift-sin.f64 99.7

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

   4. Applied rewrites 99.7%

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

   if -1.45e9 < F < 175

   1. Initial program 99.4%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 35.8

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

   4. Applied rewrites 35.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. lower-/.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. pow2 N/A

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

      8. lower-+.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-fma.f64 82.6

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

   7. Applied rewrites 82.6%

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

   if 175 < F

   1. Initial program 56.6%

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

   2. Taylor expanded in F around inf

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

      1. sub-div N/A

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

      2. lower-/.f64 N/A

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

      3. lower--.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 99.5

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

   4. Applied rewrites 99.5%

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

Alternative 8: 56.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 2.05 \cdot 10^{-13}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{-1}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666, B \cdot B, 1\right) \cdot B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 2.05e-13)
   (/ (- (* (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0)))) F) x) B)
   (+
    (/ (- x) (tan B))
    (/
     -1.0
     (*
      (fma
       (- (* 0.008333333333333333 (* B B)) 0.16666666666666666)
       (* B B)
       1.0)
      B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 2.05e-13) {
		tmp = ((sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B;
	} else {
		tmp = (-x / tan(B)) + (-1.0 / (fma(((0.008333333333333333 * (B * B)) - 0.16666666666666666), (B * B), 1.0) * B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 2.05e-13)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B);
	else
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(-1.0 / Float64(fma(Float64(Float64(0.008333333333333333 * Float64(B * B)) - 0.16666666666666666), Float64(B * B), 1.0) * B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 2.05e-13], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(N[(N[(N[(0.008333333333333333 * N[(B * B), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.0500000000000001e-13

   1. Initial program 74.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 57.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-1 N/A

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

      6. pow2 N/A

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

      7. associate-+r+ N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-fma.f64 57.7

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

   6. Applied rewrites 57.7%

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

   if 2.0500000000000001e-13 < B

   1. Initial program 84.2%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 55.8

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

   4. Applied rewrites 55.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-*r/ N/A

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

      6. distribute-neg-frac N/A

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

      7. *-rgt-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-tan.f64 55.9

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

   6. Applied rewrites 55.9%

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

   7. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. pow2 N/A

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

      9. lift-*.f64 N/A

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

      10. pow2 N/A

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

      11. lift-*.f64 54.6

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

   9. Applied rewrites 54.6%

      \[\leadsto \frac{-x}{\tan B} + \frac{-1}{\mathsf{fma}\left(0.008333333333333333 \cdot \left(B \cdot B\right) - 0.16666666666666666, B \cdot B, 1\right) \cdot \color{blue}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 56.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 2.05 \cdot 10^{-13}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 2.05e-13)
   (/ (- (* (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0)))) F) x) B)
   (+
    (- (* x (/ 1.0 (tan B))))
    (/ -1.0 (* B (+ 1.0 (* -0.16666666666666666 (* B B))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 2.05e-13) {
		tmp = ((sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B;
	} else {
		tmp = -(x * (1.0 / tan(B))) + (-1.0 / (B * (1.0 + (-0.16666666666666666 * (B * B)))));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 2.05e-13)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B);
	else
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / tan(B)))) + Float64(-1.0 / Float64(B * Float64(1.0 + Float64(-0.16666666666666666 * Float64(B * B))))));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 2.05e-13], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[((-N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / N[(B * N[(1.0 + N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.0500000000000001e-13

   1. Initial program 74.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 57.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-1 N/A

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

      6. pow2 N/A

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

      7. associate-+r+ N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-fma.f64 57.7

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

   6. Applied rewrites 57.7%

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

   if 2.0500000000000001e-13 < B

   1. Initial program 84.2%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 55.8

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

   4. Applied rewrites 55.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 53.8

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

   7. Applied rewrites 53.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 56.8% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if B < 1

   1. Initial program 74.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 57.7%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 57.8%

