VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.7% → 99.6%

Time: 7.9s

Alternatives: 22

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -3.7 \cdot 10^{+36}:\\ \;\;\;\;\frac{-1 - t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 128000000:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{F}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -3.7e+36)
     (/ (- -1.0 t_0) (sin B))
     (if (<= F 128000000.0)
       (fma
        (pow (fma 2.0 x (fma F F 2.0)) -0.5)
        (/ F (sin B))
        (/ (- x) (tan B)))
       (/ (- 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 <= -3.7e+36) {
		tmp = (-1.0 - t_0) / sin(B);
	} else if (F <= 128000000.0) {
		tmp = fma(pow(fma(2.0, x, fma(F, F, 2.0)), -0.5), (F / sin(B)), (-x / tan(B)));
	} 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 <= -3.7e+36)
		tmp = Float64(Float64(-1.0 - t_0) / sin(B));
	elseif (F <= 128000000.0)
		tmp = fma((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5), Float64(F / sin(B)), Float64(Float64(-x) / tan(B)));
	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, -3.7e+36], N[(N[(-1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 128000000.0], N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $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 < -3.70000000000000029e36

   1. Initial program 53.5%

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

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. 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 99.8

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

   5. Applied rewrites 99.8%

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

   if -3.70000000000000029e36 < F < 1.28e8

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 99.6

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

   6. Applied rewrites 99.6%

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

   8. Applied rewrites 99.7%

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

   if 1.28e8 < F

   1. Initial program 48.3%

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

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   4. 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 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.7 \cdot 10^{+36}:\\ \;\;\;\;\frac{-1 - \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 128000000:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{F}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -2.8 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1 - t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 108000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, \frac{F}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -2.8e+17)
     (/ (- -1.0 t_0) (sin B))
     (if (<= F 108000000.0)
       (fma
        (/ 1.0 (sqrt (fma F F (fma 2.0 x 2.0))))
        (/ F (sin B))
        (/ (- x) (tan B)))
       (/ (- 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 <= -2.8e+17) {
		tmp = (-1.0 - t_0) / sin(B);
	} else if (F <= 108000000.0) {
		tmp = fma((1.0 / sqrt(fma(F, F, fma(2.0, x, 2.0)))), (F / sin(B)), (-x / tan(B)));
	} 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 <= -2.8e+17)
		tmp = Float64(Float64(-1.0 - t_0) / sin(B));
	elseif (F <= 108000000.0)
		tmp = fma(Float64(1.0 / sqrt(fma(F, F, fma(2.0, x, 2.0)))), Float64(F / sin(B)), Float64(Float64(-x) / tan(B)));
	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, -2.8e+17], N[(N[(-1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 108000000.0], N[(N[(1.0 / N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[((-x) / N[Tan[B], $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 < -2.8e17

   1. Initial program 55.2%

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

   3. Taylor expanded in F around -inf

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

      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 99.8

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

   5. Applied rewrites 99.8%

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

   if -2.8e17 < F < 1.08e8

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 99.7

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

   6. Applied rewrites 99.7%

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

   8. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. pow2 N/A

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

      5. associate-+r+ N/A

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

      6. pow2 N/A

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

      7. metadata-eval N/A

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

      8. sqrt-pow1 N/A

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

      9. unpow-1 N/A

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

      10. +-commutative N/A

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

      11. pow2 N/A

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

      12. sqrt-div N/A

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

      13. metadata-eval N/A

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

      14. lower-/.f64 N/A

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

      15. +-commutative N/A

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

      16. associate-+r+ N/A

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

      17. pow2 N/A

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

      18. lower-sqrt.f64 N/A

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

   10. Applied rewrites 99.6%

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

   if 1.08e8 < F

   1. Initial program 48.3%

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

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   4. 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 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.8 \cdot 10^{+17}:\\ \;\;\;\;\frac{-1 - \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 108000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}, \frac{F}{\sin B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, -\frac{x}{B}\right)\\ t_1 := \cos B \cdot x\\ \mathbf{if}\;F \leq -2.7 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1 - t\_1}{\sin B}\\ \mathbf{elif}\;F \leq -4.2 \cdot 10^{-66}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-175}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 39000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (fma F (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (sin B)) (- (/ x B))))
        (t_1 (* (cos B) x)))
   (if (<= F -2.7e-15)
     (/ (- -1.0 t_1) (sin B))
     (if (<= F -4.2e-66)
       t_0
       (if (<= F 2.3e-175)
         (/ (* (cos B) (- x)) (sin B))
         (if (<= F 39000.0) t_0 (/ (- 1.0 t_1) (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = fma(F, (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / sin(B)), -(x / B));
	double t_1 = cos(B) * x;
	double tmp;
	if (F <= -2.7e-15) {
		tmp = (-1.0 - t_1) / sin(B);
	} else if (F <= -4.2e-66) {
		tmp = t_0;
	} else if (F <= 2.3e-175) {
		tmp = (cos(B) * -x) / sin(B);
	} else if (F <= 39000.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 - t_1) / sin(B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = fma(F, Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / sin(B)), Float64(-Float64(x / B)))
	t_1 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -2.7e-15)
		tmp = Float64(Float64(-1.0 - t_1) / sin(B));
	elseif (F <= -4.2e-66)
		tmp = t_0;
	elseif (F <= 2.3e-175)
		tmp = Float64(Float64(cos(B) * Float64(-x)) / sin(B));
	elseif (F <= 39000.0)
		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[(F * N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] + (-N[(x / B), $MachinePrecision])), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -2.7e-15], N[(N[(-1.0 - t$95$1), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -4.2e-66], t$95$0, If[LessEqual[F, 2.3e-175], N[(N[(N[Cos[B], $MachinePrecision] * (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 39000.0], 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 < -2.70000000000000009e-15

   1. Initial program 61.4%

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

   3. Taylor expanded in F around -inf

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

      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 99.2

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

   5. Applied rewrites 99.2%

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

   if -2.70000000000000009e-15 < F < -4.2000000000000001e-66 or 2.3e-175 < F < 39000

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 81.2

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

   5. Applied rewrites 81.2%

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

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. lift-pow.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-+.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-/.f64 N/A

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

   7. Applied rewrites 81.2%

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

   if -4.2000000000000001e-66 < F < 2.3e-175

   1. Initial program 99.6%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 90.1

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

   5. Applied rewrites 90.1%

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

   if 39000 < F

   1. Initial program 48.3%

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

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   4. 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 99.8

