VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.5% → 99.7%

Time: 9.5s

Alternatives: 21

Speedup: 2.2×

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

real(8) function code(f, b, x)
    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: 77.5% 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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.25 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 150000000:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot \sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2.25e+27)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 150000000.0)
       (- (/ F (* (sqrt (fma 2.0 x (fma F F 2.0))) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2.25e+27) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 150000000.0) {
		tmp = (F / (sqrt(fma(2.0, x, fma(F, F, 2.0))) * sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.25e+27)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 150000000.0)
		tmp = Float64(Float64(F / Float64(sqrt(fma(2.0, x, fma(F, F, 2.0))) * sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.25e+27], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 150000000.0], N[(N[(F / N[(N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.25e27

   1. Initial program 53.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 53.2

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

   4. Applied rewrites 53.2%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if -2.25e27 < F < 1.5e8

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.6

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.6

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

   8. Applied rewrites 99.6%

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

   if 1.5e8 < F

   1. Initial program 54.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 54.6

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

   4. Applied rewrites 54.6%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.7

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

   7. Applied rewrites 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.25 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 150000000:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot \sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.25 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)} \cdot \sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2.25e+27)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 100000000.0)
       (- (/ F (* (sqrt (fma F F 2.0)) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2.25e+27) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 100000000.0) {
		tmp = (F / (sqrt(fma(F, F, 2.0)) * sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.25e+27)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 100000000.0)
		tmp = Float64(Float64(F / Float64(sqrt(fma(F, F, 2.0)) * sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.25e+27], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 100000000.0], N[(N[(F / N[(N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision] * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.25e27

   1. Initial program 53.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 53.2

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

   4. Applied rewrites 53.2%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if -2.25e27 < F < 1e8

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.6

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.6

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in x around 0

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

       1. lower-sqrt.f64 N/A

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

       2. +-commutative N/A

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

       3. unpow2 N/A

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

       4. lower-fma.f64 99.5

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

   11. Applied rewrites 99.5%

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

   if 1e8 < F

   1. Initial program 54.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 54.6

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

   4. Applied rewrites 54.6%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.7

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

   7. Applied rewrites 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.25 \cdot 10^{+27}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)} \cdot \sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -10.2:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{\sqrt{0.5}}{\sin B} \cdot F - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -10.2)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.4)
       (- (* (/ (sqrt 0.5) (sin B)) F) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -10.2) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = ((sqrt(0.5) / sin(B)) * F) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-10.2d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.4d0) then
        tmp = ((sqrt(0.5d0) / sin(b)) * f) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -10.2) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.4) {
		tmp = ((Math.sqrt(0.5) / Math.sin(B)) * F) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -10.2:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.4:
		tmp = ((math.sqrt(0.5) / math.sin(B)) * F) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -10.2)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4)
		tmp = Float64(Float64(Float64(sqrt(0.5) / sin(B)) * F) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -10.2)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.4)
		tmp = ((sqrt(0.5) / sin(B)) * F) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -10.2], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4], N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -10.199999999999999

   1. Initial program 56.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 56.7

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

   4. Applied rewrites 56.7%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.0

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

   7. Applied rewrites 99.0%

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

   if -10.199999999999999 < F < 1.3999999999999999

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in F around 0

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

   10. Applied rewrites 99.5%

       \[\leadsto \left(-\frac{x}{\tan B}\right) + \frac{F}{\sin B} \cdot \sqrt{0.5} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-*l/ N/A

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

       4. associate-/l* N/A

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

       5. lower-*.f64 N/A

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

       6. lower-/.f64 N/A

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

   12. Applied rewrites 99.5%

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

   if 1.3999999999999999 < F

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 55.3

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

   4. Applied rewrites 55.3%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.7

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

   7. Applied rewrites 99.7%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -10.2:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{\sqrt{0.5}}{\sin B} \cdot F - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -3.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot F}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 240:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -4.6e-8)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -3.6e-109)
       (- (/ (* (sqrt (/ 1.0 (fma F F 2.0))) F) (sin B)) (/ x B))
       (if (<= F 240.0)
         (- (/ F (* (sqrt (+ (fma 2.0 x (* F F)) 2.0)) B)) t_0)
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -4.6e-8) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -3.6e-109) {
		tmp = ((sqrt((1.0 / fma(F, F, 2.0))) * F) / sin(B)) - (x / B);
	} else if (F <= 240.0) {
		tmp = (F / (sqrt((fma(2.0, x, (F * F)) + 2.0)) * B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -4.6e-8)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -3.6e-109)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) * F) / sin(B)) - Float64(x / B));
	elseif (F <= 240.0)
		tmp = Float64(Float64(F / Float64(sqrt(Float64(fma(2.0, x, Float64(F * F)) + 2.0)) * B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4.6e-8], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -3.6e-109], N[(N[(N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 240.0], N[(N[(F / N[(N[Sqrt[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.6000000000000002e-8