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

   if 1 < B

   1. Initial program 84.4%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 55.8

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

   4. Applied rewrites 55.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 53.8

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

   7. Applied rewrites 53.8%

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

   8. Taylor expanded in B around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. pow2 N/A

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

      4. lift-*.f64 53.8

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

   10. Applied rewrites 53.8%

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{B \cdot \left(\left(B \cdot B\right) \cdot -0.16666666666666666\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 56.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{2 + \mathsf{fma}\left(2, x, F \cdot F\right)}}\\ \mathbf{if}\;B \leq 0.88:\\ \;\;\;\;\frac{\mathsf{fma}\left(F, t\_0, \left(B \cdot B\right) \cdot \mathsf{fma}\left(0.16666666666666666 \cdot F, t\_0, 0.3333333333333333 \cdot x\right)\right) - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B} + \frac{-1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt (+ 2.0 (fma 2.0 x (* F F)))))))
   (if (<= B 0.88)
     (/
      (-
       (fma
        F
        t_0
        (*
         (* B B)
         (fma (* 0.16666666666666666 F) t_0 (* 0.3333333333333333 x))))
       x)
      B)
     (+ (/ (- x) (tan B)) (/ -1.0 B)))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sqrt((2.0 + fma(2.0, x, (F * F))));
	double tmp;
	if (B <= 0.88) {
		tmp = (fma(F, t_0, ((B * B) * fma((0.16666666666666666 * F), t_0, (0.3333333333333333 * x)))) - x) / B;
	} else {
		tmp = (-x / tan(B)) + (-1.0 / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sqrt(Float64(2.0 + fma(2.0, x, Float64(F * F)))))
	tmp = 0.0
	if (B <= 0.88)
		tmp = Float64(Float64(fma(F, t_0, Float64(Float64(B * B) * fma(Float64(0.16666666666666666 * F), t_0, Float64(0.3333333333333333 * x)))) - x) / B);
	else
		tmp = Float64(Float64(Float64(-x) / tan(B)) + Float64(-1.0 / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[N[(2.0 + N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 0.88], N[(N[(N[(F * t$95$0 + N[(N[(B * B), $MachinePrecision] * N[(N[(0.16666666666666666 * F), $MachinePrecision] * t$95$0 + N[(0.3333333333333333 * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 0.880000000000000004

   1. Initial program 74.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 57.7%

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

   5. Taylor expanded in B around 0

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

   7. Applied rewrites 57.8%

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

   if 0.880000000000000004 < B

   1. Initial program 84.4%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 55.8

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

   4. Applied rewrites 55.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-tan.f64 N/A

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

      5. associate-*r/ N/A

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

      6. distribute-neg-frac N/A

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

      7. *-rgt-identity N/A

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

      8. lower-/.f64 N/A

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

      9. lower-neg.f64 N/A

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

      10. lift-tan.f64 55.9

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

   6. Applied rewrites 55.9%

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

   7. Taylor expanded in B around 0

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

   9. Applied rewrites 50.7%

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

Alternative 12: 57.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{-10}:\\ \;\;\;\;\left(-\frac{x}{B}\right) + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 40000000:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.7e-10)
   (+ (- (/ x B)) (/ -1.0 (sin B)))
   (if (<= F 40000000.0)
     (/ (- (* (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0)))) F) x) B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.7e-10) {
		tmp = -(x / B) + (-1.0 / sin(B));
	} else if (F <= 40000000.0) {
		tmp = ((sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.7e-10)
		tmp = Float64(Float64(-Float64(x / B)) + Float64(-1.0 / sin(B)));
	elseif (F <= 40000000.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.7e-10], N[((-N[(x / B), $MachinePrecision]) + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 40000000.0], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.70000000000000007e-10

   1. Initial program 60.2%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 97.5

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

   4. Applied rewrites 97.5%

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

   5. Taylor expanded in B around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 73.4

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

   7. Applied rewrites 73.4%

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

   if -1.70000000000000007e-10 < F < 4e7

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 52.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-1 N/A

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

      6. pow2 N/A

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

      7. associate-+r+ N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-fma.f64 52.2

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

   6. Applied rewrites 52.2%

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

   if 4e7 < F

   1. Initial program 56.1%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in F around inf

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

   7. Applied rewrites 50.7%

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

Alternative 13: 50.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.3 \cdot 10^{-21}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, B \cdot B, 0.3333333333333333\right), B \cdot B, 1\right) \cdot B}\right) + \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)}\\ \mathbf{elif}\;F \leq 40000000:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.3e-21)
   (+
    (-
     (*
      x
      (/
       1.0
       (*
        (fma (fma 0.13333333333333333 (* B B) 0.3333333333333333) (* B B) 1.0)
        B))))
    (/ -1.0 (* B (+ 1.0 (* -0.16666666666666666 (* B B))))))
   (if (<= F 40000000.0)
     (/ (- (* (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0)))) F) x) B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.3e-21) {
		tmp = -(x * (1.0 / (fma(fma(0.13333333333333333, (B * B), 0.3333333333333333), (B * B), 1.0) * B))) + (-1.0 / (B * (1.0 + (-0.16666666666666666 * (B * B)))));
	} else if (F <= 40000000.0) {
		tmp = ((sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.3e-21)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / Float64(fma(fma(0.13333333333333333, Float64(B * B), 0.3333333333333333), Float64(B * B), 1.0) * B)))) + Float64(-1.0 / Float64(B * Float64(1.0 + Float64(-0.16666666666666666 * Float64(B * B))))));
	elseif (F <= 40000000.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.3e-21], N[((-N[(x * N[(1.0 / N[(N[(N[(0.13333333333333333 * N[(B * B), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / N[(B * N[(1.0 + N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 40000000.0], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.30000000000000009e-21