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

   5. Applied rewrites 99.8%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.7 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1 - \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq -4.2 \cdot 10^{-66}:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, -\frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-175}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 39000:\\ \;\;\;\;\mathsf{fma}\left(F, \frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B}, -\frac{x}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -2.7 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1 - t\_0}{\sin B}\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{-38}:\\ \;\;\;\;\frac{\sqrt{0.5} \cdot F}{\sin B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-167}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 9000:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -2.7e-15)
     (/ (- -1.0 t_0) (sin B))
     (if (<= F -1.3e-38)
       (/ (* (sqrt 0.5) F) (sin B))
       (if (<= F 1.05e-167)
         (/ (* (cos B) (- x)) (sin B))
         (if (<= F 9000.0)
           (* (sqrt (pow (fma F F 2.0) -1.0)) (/ F (sin B)))
           (/ (- 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 <= -2.7e-15) {
		tmp = (-1.0 - t_0) / sin(B);
	} else if (F <= -1.3e-38) {
		tmp = (sqrt(0.5) * F) / sin(B);
	} else if (F <= 1.05e-167) {
		tmp = (cos(B) * -x) / sin(B);
	} else if (F <= 9000.0) {
		tmp = sqrt(pow(fma(F, F, 2.0), -1.0)) * (F / sin(B));
	} 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 <= -2.7e-15)
		tmp = Float64(Float64(-1.0 - t_0) / sin(B));
	elseif (F <= -1.3e-38)
		tmp = Float64(Float64(sqrt(0.5) * F) / sin(B));
	elseif (F <= 1.05e-167)
		tmp = Float64(Float64(cos(B) * Float64(-x)) / sin(B));
	elseif (F <= 9000.0)
		tmp = Float64(sqrt((fma(F, F, 2.0) ^ -1.0)) * Float64(F / sin(B)));
	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, -2.7e-15], N[(N[(-1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.3e-38], N[(N[(N[Sqrt[0.5], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.05e-167], N[(N[(N[Cos[B], $MachinePrecision] * (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9000.0], N[(N[Sqrt[N[Power[N[(F * F + 2.0), $MachinePrecision], -1.0], $MachinePrecision]], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -2.70000000000000009e-15

   1. Initial program 61.4%

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

   3. Taylor expanded in F around -inf

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

      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 99.2

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

   5. Applied rewrites 99.2%

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

   if -2.70000000000000009e-15 < F < -1.30000000000000005e-38

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-/.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in F around 0

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

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2}}}{\sin B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2}}}{\sin B} \]

      3. metadata-eval N/A

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

      4. sqrt-unprod N/A

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

      5. lower-/.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2}}}{\sin B} \]

      8. metadata-eval N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2}}}{\sin B} \]

      9. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2}} \cdot F}{\sin B} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{2}} \cdot F}{\sin B} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{2}} \cdot F}{\sin B} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sqrt{\frac{1}{2}} \cdot F}{\sin B} \]

      13. lift-sin.f64 99.6

          \[\leadsto \frac{\sqrt{0.5} \cdot F}{\sin B} \]

   8. Applied rewrites 99.6%

      \[\leadsto \frac{\sqrt{0.5} \cdot F}{\color{blue}{\sin B}} \]

   if -1.30000000000000005e-38 < F < 1.05000000000000009e-167

   1. Initial program 99.6%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 88.8

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

   5. Applied rewrites 88.8%

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

   if 1.05000000000000009e-167 < F < 9e3

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-/.f64 63.4

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

   5. Applied rewrites 63.4%

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

   if 9e3 < F

   1. Initial program 48.3%

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

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   4. 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 99.8

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

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{1 - \cos B \cdot x}{\sin B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.7 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1 - \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{-38}:\\ \;\;\;\;\frac{\sqrt{0.5} \cdot F}{\sin B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-167}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 9000:\\ \;\;\;\;\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -1800000:\\ \;\;\;\;\frac{-1 - t\_0}{\sin B}\\ \mathbf{elif}\;F \leq 1.42 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{F}{B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -1800000.0)
     (/ (- -1.0 t_0) (sin B))
     (if (<= F 1.42e-36)
       (fma (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ F B) (/ (- x) (tan B)))
       (/ (- 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 <= -1800000.0) {
		tmp = (-1.0 - t_0) / sin(B);
	} else if (F <= 1.42e-36) {
		tmp = fma(pow(fma(2.0, x, fma(F, F, 2.0)), -0.5), (F / B), (-x / tan(B)));
	} 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 <= -1800000.0)
		tmp = Float64(Float64(-1.0 - t_0) / sin(B));
	elseif (F <= 1.42e-36)
		tmp = fma((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5), Float64(F / B), Float64(Float64(-x) / tan(B)));
	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, -1800000.0], N[(N[(-1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.42e-36], N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision] + N[((-x) / N[Tan[B], $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 < -1.8e6

   1. Initial program 58.2%

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

   3. Taylor expanded in F around -inf

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

      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 99.8

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

   5. Applied rewrites 99.8%

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

   if -1.8e6 < F < 1.41999999999999996e-36

   1. Initial program 99.6%

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

      1. lift-*.f64 N/A

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

      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

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-tan.f64 N/A

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

      4. associate-*r/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift-tan.f64 99.7

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

   6. Applied rewrites 99.7%

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in B around 0

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

   11. Applied rewrites 86.9%

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

   if 1.41999999999999996e-36 < F

   1. Initial program 52.3%

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

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   4. 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 96.1

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

   5. Applied rewrites 96.1%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1800000:\\ \;\;\;\;\frac{-1 - \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq 1.42 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left({\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}, \frac{F}{B}, \frac{-x}{\tan B}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos B \cdot x\\ \mathbf{if}\;F \leq -2.7 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1 - t\_0}{\sin B}\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{-38}:\\ \;\;\;\;\frac{\sqrt{0.5} \cdot F}{\sin B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-167}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - t\_0}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* (cos B) x)))
   (if (<= F -2.7e-15)
     (/ (- -1.0 t_0) (sin B))
     (if (<= F -1.3e-38)
       (/ (* (sqrt 0.5) F) (sin B))
       (if (<= F 1.05e-167)
         (/ (* (cos B) (- x)) (sin B))
         (if (<= F 1.7e-13)
           (* (sqrt 0.5) (/ F (sin B)))
           (/ (- 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 <= -2.7e-15) {
		tmp = (-1.0 - t_0) / sin(B);
	} else if (F <= -1.3e-38) {
		tmp = (sqrt(0.5) * F) / sin(B);
	} else if (F <= 1.05e-167) {
		tmp = (cos(B) * -x) / sin(B);
	} else if (F <= 1.7e-13) {
		tmp = sqrt(0.5) * (F / sin(B));
	} else {
		tmp = (1.0 - t_0) / sin(B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = cos(b) * x
    if (f <= (-2.7d-15)) then
        tmp = ((-1.0d0) - t_0) / sin(b)
    else if (f <= (-1.3d-38)) then
        tmp = (sqrt(0.5d0) * f) / sin(b)
    else if (f <= 1.05d-167) then
        tmp = (cos(b) * -x) / sin(b)
    else if (f <= 1.7d-13) then
        tmp = sqrt(0.5d0) * (f / sin(b))
    else
        tmp = (1.0d0 - t_0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.cos(B) * x;
	double tmp;
	if (F <= -2.7e-15) {
		tmp = (-1.0 - t_0) / Math.sin(B);
	} else if (F <= -1.3e-38) {
		tmp = (Math.sqrt(0.5) * F) / Math.sin(B);
	} else if (F <= 1.05e-167) {
		tmp = (Math.cos(B) * -x) / Math.sin(B);
	} else if (F <= 1.7e-13) {
		tmp = Math.sqrt(0.5) * (F / Math.sin(B));
	} else {
		tmp = (1.0 - t_0) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.cos(B) * x
	tmp = 0
	if F <= -2.7e-15:
		tmp = (-1.0 - t_0) / math.sin(B)
	elif F <= -1.3e-38:
		tmp = (math.sqrt(0.5) * F) / math.sin(B)
	elif F <= 1.05e-167:
		tmp = (math.cos(B) * -x) / math.sin(B)
	elif F <= 1.7e-13:
		tmp = math.sqrt(0.5) * (F / math.sin(B))
	else:
		tmp = (1.0 - t_0) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(cos(B) * x)
	tmp = 0.0
	if (F <= -2.7e-15)
		tmp = Float64(Float64(-1.0 - t_0) / sin(B));
	elseif (F <= -1.3e-38)
		tmp = Float64(Float64(sqrt(0.5) * F) / sin(B));
	elseif (F <= 1.05e-167)
		tmp = Float64(Float64(cos(B) * Float64(-x)) / sin(B));
	elseif (F <= 1.7e-13)
		tmp = Float64(sqrt(0.5) * Float64(F / sin(B)));
	else
		tmp = Float64(Float64(1.0 - t_0) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = cos(B) * x;
	tmp = 0.0;
	if (F <= -2.7e-15)
		tmp = (-1.0 - t_0) / sin(B);
	elseif (F <= -1.3e-38)
		tmp = (sqrt(0.5) * F) / sin(B);
	elseif (F <= 1.05e-167)
		tmp = (cos(B) * -x) / sin(B);
	elseif (F <= 1.7e-13)
		tmp = sqrt(0.5) * (F / sin(B));
	else
		tmp = (1.0 - t_0) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[Cos[B], $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -2.7e-15], N[(N[(-1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.3e-38], N[(N[(N[Sqrt[0.5], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.05e-167], N[(N[(N[Cos[B], $MachinePrecision] * (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.7e-13], N[(N[Sqrt[0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - t$95$0), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -2.70000000000000009e-15