   1. Initial program 57.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 58.0

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

   4. Applied rewrites 58.0%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.0

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

   7. Applied rewrites 99.0%

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

   if -4.6000000000000002e-8 < F < -3.6000000000000001e-109

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 93.1

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

   9. Applied rewrites 93.1%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. +-commutative N/A

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

       6. unpow2 N/A

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

       7. lower-fma.f64 93.2

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

   12. Applied rewrites 93.2%

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

   if -3.6000000000000001e-109 < F < 240

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.7

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.7

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

   8. Applied rewrites 99.7%

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

   9. Taylor expanded in B around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 85.8

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

   11. Applied rewrites 85.8%

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

   if 240 < F

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 55.3

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

   4. Applied rewrites 55.3%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.7

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

   7. Applied rewrites 99.7%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot F}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 240:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -3.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot F}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{+130}:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -4.6e-8)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -3.6e-109)
       (- (/ (* (sqrt (/ 1.0 (fma F F 2.0))) F) (sin B)) (/ x B))
       (if (<= F 7.8e+130)
         (- (/ F (* (sqrt (+ (fma 2.0 x (* F F)) 2.0)) B)) t_0)
         (- (/ 1.0 (sin B)) (/ x B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -4.6e-8) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -3.6e-109) {
		tmp = ((sqrt((1.0 / fma(F, F, 2.0))) * F) / sin(B)) - (x / B);
	} else if (F <= 7.8e+130) {
		tmp = (F / (sqrt((fma(2.0, x, (F * F)) + 2.0)) * B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -4.6e-8)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -3.6e-109)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) * F) / sin(B)) - Float64(x / B));
	elseif (F <= 7.8e+130)
		tmp = Float64(Float64(F / Float64(sqrt(Float64(fma(2.0, x, Float64(F * F)) + 2.0)) * B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4.6e-8], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -3.6e-109], N[(N[(N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.8e+130], N[(N[(F / N[(N[Sqrt[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.6000000000000002e-8

   1. Initial program 57.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 58.0

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

   4. Applied rewrites 58.0%

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

   5. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.0

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

   7. Applied rewrites 99.0%

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

   if -4.6000000000000002e-8 < F < -3.6000000000000001e-109

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 93.1

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

   9. Applied rewrites 93.1%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. +-commutative N/A

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

       6. unpow2 N/A

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

       7. lower-fma.f64 93.2

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

   12. Applied rewrites 93.2%

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

   if -3.6000000000000001e-109 < F < 7.8000000000000004e130

   1. Initial program 97.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 97.2

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

   4. Applied rewrites 97.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.6

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.6

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in B around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 84.2

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

   11. Applied rewrites 84.2%

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

   if 7.8000000000000004e130 < F

   1. Initial program 32.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 33.0

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

   4. Applied rewrites 33.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 54.6%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 35.6

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

   9. Applied rewrites 35.6%

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

   10. Taylor expanded in F around inf

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

   12. Applied rewrites 80.4%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.6 \cdot 10^{-8}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot F}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{+130}:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 78.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+120}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot F}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{+130}:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1e+120)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -3.6e-109)
     (- (/ (* (sqrt (/ 1.0 (fma F F 2.0))) F) (sin B)) (/ x B))
     (if (<= F 7.8e+130)
       (- (/ F (* (sqrt (+ (fma 2.0 x (* F F)) 2.0)) B)) (/ x (tan B)))
       (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1e+120) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -3.6e-109) {
		tmp = ((sqrt((1.0 / fma(F, F, 2.0))) * F) / sin(B)) - (x / B);
	} else if (F <= 7.8e+130) {
		tmp = (F / (sqrt((fma(2.0, x, (F * F)) + 2.0)) * B)) - (x / tan(B));
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1e+120)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -3.6e-109)
		tmp = Float64(Float64(Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) * F) / sin(B)) - Float64(x / B));
	elseif (F <= 7.8e+130)
		tmp = Float64(Float64(F / Float64(sqrt(Float64(fma(2.0, x, Float64(F * F)) + 2.0)) * B)) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1e+120], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.6e-109], N[(N[(N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * F), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.8e+130], N[(N[(F / N[(N[Sqrt[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -9.9999999999999998e119