   1. Initial program 61.2%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 96.1

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

   4. Applied rewrites 96.1%

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

   5. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 70.9

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

   7. Applied rewrites 70.9%

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

   8. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B \cdot \left(1 + {B}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {B}^{2}\right)\right)}}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \left(B \cdot B\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\left({B}^{2} \cdot \left(\frac{1}{3} + \frac{2}{15} \cdot {B}^{2}\right) + 1\right) \cdot B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \left(B \cdot B\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\left(\left(\frac{1}{3} + \frac{2}{15} \cdot {B}^{2}\right) \cdot {B}^{2} + 1\right) \cdot B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \left(B \cdot B\right)\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\mathsf{fma}\left(\frac{1}{3} + \frac{2}{15} \cdot {B}^{2}, {B}^{2}, 1\right) \cdot B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \left(B \cdot B\right)\right)} \]

      6. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\mathsf{fma}\left(\frac{2}{15} \cdot {B}^{2} + \frac{1}{3}, {B}^{2}, 1\right) \cdot B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \left(B \cdot B\right)\right)} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{15}, {B}^{2}, \frac{1}{3}\right), {B}^{2}, 1\right) \cdot B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \left(B \cdot B\right)\right)} \]

      8. pow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{15}, B \cdot B, \frac{1}{3}\right), {B}^{2}, 1\right) \cdot B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \left(B \cdot B\right)\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{15}, B \cdot B, \frac{1}{3}\right), {B}^{2}, 1\right) \cdot B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \left(B \cdot B\right)\right)} \]

      10. pow2 N/A

          \[\leadsto \left(-x \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{15}, B \cdot B, \frac{1}{3}\right), B \cdot B, 1\right) \cdot B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \left(B \cdot B\right)\right)} \]

      11. lift-*.f64 47.5

          \[\leadsto \left(-x \cdot \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, B \cdot B, 0.3333333333333333\right), B \cdot B, 1\right) \cdot B}\right) + \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)} \]

   10. Applied rewrites 47.5%

       \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.13333333333333333, B \cdot B, 0.3333333333333333\right), B \cdot B, 1\right) \cdot B}}\right) + \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)} \]

   if -3.30000000000000009e-21 < F < 4e7

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 52.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-1 N/A

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

      6. pow2 N/A

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

      7. associate-+r+ N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-fma.f64 52.2

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

   6. Applied rewrites 52.2%

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

   if 4e7 < F

   1. Initial program 56.1%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in F around inf

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

   7. Applied rewrites 50.7%

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

Alternative 14: 50.4% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.3 \cdot 10^{-21}:\\ \;\;\;\;\left(-x \cdot \frac{1}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right) + \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)}\\ \mathbf{elif}\;F \leq 40000000:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.3e-21)
   (+
    (- (* x (/ 1.0 (* (fma 0.3333333333333333 (* B B) 1.0) B))))
    (/ -1.0 (* B (+ 1.0 (* -0.16666666666666666 (* B B))))))
   (if (<= F 40000000.0)
     (/ (- (* (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0)))) F) x) B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.3e-21) {
		tmp = -(x * (1.0 / (fma(0.3333333333333333, (B * B), 1.0) * B))) + (-1.0 / (B * (1.0 + (-0.16666666666666666 * (B * B)))));
	} else if (F <= 40000000.0) {
		tmp = ((sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.3e-21)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / Float64(fma(0.3333333333333333, Float64(B * B), 1.0) * B)))) + Float64(-1.0 / Float64(B * Float64(1.0 + Float64(-0.16666666666666666 * Float64(B * B))))));
	elseif (F <= 40000000.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.3e-21], N[((-N[(x * N[(1.0 / N[(N[(0.3333333333333333 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / N[(B * N[(1.0 + N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 40000000.0], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.30000000000000009e-21