   1. Initial program 61.4%

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

   3. Taylor expanded in F around -inf

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

      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 99.2

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

   5. Applied rewrites 99.2%

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

   if -2.70000000000000009e-15 < F < -1.30000000000000005e-38

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-/.f64 99.6

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

   5. Applied rewrites 99.6%

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

   6. Taylor expanded in F around 0

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

      1. metadata-eval N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2}}}{\sin B} \]

      2. metadata-eval N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2}}}{\sin B} \]

      3. metadata-eval N/A

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

      4. sqrt-unprod N/A

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

      5. lower-/.f64 N/A

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

      6. sqrt-unprod N/A

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

      7. metadata-eval N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2}}}{\sin B} \]

      8. metadata-eval N/A

         \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2}}}{\sin B} \]

      9. *-commutative N/A

         \[\leadsto \frac{\sqrt{\frac{1}{2}} \cdot F}{\sin B} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{2}} \cdot F}{\sin B} \]

      11. lower-sqrt.f64 N/A

          \[\leadsto \frac{\sqrt{\frac{1}{2}} \cdot F}{\sin B} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\sqrt{\frac{1}{2}} \cdot F}{\sin B} \]

      13. lift-sin.f64 99.6

          \[\leadsto \frac{\sqrt{0.5} \cdot F}{\sin B} \]

   8. Applied rewrites 99.6%

      \[\leadsto \frac{\sqrt{0.5} \cdot F}{\color{blue}{\sin B}} \]

   if -1.30000000000000005e-38 < F < 1.05000000000000009e-167

   1. Initial program 99.6%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 88.8

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

   5. Applied rewrites 88.8%

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

   if 1.05000000000000009e-167 < F < 1.70000000000000008e-13

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-/.f64 62.9

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

   5. Applied rewrites 62.9%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 62.9%

      \[\leadsto \sqrt{0.5} \cdot \frac{F}{\sin B} \]

   if 1.70000000000000008e-13 < F

   1. Initial program 50.4%

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

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   4. 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 98.4

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

   5. Applied rewrites 98.4%

      \[\leadsto \color{blue}{\frac{1 - \cos B \cdot x}{\sin B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.7 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1 - \cos B \cdot x}{\sin B}\\ \mathbf{elif}\;F \leq -1.3 \cdot 10^{-38}:\\ \;\;\;\;\frac{\sqrt{0.5} \cdot F}{\sin B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-167}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -28500000000:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-167}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -28500000000.0)
   (/ (- -1.0 x) (sin B))
   (if (<= F 1.05e-167)
     (/ (* (cos B) (- x)) (sin B))
     (if (<= F 1.7e-13)
       (* (sqrt 0.5) (/ F (sin B)))
       (/ (- 1.0 (* (cos B) x)) (sin B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -28500000000.0) {
		tmp = (-1.0 - x) / sin(B);
	} else if (F <= 1.05e-167) {
		tmp = (cos(B) * -x) / sin(B);
	} else if (F <= 1.7e-13) {
		tmp = sqrt(0.5) * (F / sin(B));
	} else {
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-28500000000.0d0)) then
        tmp = ((-1.0d0) - x) / sin(b)
    else if (f <= 1.05d-167) then
        tmp = (cos(b) * -x) / sin(b)
    else if (f <= 1.7d-13) then
        tmp = sqrt(0.5d0) * (f / sin(b))
    else
        tmp = (1.0d0 - (cos(b) * x)) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -28500000000.0) {
		tmp = (-1.0 - x) / Math.sin(B);
	} else if (F <= 1.05e-167) {
		tmp = (Math.cos(B) * -x) / Math.sin(B);
	} else if (F <= 1.7e-13) {
		tmp = Math.sqrt(0.5) * (F / Math.sin(B));
	} else {
		tmp = (1.0 - (Math.cos(B) * x)) / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -28500000000.0:
		tmp = (-1.0 - x) / math.sin(B)
	elif F <= 1.05e-167:
		tmp = (math.cos(B) * -x) / math.sin(B)
	elif F <= 1.7e-13:
		tmp = math.sqrt(0.5) * (F / math.sin(B))
	else:
		tmp = (1.0 - (math.cos(B) * x)) / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -28500000000.0)
		tmp = Float64(Float64(-1.0 - x) / sin(B));
	elseif (F <= 1.05e-167)
		tmp = Float64(Float64(cos(B) * Float64(-x)) / sin(B));
	elseif (F <= 1.7e-13)
		tmp = Float64(sqrt(0.5) * Float64(F / sin(B)));
	else
		tmp = Float64(Float64(1.0 - Float64(cos(B) * x)) / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -28500000000.0)
		tmp = (-1.0 - x) / sin(B);
	elseif (F <= 1.05e-167)
		tmp = (cos(B) * -x) / sin(B);
	elseif (F <= 1.7e-13)
		tmp = sqrt(0.5) * (F / sin(B));
	else
		tmp = (1.0 - (cos(B) * x)) / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -28500000000.0], N[(N[(-1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.05e-167], N[(N[(N[Cos[B], $MachinePrecision] * (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.7e-13], N[(N[Sqrt[0.5], $MachinePrecision] * N[(F / N[Sin[B], $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 < -2.85e10