   1. Initial program 33.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 33.7

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

   4. Applied rewrites 33.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 50.6%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 26.3

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

   9. Applied rewrites 26.3%

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

   10. Taylor expanded in F around -inf

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

   12. Applied rewrites 76.2%

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

   if -9.9999999999999998e119 < F < -3.6000000000000001e-109

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 99.6%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 90.4

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

   9. Applied rewrites 90.4%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. lower-/.f64 N/A

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

       5. +-commutative N/A

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

       6. unpow2 N/A

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

       7. lower-fma.f64 90.5

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

   12. Applied rewrites 90.5%

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

   if -3.6000000000000001e-109 < F < 7.8000000000000004e130

   1. Initial program 97.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 97.2

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

   4. Applied rewrites 97.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.6

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.6

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in B around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 84.2

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

   11. Applied rewrites 84.2%

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

   if 7.8000000000000004e130 < F

   1. Initial program 32.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 33.0

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

   4. Applied rewrites 33.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 54.6%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 35.6

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

   9. Applied rewrites 35.6%

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

   10. Taylor expanded in F around inf

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

   12. Applied rewrites 80.4%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{+120}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot F}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{+130}:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{+46}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.6 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{+130}:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.4e+46)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -3.6e-109)
     (- (* (sqrt (/ 1.0 (fma F F 2.0))) (/ F (sin B))) (/ x B))
     (if (<= F 7.8e+130)
       (- (/ F (* (sqrt (+ (fma 2.0 x (* F F)) 2.0)) B)) (/ x (tan B)))
       (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.4e+46) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -3.6e-109) {
		tmp = (sqrt((1.0 / fma(F, F, 2.0))) * (F / sin(B))) - (x / B);
	} else if (F <= 7.8e+130) {
		tmp = (F / (sqrt((fma(2.0, x, (F * F)) + 2.0)) * B)) - (x / tan(B));
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.4e+46)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -3.6e-109)
		tmp = Float64(Float64(sqrt(Float64(1.0 / fma(F, F, 2.0))) * Float64(F / sin(B))) - Float64(x / B));
	elseif (F <= 7.8e+130)
		tmp = Float64(Float64(F / Float64(sqrt(Float64(fma(2.0, x, Float64(F * F)) + 2.0)) * B)) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.4e+46], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.6e-109], N[(N[(N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.8e+130], N[(N[(F / N[(N[Sqrt[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.40000000000000008e46

   1. Initial program 50.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 50.8

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

   4. Applied rewrites 50.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 63.3%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 43.6

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

   9. Applied rewrites 43.6%

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

   10. Taylor expanded in F around -inf

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

   12. Applied rewrites 80.6%

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

   if -2.40000000000000008e46 < F < -3.6000000000000001e-109

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in B around 0

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

      1. lower-/.f64 89.3

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

   10. Applied rewrites 89.3%

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

   if -3.6000000000000001e-109 < F < 7.8000000000000004e130

   1. Initial program 97.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 97.2

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

   4. Applied rewrites 97.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.6

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.6

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in B around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 84.2

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

   11. Applied rewrites 84.2%

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

   if 7.8000000000000004e130 < F

   1. Initial program 32.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 33.0

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

   4. Applied rewrites 33.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 54.6%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 35.6