   1. Initial program 61.2%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 96.1

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

   4. Applied rewrites 96.1%

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

   5. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 70.9

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

   7. Applied rewrites 70.9%

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

   8. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\color{blue}{B \cdot \left(1 + \frac{1}{3} \cdot {B}^{2}\right)}}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \left(B \cdot B\right)\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\mathsf{fma}\left(\frac{1}{3}, {B}^{2}, 1\right) \cdot B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \left(B \cdot B\right)\right)} \]

      5. pow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\mathsf{fma}\left(\frac{1}{3}, B \cdot B, 1\right) \cdot B}\right) + \frac{-1}{B \cdot \left(1 + \frac{-1}{6} \cdot \left(B \cdot B\right)\right)} \]

      6. lift-*.f64 47.5

         \[\leadsto \left(-x \cdot \frac{1}{\mathsf{fma}\left(0.3333333333333333, B \cdot B, 1\right) \cdot B}\right) + \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)} \]

   10. Applied rewrites 47.5%

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

   if -3.30000000000000009e-21 < F < 4e7

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 52.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-1 N/A

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

      6. pow2 N/A

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

      7. associate-+r+ N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-fma.f64 52.2

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

   6. Applied rewrites 52.2%

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

   if 4e7 < F

   1. Initial program 56.1%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in F around inf

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

   7. Applied rewrites 50.7%

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

Alternative 15: 50.5% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.3 \cdot 10^{-21}:\\ \;\;\;\;\left(-x \cdot \frac{1}{B}\right) + \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)}\\ \mathbf{elif}\;F \leq 40000000:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.3e-21)
   (+
    (- (* x (/ 1.0 B)))
    (/ -1.0 (* B (+ 1.0 (* -0.16666666666666666 (* B B))))))
   (if (<= F 40000000.0)
     (/ (- (* (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0)))) F) x) B)
     (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.3e-21) {
		tmp = -(x * (1.0 / B)) + (-1.0 / (B * (1.0 + (-0.16666666666666666 * (B * B)))));
	} else if (F <= 40000000.0) {
		tmp = ((sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.3e-21)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / B))) + Float64(-1.0 / Float64(B * Float64(1.0 + Float64(-0.16666666666666666 * Float64(B * B))))));
	elseif (F <= 40000000.0)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.3e-21], N[((-N[(x * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / N[(B * N[(1.0 + N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 40000000.0], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.30000000000000009e-21

   1. Initial program 61.2%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 96.1

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

   4. Applied rewrites 96.1%

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

   5. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 70.9

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

   7. Applied rewrites 70.9%

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

   8. Taylor expanded in B around 0

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

   10. Applied rewrites 47.7%

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

   if -3.30000000000000009e-21 < F < 4e7

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 52.2%

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

      1. lift-pow.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. unpow-1 N/A

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

      6. pow2 N/A

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

      7. associate-+r+ N/A

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

      8. pow2 N/A

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

      9. lower-/.f64 N/A

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

      10. lift-fma.f64 N/A

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

      11. lift-fma.f64 52.2

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

   6. Applied rewrites 52.2%

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

   if 4e7 < F

   1. Initial program 56.1%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 37.5%

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

   5. Taylor expanded in F around inf

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

   7. Applied rewrites 50.7%

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

Alternative 16: 50.4% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.3 \cdot 10^{-21}:\\ \;\;\;\;\left(-x \cdot \frac{1}{B}\right) + \frac{-1}{B \cdot \left(1 + -0.16666666666666666 \cdot \left(B \cdot B\right)\right)}\\ \mathbf{elif}\;F \leq 5:\\ \;\;\;\;\frac{F \cdot \frac{1}{\sqrt{2 + 2 \cdot x}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -0.5, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.3e-21)
   (+
    (- (* x (/ 1.0 B)))
    (/ -1.0 (* B (+ 1.0 (* -0.16666666666666666 (* B B))))))
   (if (<= F 5.0)
     (/ (- (* F (/ 1.0 (sqrt (+ 2.0 (* 2.0 x))))) x) B)
     (/ (- (fma (/ (fma 2.0 x 2.0) (* F F)) -0.5 1.0) x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.3e-21) {
		tmp = -(x * (1.0 / B)) + (-1.0 / (B * (1.0 + (-0.16666666666666666 * (B * B)))));
	} else if (F <= 5.0) {
		tmp = ((F * (1.0 / sqrt((2.0 + (2.0 * x))))) - x) / B;
	} else {
		tmp = (fma((fma(2.0, x, 2.0) / (F * F)), -0.5, 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.3e-21)
		tmp = Float64(Float64(-Float64(x * Float64(1.0 / B))) + Float64(-1.0 / Float64(B * Float64(1.0 + Float64(-0.16666666666666666 * Float64(B * B))))));
	elseif (F <= 5.0)
		tmp = Float64(Float64(Float64(F * Float64(1.0 / sqrt(Float64(2.0 + Float64(2.0 * x))))) - x) / B);
	else
		tmp = Float64(Float64(fma(Float64(fma(2.0, x, 2.0) / Float64(F * F)), -0.5, 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.3e-21], N[((-N[(x * N[(1.0 / B), $MachinePrecision]), $MachinePrecision]) + N[(-1.0 / N[(B * N[(1.0 + N[(-0.16666666666666666 * N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.0], N[(N[(N[(F * N[(1.0 / N[Sqrt[N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.30000000000000009e-21