   1. Initial program 57.5%

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

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. 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 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 88.1%

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

   if -2.85e10 < F < 1.05000000000000009e-167

   1. Initial program 99.6%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 81.9

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

   5. Applied rewrites 81.9%

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

   if 1.05000000000000009e-167 < F < 1.70000000000000008e-13

   1. Initial program 99.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-/.f64 62.9

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

   5. Applied rewrites 62.9%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 62.9%

      \[\leadsto \sqrt{0.5} \cdot \frac{F}{\sin B} \]

   if 1.70000000000000008e-13 < F

   1. Initial program 50.4%

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

   3. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x \cdot \cos B}{\sin B}} \]
   4. 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 98.4

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

   5. Applied rewrites 98.4%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -28500000000:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-167}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-13}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos B \cdot x}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -28500000000:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-167}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.06 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;F \leq 4 \cdot 10^{+166}:\\ \;\;\;\;\frac{\frac{1 - x \cdot x}{1 + x}}{B}\\ \mathbf{else}:\\ \;\;\;\;{\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -28500000000.0)
   (/ (- -1.0 x) (sin B))
   (if (<= F 1.05e-167)
     (/ (* (cos B) (- x)) (sin B))
     (if (<= F 1.06e+77)
       (* (/ 1.0 (sqrt (fma F F 2.0))) (/ F (sin B)))
       (if (<= F 4e+166)
         (/ (/ (- 1.0 (* x x)) (+ 1.0 x)) B)
         (pow (sin B) -1.0))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -28500000000.0) {
		tmp = (-1.0 - x) / sin(B);
	} else if (F <= 1.05e-167) {
		tmp = (cos(B) * -x) / sin(B);
	} else if (F <= 1.06e+77) {
		tmp = (1.0 / sqrt(fma(F, F, 2.0))) * (F / sin(B));
	} else if (F <= 4e+166) {
		tmp = ((1.0 - (x * x)) / (1.0 + x)) / B;
	} else {
		tmp = pow(sin(B), -1.0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -28500000000.0)
		tmp = Float64(Float64(-1.0 - x) / sin(B));
	elseif (F <= 1.05e-167)
		tmp = Float64(Float64(cos(B) * Float64(-x)) / sin(B));
	elseif (F <= 1.06e+77)
		tmp = Float64(Float64(1.0 / sqrt(fma(F, F, 2.0))) * Float64(F / sin(B)));
	elseif (F <= 4e+166)
		tmp = Float64(Float64(Float64(1.0 - Float64(x * x)) / Float64(1.0 + x)) / B);
	else
		tmp = sin(B) ^ -1.0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -28500000000.0], N[(N[(-1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.05e-167], N[(N[(N[Cos[B], $MachinePrecision] * (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.06e+77], N[(N[(1.0 / N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4e+166], N[(N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -2.85e10

   1. Initial program 57.5%

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

   3. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x \cdot \cos B}{\sin B}\right)} \]
   4. 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 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 88.1%

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

   if -2.85e10 < F < 1.05000000000000009e-167

   1. Initial program 99.6%

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

   3. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-cos.f64 N/A

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

      7. lift-sin.f64 81.9

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

   5. Applied rewrites 81.9%

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

   if 1.05000000000000009e-167 < F < 1.06000000000000003e77

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-/.f64 65.7

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

   5. Applied rewrites 65.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. unpow-1 N/A

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

      5. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{{F}^{2} + 2}} \cdot \frac{F}{\sin B} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\sin B} \]

      7. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{2 + {F}^{2}}} \cdot \frac{F}{\sin B} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{2 + {F}^{2}}} \cdot \frac{F}{\sin B} \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{\sqrt{{F}^{2} + 2}} \cdot \frac{F}{\sin B} \]

      12. pow2 N/A

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

      13. lift-fma.f64 65.6

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

   7. Applied rewrites 65.6%

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

   if 1.06000000000000003e77 < F < 3.99999999999999976e166

   1. Initial program 71.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 60.1%

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

   6. Taylor expanded in F around inf

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

      1. lower--.f64 60.2

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

   8. Applied rewrites 60.2%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 65.6

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

   10. Applied rewrites 65.6%

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

   if 3.99999999999999976e166 < F

   1. Initial program 15.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-/.f64 1.9

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

   5. Applied rewrites 1.9%

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

   6. Taylor expanded in F around inf

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. lift-sin.f64 67.5

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

   8. Applied rewrites 67.5%

      \[\leadsto {\sin B}^{\color{blue}{-1}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -28500000000:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 1.05 \cdot 10^{-167}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 1.06 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;F \leq 4 \cdot 10^{+166}:\\ \;\;\;\;\frac{\frac{1 - x \cdot x}{1 + x}}{B}\\ \mathbf{else}:\\ \;\;\;\;{\sin B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 1.02 \cdot 10^{-167}:\\ \;\;\;\;\frac{F}{B} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.06 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;F \leq 4 \cdot 10^{+166}:\\ \;\;\;\;\frac{\frac{1 - x \cdot x}{1 + x}}{B}\\ \mathbf{else}:\\ \;\;\;\;{\sin B}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.5e-15)
   (/ (- -1.0 x) (sin B))
   (if (<= F 1.02e-167)
     (- (* (/ F B) (pow (fma 2.0 x (fma F F 2.0)) -0.5)) (/ x B))
     (if (<= F 1.06e+77)
       (* (/ 1.0 (sqrt (fma F F 2.0))) (/ F (sin B)))
       (if (<= F 4e+166)
         (/ (/ (- 1.0 (* x x)) (+ 1.0 x)) B)
         (pow (sin B) -1.0))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.5e-15) {
		tmp = (-1.0 - x) / sin(B);
	} else if (F <= 1.02e-167) {
		tmp = ((F / B) * pow(fma(2.0, x, fma(F, F, 2.0)), -0.5)) - (x / B);
	} else if (F <= 1.06e+77) {
		tmp = (1.0 / sqrt(fma(F, F, 2.0))) * (F / sin(B));
	} else if (F <= 4e+166) {
		tmp = ((1.0 - (x * x)) / (1.0 + x)) / B;
	} else {
		tmp = pow(sin(B), -1.0);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.5e-15)
		tmp = Float64(Float64(-1.0 - x) / sin(B));
	elseif (F <= 1.02e-167)
		tmp = Float64(Float64(Float64(F / B) * (fma(2.0, x, fma(F, F, 2.0)) ^ -0.5)) - Float64(x / B));
	elseif (F <= 1.06e+77)
		tmp = Float64(Float64(1.0 / sqrt(fma(F, F, 2.0))) * Float64(F / sin(B)));
	elseif (F <= 4e+166)
		tmp = Float64(Float64(Float64(1.0 - Float64(x * x)) / Float64(1.0 + x)) / B);
	else
		tmp = sin(B) ^ -1.0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.5e-15], N[(N[(-1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.02e-167], N[(N[(N[(F / B), $MachinePrecision] * N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.06e+77], N[(N[(1.0 / N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4e+166], N[(N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], N[Power[N[Sin[B], $MachinePrecision], -1.0], $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -2.5e-15