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

   9. Applied rewrites 35.6%

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

   10. Taylor expanded in F around inf

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

   12. Applied rewrites 80.4%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{+46}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.6 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}} \cdot \frac{F}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{+130}:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7600:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot \sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{+130}:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7600.0)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -3.6e-109)
     (- (/ F (* (sqrt (fma 2.0 x (fma F F 2.0))) (sin B))) (/ x B))
     (if (<= F 7.8e+130)
       (- (/ F (* (sqrt (+ (fma 2.0 x (* F F)) 2.0)) B)) (/ x (tan B)))
       (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7600.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -3.6e-109) {
		tmp = (F / (sqrt(fma(2.0, x, fma(F, F, 2.0))) * sin(B))) - (x / B);
	} else if (F <= 7.8e+130) {
		tmp = (F / (sqrt((fma(2.0, x, (F * F)) + 2.0)) * B)) - (x / tan(B));
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7600.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -3.6e-109)
		tmp = Float64(Float64(F / Float64(sqrt(fma(2.0, x, fma(F, F, 2.0))) * sin(B))) - Float64(x / B));
	elseif (F <= 7.8e+130)
		tmp = Float64(Float64(F / Float64(sqrt(Float64(fma(2.0, x, Float64(F * F)) + 2.0)) * B)) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7600.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.6e-109], N[(N[(F / N[(N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.8e+130], N[(N[(F / N[(N[Sqrt[N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7600

   1. Initial program 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 55.4

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

   4. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 66.7%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 47.3

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

   9. Applied rewrites 47.3%

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

   10. Taylor expanded in F around -inf

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

   12. Applied rewrites 80.8%

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

   if -7600 < F < -3.6000000000000001e-109

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.4

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.4

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in B around 0

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

       1. lower-/.f64 90.4

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

   11. Applied rewrites 90.4%

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

   if -3.6000000000000001e-109 < F < 7.8000000000000004e130

   1. Initial program 97.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 97.2

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

   4. Applied rewrites 97.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.6

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.6

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

   8. Applied rewrites 99.6%

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

   9. Taylor expanded in B around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-sqrt.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-+.f64 N/A

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

       6. lower-fma.f64 N/A

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

       7. unpow2 N/A

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

       8. lower-*.f64 84.2

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

   11. Applied rewrites 84.2%

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

   if 7.8000000000000004e130 < F

   1. Initial program 32.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 33.0

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

   4. Applied rewrites 33.0%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 54.6%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 35.6

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

   9. Applied rewrites 35.6%

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

   10. Taylor expanded in F around inf

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

   12. Applied rewrites 80.4%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7600:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot \sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{+130}:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, F \cdot F\right) + 2} \cdot B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7600:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot \sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, 2\right)} \cdot B} - \frac{1}{\tan B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} + \frac{-1}{\tan B} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7600.0)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -3.6e-109)
     (- (/ F (* (sqrt (fma 2.0 x (fma F F 2.0))) (sin B))) (/ x B))
     (if (<= F 1.4)
       (- (/ F (* (sqrt (fma 2.0 x 2.0)) B)) (* (/ 1.0 (tan B)) x))
       (+ (/ 1.0 B) (* (/ -1.0 (tan B)) x))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7600.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -3.6e-109) {
		tmp = (F / (sqrt(fma(2.0, x, fma(F, F, 2.0))) * sin(B))) - (x / B);
	} else if (F <= 1.4) {
		tmp = (F / (sqrt(fma(2.0, x, 2.0)) * B)) - ((1.0 / tan(B)) * x);
	} else {
		tmp = (1.0 / B) + ((-1.0 / tan(B)) * x);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7600.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -3.6e-109)
		tmp = Float64(Float64(F / Float64(sqrt(fma(2.0, x, fma(F, F, 2.0))) * sin(B))) - Float64(x / B));
	elseif (F <= 1.4)
		tmp = Float64(Float64(F / Float64(sqrt(fma(2.0, x, 2.0)) * B)) - Float64(Float64(1.0 / tan(B)) * x));
	else
		tmp = Float64(Float64(1.0 / B) + Float64(Float64(-1.0 / tan(B)) * x));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7600.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.6e-109], N[(N[(F / N[(N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4], N[(N[(F / N[(N[Sqrt[N[(2.0 * x + 2.0), $MachinePrecision]], $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] + N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7600

   1. Initial program 55.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 55.4

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

   4. Applied rewrites 55.4%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 66.7%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 47.3

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

   9. Applied rewrites 47.3%

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

   10. Taylor expanded in F around -inf

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

   12. Applied rewrites 80.8%

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

   if -7600 < F < -3.6000000000000001e-109

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 99.4

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

      7. lift-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 99.4

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

   8. Applied rewrites 99.4%

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

   9. Taylor expanded in B around 0

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

       1. lower-/.f64 90.4

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

   11. Applied rewrites 90.4%

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

   if -3.6000000000000001e-109 < F < 1.3999999999999999

   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 B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 85.6