   1. Initial program 61.2%

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

   2. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 96.1

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

   4. Applied rewrites 96.1%

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

   5. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-+.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 70.9

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

   7. Applied rewrites 70.9%

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

   8. Taylor expanded in B around 0

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

   10. Applied rewrites 47.7%

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

   if -3.30000000000000009e-21 < F < 5

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 52.2%

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

   5. Taylor expanded in B around 0

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

   7. Applied rewrites 52.2%

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

   8. Taylor expanded in F around 0

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

      1. lower-*.f64 52.1

         \[\leadsto \frac{F \cdot \frac{1}{\sqrt{2 + 2 \cdot x}} - x}{B} \]

   10. Applied rewrites 52.1%

       \[\leadsto \frac{F \cdot \frac{1}{\sqrt{2 + 2 \cdot x}} - x}{B} \]

   if 5 < F

   1. Initial program 56.8%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 37.7%

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

   5. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 26.1

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

   7. Applied rewrites 26.1%

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

   8. Taylor expanded in F around inf

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

      1. lower--.f64 N/A

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

   10. Applied rewrites 50.5%

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

Alternative 17: 50.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1450000000:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 5:\\ \;\;\;\;\frac{F \cdot \frac{1}{\sqrt{2 + 2 \cdot x}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -0.5, 1\right) - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1450000000.0)
   (/ (- -1.0 x) B)
   (if (<= F 5.0)
     (/ (- (* F (/ 1.0 (sqrt (+ 2.0 (* 2.0 x))))) x) B)
     (/ (- (fma (/ (fma 2.0 x 2.0) (* F F)) -0.5 1.0) x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1450000000.0) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 5.0) {
		tmp = ((F * (1.0 / sqrt((2.0 + (2.0 * x))))) - x) / B;
	} else {
		tmp = (fma((fma(2.0, x, 2.0) / (F * F)), -0.5, 1.0) - x) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1450000000.0)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 5.0)
		tmp = Float64(Float64(Float64(F * Float64(1.0 / sqrt(Float64(2.0 + Float64(2.0 * x))))) - x) / B);
	else
		tmp = Float64(Float64(fma(Float64(fma(2.0, x, 2.0) / Float64(F * F)), -0.5, 1.0) - x) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1450000000.0], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 5.0], N[(N[(N[(F * N[(1.0 / N[Sqrt[N[(2.0 + N[(2.0 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.45e9

   1. Initial program 58.3%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 36.0%

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

   5. Taylor expanded in F around -inf

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

   7. Applied rewrites 49.1%

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

   if -1.45e9 < F < 5

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 52.1%

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

   5. Taylor expanded in B around 0

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

      1. associate-*r/ N/A

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

      2. metadata-eval N/A

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

      3. metadata-eval N/A

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

   7. Applied rewrites 52.0%

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

   8. Taylor expanded in F around 0

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

      1. lower-*.f64 51.6

         \[\leadsto \frac{F \cdot \frac{1}{\sqrt{2 + 2 \cdot x}} - x}{B} \]

   10. Applied rewrites 51.6%

       \[\leadsto \frac{F \cdot \frac{1}{\sqrt{2 + 2 \cdot x}} - x}{B} \]