   1. Initial program 61.4%

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

   3. Taylor expanded in F around -inf

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

      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 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 81.7%

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

   if -2.5e-15 < F < 1.0199999999999999e-167

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 49.6%

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

   7. Applied rewrites 49.6%

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

   if 1.0199999999999999e-167 < F < 1.06000000000000003e77

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-/.f64 65.7

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

   5. Applied rewrites 65.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. unpow-1 N/A

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

      5. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{{F}^{2} + 2}} \cdot \frac{F}{\sin B} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\sin B} \]

      7. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{2 + {F}^{2}}} \cdot \frac{F}{\sin B} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{2 + {F}^{2}}} \cdot \frac{F}{\sin B} \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{\sqrt{{F}^{2} + 2}} \cdot \frac{F}{\sin B} \]

      12. pow2 N/A

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

      13. lift-fma.f64 65.6

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

   7. Applied rewrites 65.6%

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

   if 1.06000000000000003e77 < F < 3.99999999999999976e166

   1. Initial program 71.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 60.1%

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

   6. Taylor expanded in F around inf

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

      1. lower--.f64 60.2

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

   8. Applied rewrites 60.2%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 65.6

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

   10. Applied rewrites 65.6%

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

   if 3.99999999999999976e166 < F

   1. Initial program 15.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-/.f64 1.9

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

   5. Applied rewrites 1.9%

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

   6. Taylor expanded in F around inf

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. lift-sin.f64 67.5

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

   8. Applied rewrites 67.5%

      \[\leadsto {\sin B}^{\color{blue}{-1}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 1.02 \cdot 10^{-167}:\\ \;\;\;\;\frac{F}{B} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.06 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \mathbf{elif}\;F \leq 4 \cdot 10^{+166}:\\ \;\;\;\;\frac{\frac{1 - x \cdot x}{1 + x}}{B}\\ \mathbf{else}:\\ \;\;\;\;{\sin B}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 1.02 \cdot 10^{-167}:\\ \;\;\;\;\frac{F}{B} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.06 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - x \cdot x}{1 + x}}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.5e-15)
   (/ (- -1.0 x) (sin B))
   (if (<= F 1.02e-167)
     (- (* (/ F B) (pow (fma 2.0 x (fma F F 2.0)) -0.5)) (/ x B))
     (if (<= F 1.06e+77)
       (* (/ 1.0 (sqrt (fma F F 2.0))) (/ F (sin B)))
       (/ (/ (- 1.0 (* x x)) (+ 1.0 x)) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.5e-15) {
		tmp = (-1.0 - x) / sin(B);
	} else if (F <= 1.02e-167) {
		tmp = ((F / B) * pow(fma(2.0, x, fma(F, F, 2.0)), -0.5)) - (x / B);
	} else if (F <= 1.06e+77) {
		tmp = (1.0 / sqrt(fma(F, F, 2.0))) * (F / sin(B));
	} else {
		tmp = ((1.0 - (x * x)) / (1.0 + x)) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.5e-15)
		tmp = Float64(Float64(-1.0 - x) / sin(B));
	elseif (F <= 1.02e-167)
		tmp = Float64(Float64(Float64(F / B) * (fma(2.0, x, fma(F, F, 2.0)) ^ -0.5)) - Float64(x / B));
	elseif (F <= 1.06e+77)
		tmp = Float64(Float64(1.0 / sqrt(fma(F, F, 2.0))) * Float64(F / sin(B)));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(x * x)) / Float64(1.0 + x)) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.5e-15], N[(N[(-1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.02e-167], N[(N[(N[(F / B), $MachinePrecision] * N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.06e+77], N[(N[(1.0 / N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.5e-15

   1. Initial program 61.4%

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

   3. Taylor expanded in F around -inf

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

      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 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 81.7%

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

   if -2.5e-15 < F < 1.0199999999999999e-167

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 49.6%

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

   7. Applied rewrites 49.6%

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

   if 1.0199999999999999e-167 < F < 1.06000000000000003e77

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-/.f64 65.7

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

   5. Applied rewrites 65.7%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. unpow-1 N/A

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

      5. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{{F}^{2} + 2}} \cdot \frac{F}{\sin B} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\sin B} \]

      7. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{2 + {F}^{2}}} \cdot \frac{F}{\sin B} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{2 + {F}^{2}}} \cdot \frac{F}{\sin B} \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{\sqrt{{F}^{2} + 2}} \cdot \frac{F}{\sin B} \]

      12. pow2 N/A

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

      13. lift-fma.f64 65.6

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

   7. Applied rewrites 65.6%

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

   if 1.06000000000000003e77 < F

   1. Initial program 32.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 32.9%

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

   6. Taylor expanded in F around inf

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

      1. lower--.f64 49.3

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

   8. Applied rewrites 49.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 51.1

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

   10. Applied rewrites 51.1%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 1.02 \cdot 10^{-167}:\\ \;\;\;\;\frac{F}{B} \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.06 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - x \cdot x}{1 + x}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-164}:\\ \;\;\;\;\frac{\sqrt{0.5} \cdot F - x}{B}\\ \mathbf{elif}\;F \leq 1.06 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - x \cdot x}{1 + x}}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.5e-15)
   (/ (- -1.0 x) (sin B))
   (if (<= F 3.5e-164)
     (/ (- (* (sqrt 0.5) F) x) B)
     (if (<= F 1.06e+77)
       (* (/ 1.0 (sqrt (fma F F 2.0))) (/ F (sin B)))
       (/ (/ (- 1.0 (* x x)) (+ 1.0 x)) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.5e-15) {
		tmp = (-1.0 - x) / sin(B);
	} else if (F <= 3.5e-164) {
		tmp = ((sqrt(0.5) * F) - x) / B;
	} else if (F <= 1.06e+77) {
		tmp = (1.0 / sqrt(fma(F, F, 2.0))) * (F / sin(B));
	} else {
		tmp = ((1.0 - (x * x)) / (1.0 + x)) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.5e-15)
		tmp = Float64(Float64(-1.0 - x) / sin(B));
	elseif (F <= 3.5e-164)
		tmp = Float64(Float64(Float64(sqrt(0.5) * F) - x) / B);
	elseif (F <= 1.06e+77)
		tmp = Float64(Float64(1.0 / sqrt(fma(F, F, 2.0))) * Float64(F / sin(B)));
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(x * x)) / Float64(1.0 + x)) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.5e-15], N[(N[(-1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.5e-164], N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.06e+77], N[(N[(1.0 / N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.5e-15

   1. Initial program 61.4%

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

   3. Taylor expanded in F around -inf

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

      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 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 81.7%