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 85.6%

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

   10. Applied rewrites 85.7%

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

   if 1.3999999999999999 < F

   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 B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 45.4

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

   5. Applied rewrites 45.4%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 77.4%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7600:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot \sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, 2\right)} \cdot B} - \frac{1}{\tan B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} + \frac{-1}{\tan B} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 78.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -10.2:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, 2\right)} \cdot B} - \frac{1}{\tan B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} + \frac{-1}{\tan B} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -10.2)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -3.6e-109)
     (- (* (/ F (sin B)) (sqrt 0.5)) (/ x B))
     (if (<= F 1.4)
       (- (/ F (* (sqrt (fma 2.0 x 2.0)) B)) (* (/ 1.0 (tan B)) x))
       (+ (/ 1.0 B) (* (/ -1.0 (tan B)) x))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -10.2) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -3.6e-109) {
		tmp = ((F / sin(B)) * sqrt(0.5)) - (x / B);
	} else if (F <= 1.4) {
		tmp = (F / (sqrt(fma(2.0, x, 2.0)) * B)) - ((1.0 / tan(B)) * x);
	} else {
		tmp = (1.0 / B) + ((-1.0 / tan(B)) * x);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -10.2)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -3.6e-109)
		tmp = Float64(Float64(Float64(F / sin(B)) * sqrt(0.5)) - Float64(x / B));
	elseif (F <= 1.4)
		tmp = Float64(Float64(F / Float64(sqrt(fma(2.0, x, 2.0)) * B)) - Float64(Float64(1.0 / tan(B)) * x));
	else
		tmp = Float64(Float64(1.0 / B) + Float64(Float64(-1.0 / tan(B)) * x));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -10.2], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.6e-109], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4], N[(N[(F / N[(N[Sqrt[N[(2.0 * x + 2.0), $MachinePrecision]], $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] - N[(N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] + N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -10.199999999999999

   1. Initial program 56.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 56.7

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

   4. Applied rewrites 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 67.7%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 48.9

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

   9. Applied rewrites 48.9%

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

   10. Taylor expanded in F around -inf

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

   12. Applied rewrites 80.5%

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

   if -10.199999999999999 < F < -3.6000000000000001e-109

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in F around 0

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

   10. Applied rewrites 99.5%

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

   11. Taylor expanded in B around 0

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

       1. lower-/.f64 89.8

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

   13. Applied rewrites 89.8%

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

   if -3.6000000000000001e-109 < F < 1.3999999999999999

   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 B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 85.6

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 85.6%

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

   10. Applied rewrites 85.7%

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

   if 1.3999999999999999 < F

   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 B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 45.4

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

   5. Applied rewrites 45.4%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 77.4%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -10.2:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, 2\right)} \cdot B} - \frac{1}{\tan B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} + \frac{-1}{\tan B} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 78.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -10.2:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{\sqrt{0.5} \cdot F}{B} - \frac{1}{\tan B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} + \frac{-1}{\tan B} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -10.2)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -3.6e-109)
     (- (* (/ F (sin B)) (sqrt 0.5)) (/ x B))
     (if (<= F 1.4)
       (- (/ (* (sqrt 0.5) F) B) (* (/ 1.0 (tan B)) x))
       (+ (/ 1.0 B) (* (/ -1.0 (tan B)) x))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -10.2) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -3.6e-109) {
		tmp = ((F / sin(B)) * sqrt(0.5)) - (x / B);
	} else if (F <= 1.4) {
		tmp = ((sqrt(0.5) * F) / B) - ((1.0 / tan(B)) * x);
	} else {
		tmp = (1.0 / B) + ((-1.0 / tan(B)) * x);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-10.2d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-3.6d-109)) then
        tmp = ((f / sin(b)) * sqrt(0.5d0)) - (x / b)
    else if (f <= 1.4d0) then
        tmp = ((sqrt(0.5d0) * f) / b) - ((1.0d0 / tan(b)) * x)
    else
        tmp = (1.0d0 / b) + (((-1.0d0) / tan(b)) * x)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -10.2) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -3.6e-109) {
		tmp = ((F / Math.sin(B)) * Math.sqrt(0.5)) - (x / B);
	} else if (F <= 1.4) {
		tmp = ((Math.sqrt(0.5) * F) / B) - ((1.0 / Math.tan(B)) * x);
	} else {
		tmp = (1.0 / B) + ((-1.0 / Math.tan(B)) * x);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -10.2:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -3.6e-109:
		tmp = ((F / math.sin(B)) * math.sqrt(0.5)) - (x / B)
	elif F <= 1.4:
		tmp = ((math.sqrt(0.5) * F) / B) - ((1.0 / math.tan(B)) * x)
	else:
		tmp = (1.0 / B) + ((-1.0 / math.tan(B)) * x)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -10.2)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -3.6e-109)
		tmp = Float64(Float64(Float64(F / sin(B)) * sqrt(0.5)) - Float64(x / B));
	elseif (F <= 1.4)
		tmp = Float64(Float64(Float64(sqrt(0.5) * F) / B) - Float64(Float64(1.0 / tan(B)) * x));
	else
		tmp = Float64(Float64(1.0 / B) + Float64(Float64(-1.0 / tan(B)) * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -10.2)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -3.6e-109)
		tmp = ((F / sin(B)) * sqrt(0.5)) - (x / B);
	elseif (F <= 1.4)
		tmp = ((sqrt(0.5) * F) / B) - ((1.0 / tan(B)) * x);
	else
		tmp = (1.0 / B) + ((-1.0 / tan(B)) * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -10.2], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.6e-109], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4], N[(N[(N[(N[Sqrt[0.5], $MachinePrecision] * F), $MachinePrecision] / B), $MachinePrecision] - N[(N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] + N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -10.199999999999999