   if 5 < F

   1. Initial program 56.8%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 37.7%

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

   5. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 26.1

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

   7. Applied rewrites 26.1%

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

   8. Taylor expanded in F around inf

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

      1. lower--.f64 N/A

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

   10. Applied rewrites 50.5%

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

Alternative 18: 43.6% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-37}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, \frac{2}{\left(F \cdot F\right) \cdot B}, \frac{1 - x}{B}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8.2e-42)
   (/ (- -1.0 x) B)
   (if (<= F 7.5e-54)
     (/ (- x) B)
     (if (<= F 7.5e-37)
       (* (sqrt 0.5) (/ F B))
       (fma -0.5 (/ 2.0 (* (* F F) B)) (/ (- 1.0 x) B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.2e-42) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 7.5e-54) {
		tmp = -x / B;
	} else if (F <= 7.5e-37) {
		tmp = sqrt(0.5) * (F / B);
	} else {
		tmp = fma(-0.5, (2.0 / ((F * F) * B)), ((1.0 - x) / B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8.2e-42)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 7.5e-54)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 7.5e-37)
		tmp = Float64(sqrt(0.5) * Float64(F / B));
	else
		tmp = fma(-0.5, Float64(2.0 / Float64(Float64(F * F) * B)), Float64(Float64(1.0 - x) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8.2e-42], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 7.5e-54], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 7.5e-37], N[(N[Sqrt[0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(2.0 / N[(N[(F * F), $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -8.2000000000000003e-42

   1. Initial program 63.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 37.7%

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

   5. Taylor expanded in F around -inf

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

   7. Applied rewrites 46.0%

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

   if -8.2000000000000003e-42 < F < 7.5000000000000005e-54

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 52.3%

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

   5. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 39.1

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

   7. Applied rewrites 39.1%

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

   if 7.5000000000000005e-54 < F < 7.5000000000000004e-37

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 46.9%

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

   5. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 22.1

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

   7. Applied rewrites 22.1%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\color{blue}{B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\color{blue}{B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{B} \]

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lift-fma.f64 N/A

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

      9. lower-/.f64 27.4

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

   10. Applied rewrites 27.4%

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

   11. Taylor expanded in F around 0

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

   13. Applied rewrites 27.4%

       \[\leadsto \sqrt{0.5} \cdot \frac{F}{B} \]

   if 7.5000000000000004e-37 < F

   1. Initial program 60.5%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 39.2%

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

   5. Taylor expanded in F around inf

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

      1. associate--l+ N/A

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

      2. div-sub N/A

         \[\leadsto \frac{-1}{2} \cdot \frac{2 + 2 \cdot x}{B \cdot {F}^{2}} + \frac{1 - x}{B} \]

      3. lower-fma.f64 N/A

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

      4. associate-/r* N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 40.1

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

   7. Applied rewrites 40.1%

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

   8. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 47.7

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

   10. Applied rewrites 47.7%

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

Alternative 19: 43.7% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-37}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8.2e-42)
   (/ (- -1.0 x) B)
   (if (<= F 7.5e-54)
     (/ (- x) B)
     (if (<= F 7.5e-37) (* (sqrt 0.5) (/ F B)) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.2e-42) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 7.5e-54) {
		tmp = -x / B;
	} else if (F <= 7.5e-37) {
		tmp = sqrt(0.5) * (F / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-8.2d-42)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 7.5d-54) then
        tmp = -x / b
    else if (f <= 7.5d-37) then
        tmp = sqrt(0.5d0) * (f / b)
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.2e-42) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 7.5e-54) {
		tmp = -x / B;
	} else if (F <= 7.5e-37) {
		tmp = Math.sqrt(0.5) * (F / B);
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -8.2e-42:
		tmp = (-1.0 - x) / B
	elif F <= 7.5e-54:
		tmp = -x / B
	elif F <= 7.5e-37:
		tmp = math.sqrt(0.5) * (F / B)
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8.2e-42)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 7.5e-54)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 7.5e-37)
		tmp = Float64(sqrt(0.5) * Float64(F / B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -8.2e-42)
		tmp = (-1.0 - x) / B;
	elseif (F <= 7.5e-54)
		tmp = -x / B;
	elseif (F <= 7.5e-37)
		tmp = sqrt(0.5) * (F / B);
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8.2e-42], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 7.5e-54], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 7.5e-37], N[(N[Sqrt[0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -8.2000000000000003e-42

   1. Initial program 63.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 37.7%

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

   5. Taylor expanded in F around -inf

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

   7. Applied rewrites 46.0%

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

   if -8.2000000000000003e-42 < F < 7.5000000000000005e-54

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 52.3%

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

   5. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 39.1

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

   7. Applied rewrites 39.1%

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

   if 7.5000000000000005e-54 < F < 7.5000000000000004e-37

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 46.9%

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

   5. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 22.1

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

   7. Applied rewrites 22.1%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\color{blue}{B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\color{blue}{B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{B} \]