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

   if -2.5e-15 < F < 3.5e-164

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 50.7%

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

   6. Taylor expanded in F around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 50.7

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

   8. Applied rewrites 50.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 50.7%

       \[\leadsto \frac{\sqrt{0.5} \cdot F - x}{B} \]

   if 3.5e-164 < F < 1.06000000000000003e77

   1. Initial program 99.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \color{blue}{\frac{F}{\sin B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      4. inv-pow N/A

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

      5. lower-pow.f64 N/A

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

      6. +-commutative N/A

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

      7. pow2 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift-sin.f64 N/A

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

      10. lift-/.f64 64.3

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

   5. Applied rewrites 64.3%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lift-fma.f64 N/A

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

      4. unpow-1 N/A

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

      5. pow2 N/A

         \[\leadsto \sqrt{\frac{1}{{F}^{2} + 2}} \cdot \frac{F}{\sin B} \]

      6. +-commutative N/A

         \[\leadsto \sqrt{\frac{1}{2 + {F}^{2}}} \cdot \frac{F}{\sin B} \]

      7. sqrt-div N/A

         \[\leadsto \frac{\sqrt{1}}{\sqrt{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      8. metadata-eval N/A

         \[\leadsto \frac{1}{\sqrt{2 + {F}^{2}}} \cdot \frac{F}{\sin B} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{1}{\sqrt{2 + {F}^{2}}} \cdot \frac{\color{blue}{F}}{\sin B} \]

      10. lower-sqrt.f64 N/A

          \[\leadsto \frac{1}{\sqrt{2 + {F}^{2}}} \cdot \frac{F}{\sin B} \]

      11. +-commutative N/A

          \[\leadsto \frac{1}{\sqrt{{F}^{2} + 2}} \cdot \frac{F}{\sin B} \]

      12. pow2 N/A

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

      13. lift-fma.f64 64.2

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

   7. Applied rewrites 64.2%

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

   if 1.06000000000000003e77 < F

   1. Initial program 32.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 32.9%

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

   6. Taylor expanded in F around inf

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

      1. lower--.f64 49.3

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

   8. Applied rewrites 49.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 51.1

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

   10. Applied rewrites 51.1%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-164}:\\ \;\;\;\;\frac{\sqrt{0.5} \cdot F - x}{B}\\ \mathbf{elif}\;F \leq 1.06 \cdot 10^{+77}:\\ \;\;\;\;\frac{1}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - x \cdot x}{1 + x}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{+37}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - x \cdot x}{1 + x}}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.5e-15)
   (/ (- -1.0 x) (sin B))
   (if (<= F 1.35e+37)
     (/ (- (* (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0)))) F) x) B)
     (/ (/ (- 1.0 (* x x)) (+ 1.0 x)) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.5e-15) {
		tmp = (-1.0 - x) / sin(B);
	} else if (F <= 1.35e+37) {
		tmp = ((sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B;
	} else {
		tmp = ((1.0 - (x * x)) / (1.0 + x)) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.5e-15)
		tmp = Float64(Float64(-1.0 - x) / sin(B));
	elseif (F <= 1.35e+37)
		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(1.0 - Float64(x * x)) / Float64(1.0 + x)) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.5e-15], N[(N[(-1.0 - x), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.35e+37], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.5e-15

   1. Initial program 61.4%

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

   3. Taylor expanded in F around -inf

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

      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 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in B around 0

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

   8. Applied rewrites 81.7%

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

   if -2.5e-15 < F < 1.34999999999999993e37

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 48.7%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]
   6. 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. pow2 N/A

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

      6. +-commutative N/A

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

      7. inv-pow N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. pow2 N/A

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

      11. +-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-fma.f64 48.7

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

   7. Applied rewrites 48.7%

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

   if 1.34999999999999993e37 < F

   1. Initial program 42.9%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 32.1%

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

   6. Taylor expanded in F around inf

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

      1. lower--.f64 46.0

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

   8. Applied rewrites 46.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 47.5

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

   10. Applied rewrites 47.5%

       \[\leadsto \frac{\frac{1 - x \cdot x}{1 + x}}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1 - x}{\sin B}\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{+37}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - x \cdot x}{1 + x}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.9% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.65 \cdot 10^{+66}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{+37}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - x \cdot x}{1 + x}}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.65e+66)
   (/ -1.0 (sin B))
   (if (<= F 1.35e+37)
     (/ (- (* (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0)))) F) x) B)
     (/ (/ (- 1.0 (* x x)) (+ 1.0 x)) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.65e+66) {
		tmp = -1.0 / sin(B);
	} else if (F <= 1.35e+37) {
		tmp = ((sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B;
	} else {
		tmp = ((1.0 - (x * x)) / (1.0 + x)) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.65e+66)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 1.35e+37)
		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(1.0 - Float64(x * x)) / Float64(1.0 + x)) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.65e+66], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.35e+37], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.6500000000000001e66

   1. Initial program 48.7%

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

   3. Taylor expanded in F around -inf

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

      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 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lift-sin.f64 67.5

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

   8. Applied rewrites 67.5%

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

   if -1.6500000000000001e66 < F < 1.34999999999999993e37

   1. Initial program 98.8%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 49.1%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]
   6. 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. pow2 N/A

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

      6. +-commutative N/A

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

      7. inv-pow N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. pow2 N/A

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

      11. +-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-fma.f64 49.1

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

   7. Applied rewrites 49.1%

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

   if 1.34999999999999993e37 < F

   1. Initial program 42.9%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 32.1%

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

   6. Taylor expanded in F around inf

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

      1. lower--.f64 46.0

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

   8. Applied rewrites 46.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 47.5

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

   10. Applied rewrites 47.5%

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

Alternative 14: 50.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.7 \cdot 10^{-15}:\\ \;\;\;\;\frac{\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F} \cdot 0.5 - 1\right) - x}{B}\\ \mathbf{elif}\;F \leq 1.35 \cdot 10^{+37}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - x \cdot x}{1 + x}}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.7e-15)
   (/ (- (- (* (/ (fma 2.0 x 2.0) (* F F)) 0.5) 1.0) x) B)
   (if (<= F 1.35e+37)
     (/ (- (* (sqrt (/ 1.0 (fma 2.0 x (fma F F 2.0)))) F) x) B)
     (/ (/ (- 1.0 (* x x)) (+ 1.0 x)) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.7e-15) {
		tmp = ((((fma(2.0, x, 2.0) / (F * F)) * 0.5) - 1.0) - x) / B;
	} else if (F <= 1.35e+37) {
		tmp = ((sqrt((1.0 / fma(2.0, x, fma(F, F, 2.0)))) * F) - x) / B;
	} else {
		tmp = ((1.0 - (x * x)) / (1.0 + x)) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.7e-15)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(2.0, x, 2.0) / Float64(F * F)) * 0.5) - 1.0) - x) / B);
	elseif (F <= 1.35e+37)
		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(1.0 - Float64(x * x)) / Float64(1.0 + x)) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.7e-15], N[(N[(N[(N[(N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] - 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.35e+37], N[(N[(N[(N[Sqrt[N[(1.0 / N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.70000000000000009e-15