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lift-fma.f64 N/A

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

      9. lower-/.f64 27.4

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

   10. Applied rewrites 27.4%

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

   11. Taylor expanded in F around 0

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

   13. Applied rewrites 27.4%

       \[\leadsto \sqrt{0.5} \cdot \frac{F}{B} \]

   if 7.5000000000000004e-37 < F

   1. Initial program 60.5%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 39.2%

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

   5. Taylor expanded in F around inf

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

   7. Applied rewrites 48.1%

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

Alternative 20: 43.7% accurate, 8.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-54}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{elif}\;F \leq 7.5 \cdot 10^{-37}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8.2e-42)
   (/ (- -1.0 x) B)
   (if (<= F 7.5e-54)
     (/ (- x) B)
     (if (<= F 7.5e-37) (/ (* F (sqrt 0.5)) B) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.2e-42) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 7.5e-54) {
		tmp = -x / B;
	} else if (F <= 7.5e-37) {
		tmp = (F * sqrt(0.5)) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-8.2d-42)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 7.5d-54) then
        tmp = -x / b
    else if (f <= 7.5d-37) then
        tmp = (f * sqrt(0.5d0)) / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.2e-42) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 7.5e-54) {
		tmp = -x / B;
	} else if (F <= 7.5e-37) {
		tmp = (F * Math.sqrt(0.5)) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -8.2e-42:
		tmp = (-1.0 - x) / B
	elif F <= 7.5e-54:
		tmp = -x / B
	elif F <= 7.5e-37:
		tmp = (F * math.sqrt(0.5)) / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8.2e-42)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 7.5e-54)
		tmp = Float64(Float64(-x) / B);
	elseif (F <= 7.5e-37)
		tmp = Float64(Float64(F * sqrt(0.5)) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -8.2e-42)
		tmp = (-1.0 - x) / B;
	elseif (F <= 7.5e-54)
		tmp = -x / B;
	elseif (F <= 7.5e-37)
		tmp = (F * sqrt(0.5)) / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8.2e-42], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 7.5e-54], N[((-x) / B), $MachinePrecision], If[LessEqual[F, 7.5e-37], N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -8.2000000000000003e-42

   1. Initial program 63.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 37.7%

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

   5. Taylor expanded in F around -inf

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

   7. Applied rewrites 46.0%

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

   if -8.2000000000000003e-42 < F < 7.5000000000000005e-54

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 52.3%

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

   5. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 39.1

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

   7. Applied rewrites 39.1%

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

   if 7.5000000000000005e-54 < F < 7.5000000000000004e-37

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   4. Applied rewrites 46.9%

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

   5. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 22.1

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

   7. Applied rewrites 22.1%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\color{blue}{B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\color{blue}{B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{B} \]

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lift-fma.f64 N/A

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

      9. lower-/.f64 27.4

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

   10. Applied rewrites 27.4%

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

   11. Taylor expanded in F around 0

       \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2}}}{B} \]
   12. Step-by-step derivation

       1. metadata-eval N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2}}}{B} \]

       2. metadata-eval N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2}}}{B} \]

       3. metadata-eval N/A

          \[\leadsto \frac{F \cdot \sqrt{-1 \cdot \frac{-1}{2}}}{B} \]

       4. sqrt-unprod N/A

          \[\leadsto \frac{F \cdot \left(\sqrt{-1} \cdot \sqrt{\frac{-1}{2}}\right)}{B} \]

       5. lower-/.f64 N/A

          \[\leadsto \frac{F \cdot \left(\sqrt{-1} \cdot \sqrt{\frac{-1}{2}}\right)}{B} \]

       6. sqrt-unprod N/A

          \[\leadsto \frac{F \cdot \sqrt{-1 \cdot \frac{-1}{2}}}{B} \]

       7. metadata-eval N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2}}}{B} \]

       8. metadata-eval N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2}}}{B} \]

       9. lower-*.f64 N/A

          \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2}}}{B} \]

       10. lower-sqrt.f64 N/A

           \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2}}}{B} \]

       11. metadata-eval 27.3

           \[\leadsto \frac{F \cdot \sqrt{0.5}}{B} \]

   13. Applied rewrites 27.3%

       \[\leadsto \frac{F \cdot \sqrt{0.5}}{B} \]

   if 7.5000000000000004e-37 < F

   1. Initial program 60.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 39.2%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 48.1%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 21: 43.6% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{-55}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8.2e-42)
   (/ (- -1.0 x) B)
   (if (<= F 4.1e-55) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.2e-42) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.1e-55) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-8.2d-42)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 4.1d-55) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.2e-42) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.1e-55) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -8.2e-42:
		tmp = (-1.0 - x) / B
	elif F <= 4.1e-55:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8.2e-42)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 4.1e-55)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -8.2e-42)
		tmp = (-1.0 - x) / B;
	elseif (F <= 4.1e-55)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8.2e-42], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4.1e-55], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.2000000000000003e-42