   1. Initial program 61.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 35.6%

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

   6. Taylor expanded in F around -inf

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

      1. metadata-eval N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

   8. Applied rewrites 53.7%

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

   if -2.70000000000000009e-15 < F < 1.34999999999999993e37

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 48.7%

      \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]
   6. 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. pow2 N/A

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

      6. +-commutative N/A

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

      7. inv-pow N/A

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

      8. +-commutative N/A

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

      9. associate-+r+ N/A

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

      10. pow2 N/A

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

      11. +-commutative N/A

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

      12. lower-/.f64 N/A

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

      13. +-commutative N/A

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

      14. lift-fma.f64 N/A

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

      15. lift-fma.f64 48.7

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

   7. Applied rewrites 48.7%

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

   if 1.34999999999999993e37 < F

   1. Initial program 42.9%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 32.1%

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

   6. Taylor expanded in F around inf

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

      1. lower--.f64 46.0

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

   8. Applied rewrites 46.0%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 47.5

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

   10. Applied rewrites 47.5%

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

Alternative 15: 50.6% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.7 \cdot 10^{-15}:\\ \;\;\;\;\frac{\left(\frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F} \cdot 0.5 - 1\right) - x}{B}\\ \mathbf{elif}\;F \leq 205000000:\\ \;\;\;\;\frac{\sqrt{0.5} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - x \cdot x}{1 + x}}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.7e-15)
   (/ (- (- (* (/ (fma 2.0 x 2.0) (* F F)) 0.5) 1.0) x) B)
   (if (<= F 205000000.0)
     (/ (- (* (sqrt 0.5) F) x) B)
     (/ (/ (- 1.0 (* x x)) (+ 1.0 x)) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.7e-15) {
		tmp = ((((fma(2.0, x, 2.0) / (F * F)) * 0.5) - 1.0) - x) / B;
	} else if (F <= 205000000.0) {
		tmp = ((sqrt(0.5) * F) - x) / B;
	} else {
		tmp = ((1.0 - (x * x)) / (1.0 + x)) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.7e-15)
		tmp = Float64(Float64(Float64(Float64(Float64(fma(2.0, x, 2.0) / Float64(F * F)) * 0.5) - 1.0) - x) / B);
	elseif (F <= 205000000.0)
		tmp = Float64(Float64(Float64(sqrt(0.5) * F) - x) / B);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(x * x)) / Float64(1.0 + x)) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.7e-15], N[(N[(N[(N[(N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] - 1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 205000000.0], N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.70000000000000009e-15

   1. Initial program 61.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 35.6%

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

   6. Taylor expanded in F around -inf

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

      1. metadata-eval N/A

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

      2. associate--r+ N/A

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

      3. lower--.f64 N/A

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

   8. Applied rewrites 53.7%

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

   if -2.70000000000000009e-15 < F < 2.05e8

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 49.4%

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

   6. Taylor expanded in F around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 49.4

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

   8. Applied rewrites 49.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 49.4%

       \[\leadsto \frac{\sqrt{0.5} \cdot F - x}{B} \]

   if 2.05e8 < F

   1. Initial program 47.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 32.5%

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

   6. Taylor expanded in F around inf

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

      1. lower--.f64 45.2

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

   8. Applied rewrites 45.2%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 46.6

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

   10. Applied rewrites 46.6%

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

Alternative 16: 50.8% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.7 \cdot 10^{-15}:\\ \;\;\;\;\frac{\left(-0.16666666666666666 \cdot x - \mathsf{fma}\left(-0.5, x, 0.16666666666666666\right)\right) \cdot \left(B \cdot B\right) - \left(1 + x\right)}{B}\\ \mathbf{elif}\;F \leq 205000000:\\ \;\;\;\;\frac{\sqrt{0.5} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - x \cdot x}{1 + x}}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.7e-15)
   (/
    (-
     (*
      (- (* -0.16666666666666666 x) (fma -0.5 x 0.16666666666666666))
      (* B B))
     (+ 1.0 x))
    B)
   (if (<= F 205000000.0)
     (/ (- (* (sqrt 0.5) F) x) B)
     (/ (/ (- 1.0 (* x x)) (+ 1.0 x)) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.7e-15) {
		tmp = ((((-0.16666666666666666 * x) - fma(-0.5, x, 0.16666666666666666)) * (B * B)) - (1.0 + x)) / B;
	} else if (F <= 205000000.0) {
		tmp = ((sqrt(0.5) * F) - x) / B;
	} else {
		tmp = ((1.0 - (x * x)) / (1.0 + x)) / B;
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.7e-15)
		tmp = Float64(Float64(Float64(Float64(Float64(-0.16666666666666666 * x) - fma(-0.5, x, 0.16666666666666666)) * Float64(B * B)) - Float64(1.0 + x)) / B);
	elseif (F <= 205000000.0)
		tmp = Float64(Float64(Float64(sqrt(0.5) * F) - x) / B);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(x * x)) / Float64(1.0 + x)) / B);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.7e-15], N[(N[(N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] - N[(-0.5 * x + 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(B * B), $MachinePrecision]), $MachinePrecision] - N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 205000000.0], N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.70000000000000009e-15

   1. Initial program 61.4%

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

   3. Taylor expanded in F around -inf

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

      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 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. unpow2 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-+.f64 52.7

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

   8. Applied rewrites 52.7%

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

   if -2.70000000000000009e-15 < F < 2.05e8

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 49.4%

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

   6. Taylor expanded in F around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 49.4

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

   8. Applied rewrites 49.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 49.4%

       \[\leadsto \frac{\sqrt{0.5} \cdot F - x}{B} \]

   if 2.05e8 < F

   1. Initial program 47.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 32.5%

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

   6. Taylor expanded in F around inf

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

      1. lower--.f64 45.2

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

   8. Applied rewrites 45.2%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 46.6

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

   10. Applied rewrites 46.6%

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

Alternative 17: 51.1% accurate, 8.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 205000000:\\ \;\;\;\;\frac{\sqrt{0.5} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - x \cdot x}{1 + x}}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.4)
   (/ (- -1.0 x) B)
   (if (<= F 205000000.0)
     (/ (- (* (sqrt 0.5) F) x) B)
     (/ (/ (- 1.0 (* x x)) (+ 1.0 x)) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 205000000.0) {
		tmp = ((sqrt(0.5) * F) - x) / B;
	} else {
		tmp = ((1.0 - (x * x)) / (1.0 + x)) / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.4d0)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 205000000.0d0) then
        tmp = ((sqrt(0.5d0) * f) - x) / b
    else
        tmp = ((1.0d0 - (x * x)) / (1.0d0 + x)) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 205000000.0) {
		tmp = ((Math.sqrt(0.5) * F) - x) / B;
	} else {
		tmp = ((1.0 - (x * x)) / (1.0 + x)) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.4:
		tmp = (-1.0 - x) / B
	elif F <= 205000000.0:
		tmp = ((math.sqrt(0.5) * F) - x) / B
	else:
		tmp = ((1.0 - (x * x)) / (1.0 + x)) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 205000000.0)
		tmp = Float64(Float64(Float64(sqrt(0.5) * F) - x) / B);
	else
		tmp = Float64(Float64(Float64(1.0 - Float64(x * x)) / Float64(1.0 + x)) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.4)
		tmp = (-1.0 - x) / B;
	elseif (F <= 205000000.0)
		tmp = ((sqrt(0.5) * F) - x) / B;
	else
		tmp = ((1.0 - (x * x)) / (1.0 + x)) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.4], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 205000000.0], N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(1.0 - N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(1.0 + x), $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 59.5%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 37.3%