   1. Initial program 63.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 37.7%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 46.0%

      \[\leadsto \frac{-1 - x}{B} \]

   if -8.2000000000000003e-42 < F < 4.0999999999999998e-55

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 52.3%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

      2. lower-neg.f64 39.1

         \[\leadsto \frac{-x}{B} \]

   7. Applied rewrites 39.1%

      \[\leadsto \frac{-x}{B} \]

   if 4.0999999999999998e-55 < F

   1. Initial program 62.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 39.6%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around inf

      \[\leadsto \frac{1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 46.6%

      \[\leadsto \frac{1 - x}{B} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 22: 37.0% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8.2e-42) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.2e-42) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-8.2d-42)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = -x / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.2e-42) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -8.2e-42:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8.2e-42)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -8.2e-42)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8.2e-42], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -8.2000000000000003e-42

   1. Initial program 63.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 37.7%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around -inf

      \[\leadsto \frac{-1 - x}{B} \]
   6. Step-by-step derivation

   7. Applied rewrites 46.0%

      \[\leadsto \frac{-1 - x}{B} \]

   if -8.2000000000000003e-42 < F

   1. Initial program 82.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   2. Taylor expanded in B around 0

      \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

   4. Applied rewrites 46.6%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

   5. Taylor expanded in F around 0

      \[\leadsto \frac{-1 \cdot x}{B} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

      2. lower-neg.f64 33.1

         \[\leadsto \frac{-x}{B} \]

   7. Applied rewrites 33.1%

      \[\leadsto \frac{-x}{B} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 30.4% accurate, 26.3× speedup

Math

\[\begin{array}{l} \\ \frac{-x}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ (- x) B))

C

double code(double F, double B, double x) {
	return -x / B;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -x / b
end function

Java

public static double code(double F, double B, double x) {
	return -x / B;
}

Python

def code(F, B, x):
	return -x / B

Julia

function code(F, B, x)
	return Float64(Float64(-x) / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -x / B;
end

Wolfram

code[F_, B_, x_] := N[((-x) / B), $MachinePrecision]

Derivation

1. Initial program 76.9%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

4. Applied rewrites 43.9%

   \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

5. Taylor expanded in F around 0

   \[\leadsto \frac{-1 \cdot x}{B} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

   2. lower-neg.f64 30.4

      \[\leadsto \frac{-x}{B} \]

7. Applied rewrites 30.4%

   \[\leadsto \frac{-x}{B} \]
8. Add Preprocessing

Alternative 24: 10.0% accurate, 30.7× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 76.9%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

2. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

4. Applied rewrites 43.9%

   \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

5. Taylor expanded in F around 0

   \[\leadsto \frac{-1 \cdot x}{B} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \frac{\mathsf{neg}\left(x\right)}{B} \]

   2. lower-neg.f64 30.4

      \[\leadsto \frac{-x}{B} \]

7. Applied rewrites 30.4%

   \[\leadsto \frac{-x}{B} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\color{blue}{B}} \]

   2. lower-*.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\color{blue}{B}} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{B} \]

   4. inv-pow N/A

      \[\leadsto \sqrt{{\left(2 + {F}^{2}\right)}^{-1}} \cdot \frac{F}{B} \]

   5. lower-pow.f64 N/A

      \[\leadsto \sqrt{{\left(2 + {F}^{2}\right)}^{-1}} \cdot \frac{F}{B} \]

   6. +-commutative N/A

      \[\leadsto \sqrt{{\left({F}^{2} + 2\right)}^{-1}} \cdot \frac{F}{B} \]

   7. pow2 N/A

      \[\leadsto \sqrt{{\left(F \cdot F + 2\right)}^{-1}} \cdot \frac{F}{B} \]

   8. lift-fma.f64 N/A

      \[\leadsto \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{B} \]

   9. lower-/.f64 14.5

      \[\leadsto \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{B} \]

10. Applied rewrites 14.5%

    \[\leadsto \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \color{blue}{\frac{F}{B}} \]

11. Taylor expanded in F around -inf

    \[\leadsto \frac{-1}{B} \]
12. Step-by-step derivation

    1. lower-/.f64 10.0

       \[\leadsto \frac{-1}{B} \]

13. Applied rewrites 10.0%

    \[\leadsto \frac{-1}{B} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025097 
(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))))))