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

   6. Taylor expanded in F around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 11.3

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

   8. Applied rewrites 11.3%

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

   9. Taylor expanded in F around -inf

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

   11. Applied rewrites 55.2%

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

   if -1.3999999999999999 < F < 2.05e8

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 48.2%

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

   6. Taylor expanded in F around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 48.2

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

   8. Applied rewrites 48.2%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 48.2%

       \[\leadsto \frac{\sqrt{0.5} \cdot F - x}{B} \]

   if 2.05e8 < F

   1. Initial program 47.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 32.5%

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

   6. Taylor expanded in F around inf

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

      1. lower--.f64 45.2

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

   8. Applied rewrites 45.2%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. unpow2 N/A

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

      6. lower--.f64 N/A

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

      7. unpow2 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-+.f64 46.6

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

   10. Applied rewrites 46.6%

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

Alternative 18: 51.7% accurate, 8.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-13}:\\ \;\;\;\;\frac{\sqrt{0.5} \cdot F - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.4)
   (/ (- -1.0 x) B)
   (if (<= F 1.75e-13) (/ (- (* (sqrt 0.5) F) x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.75e-13) {
		tmp = ((sqrt(0.5) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.4d0)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.75d-13) then
        tmp = ((sqrt(0.5d0) * f) - x) / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.75e-13) {
		tmp = ((Math.sqrt(0.5) * F) - x) / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.4:
		tmp = (-1.0 - x) / B
	elif F <= 1.75e-13:
		tmp = ((math.sqrt(0.5) * F) - x) / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.75e-13)
		tmp = Float64(Float64(Float64(sqrt(0.5) * F) - x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.4)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.75e-13)
		tmp = ((sqrt(0.5) * F) - x) / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.4], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.75e-13], N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * F), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 59.5%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 37.3%

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

   6. Taylor expanded in F around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 11.3

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

   8. Applied rewrites 11.3%

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

   9. Taylor expanded in F around -inf

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

   11. Applied rewrites 55.2%

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

   if -1.3999999999999999 < F < 1.7500000000000001e-13

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 49.7%

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

   6. Taylor expanded in F around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 49.7

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

   8. Applied rewrites 49.7%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 49.7%

       \[\leadsto \frac{\sqrt{0.5} \cdot F - x}{B} \]

   if 1.7500000000000001e-13 < F

   1. Initial program 50.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in F around inf

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

      1. lower--.f64 42.9

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

   8. Applied rewrites 42.9%

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

Alternative 19: 44.2% accurate, 13.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.6 \cdot 10^{-59}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-63}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.6e-59)
   (/ (- -1.0 x) B)
   (if (<= F 1.2e-63) (/ (- x) B) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.6e-59) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.2e-63) {
		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 <= (-3.6d-59)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.2d-63) 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 <= -3.6e-59) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.2e-63) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.6e-59:
		tmp = (-1.0 - x) / B
	elif F <= 1.2e-63:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.6e-59)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.2e-63)
		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 <= -3.6e-59)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.2e-63)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.6e-59], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.2e-63], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.6e-59

   1. Initial program 66.5%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 36.6%

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

   6. Taylor expanded in F around 0

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

      1. inv-pow N/A

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

      2. lower-pow.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-fma.f64 15.2

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

   8. Applied rewrites 15.2%

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

   9. Taylor expanded in F around -inf

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

   11. Applied rewrites 47.8%

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

   if -3.6e-59 < F < 1.2e-63

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 51.7%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 40.1

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

   8. Applied rewrites 40.1%

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

   if 1.2e-63 < F

   1. Initial program 56.2%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 33.3%

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

   6. Taylor expanded in F around inf

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

      1. lower--.f64 39.6

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

   8. Applied rewrites 39.6%

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

Alternative 20: 36.5% accurate, 17.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 1.2 \cdot 10^{-63}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F 1.2e-63) (/ (- x) B) (/ (- 1.0 x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 1.2e-63) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= 1.2d-63) then
        tmp = -x / b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 1.2e-63) {
		tmp = -x / B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 1.2e-63:
		tmp = -x / B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 1.2e-63)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 1.2e-63)
		tmp = -x / B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 1.2e-63], N[((-x) / B), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.2e-63

   1. Initial program 85.3%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 45.2%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 31.7

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

   8. Applied rewrites 31.7%

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

   if 1.2e-63 < F

   1. Initial program 56.2%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 33.3%

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

   6. Taylor expanded in F around inf

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

      1. lower--.f64 39.6

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

   8. Applied rewrites 39.6%

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

Alternative 21: 29.8% accurate, 18.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq 2.05 \cdot 10^{-20}:\\ \;\;\;\;\frac{-x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (if (<= F 2.05e-20) (/ (- x) B) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= 2.05e-20) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(f, b, x)
use fmin_fmax_functions
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= 2.05d-20) then
        tmp = -x / b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= 2.05e-20) {
		tmp = -x / B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= 2.05e-20:
		tmp = -x / B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= 2.05e-20)
		tmp = Float64(Float64(-x) / B);
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= 2.05e-20)
		tmp = -x / B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, 2.05e-20], N[((-x) / B), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 2.05e-20

   1. Initial program 86.1%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 45.5%

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

   6. Taylor expanded in F around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 30.7

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

   8. Applied rewrites 30.7%

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

   if 2.05e-20 < F

   1. Initial program 50.4%

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

   3. Taylor expanded in B around 0

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

      1. lower-/.f64 N/A

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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in F around inf

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

      1. lower--.f64 42.9

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

   8. Applied rewrites 42.9%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 25.7%

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

Alternative 22: 9.5% 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 75.6%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in B around 0

   \[\leadsto \color{blue}{\frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{B}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{F \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} - x}{\color{blue}{B}} \]

5. Applied rewrites 41.2%

   \[\leadsto \color{blue}{\frac{\sqrt{{\left(\mathsf{fma}\left(2, x, F \cdot F\right) + 2\right)}^{-1}} \cdot F - x}{B}} \]

6. Taylor expanded in F around inf

   \[\leadsto \frac{1 - x}{B} \]
7. Step-by-step derivation

   1. lower--.f64 25.3

      \[\leadsto \frac{1 - x}{B} \]

8. Applied rewrites 25.3%

   \[\leadsto \frac{1 - x}{B} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{1}{B} \]
10. Step-by-step derivation

11. Applied rewrites 9.7%

    \[\leadsto \frac{1}{B} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025085 
(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))))))