   1. Initial program 56.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 56.7

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

   4. Applied rewrites 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 67.7%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 48.9

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

   9. Applied rewrites 48.9%

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

   10. Taylor expanded in F around -inf

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

   12. Applied rewrites 80.5%

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

   if -10.199999999999999 < F < -3.6000000000000001e-109

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in F around 0

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

   10. Applied rewrites 99.5%

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

   11. Taylor expanded in B around 0

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

       1. lower-/.f64 89.8

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

   13. Applied rewrites 89.8%

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

   if -3.6000000000000001e-109 < F < 1.3999999999999999

   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 B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 85.6

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in F around 0

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

   8. Applied rewrites 85.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 85.7%

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\sqrt{0.5} \cdot F}{B} \]

   if 1.3999999999999999 < F

   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 B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 45.4

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

   5. Applied rewrites 45.4%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 77.4%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -10.2:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.6 \cdot 10^{-109}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.4:\\ \;\;\;\;\frac{\sqrt{0.5} \cdot F}{B} - \frac{1}{\tan B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} + \frac{-1}{\tan B} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -10.2:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2 \cdot 10^{-295}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} + \frac{-1}{\tan B} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -10.2)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -2e-295)
     (- (* (/ F (sin B)) (sqrt 0.5)) (/ x B))
     (+ (/ 1.0 B) (* (/ -1.0 (tan B)) x)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -10.2) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -2e-295) {
		tmp = ((F / sin(B)) * sqrt(0.5)) - (x / B);
	} else {
		tmp = (1.0 / B) + ((-1.0 / tan(B)) * x);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-10.2d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-2d-295)) then
        tmp = ((f / sin(b)) * sqrt(0.5d0)) - (x / b)
    else
        tmp = (1.0d0 / b) + (((-1.0d0) / tan(b)) * x)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -10.2) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -2e-295) {
		tmp = ((F / Math.sin(B)) * Math.sqrt(0.5)) - (x / B);
	} else {
		tmp = (1.0 / B) + ((-1.0 / Math.tan(B)) * x);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -10.2:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -2e-295:
		tmp = ((F / math.sin(B)) * math.sqrt(0.5)) - (x / B)
	else:
		tmp = (1.0 / B) + ((-1.0 / math.tan(B)) * x)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -10.2)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -2e-295)
		tmp = Float64(Float64(Float64(F / sin(B)) * sqrt(0.5)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) + Float64(Float64(-1.0 / tan(B)) * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -10.2)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -2e-295)
		tmp = ((F / sin(B)) * sqrt(0.5)) - (x / B);
	else
		tmp = (1.0 / B) + ((-1.0 / tan(B)) * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -10.2], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2e-295], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] + N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -10.199999999999999

   1. Initial program 56.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 56.7

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

   4. Applied rewrites 56.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 67.7%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 48.9

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

   9. Applied rewrites 48.9%

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

   10. Taylor expanded in F around -inf

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

   12. Applied rewrites 80.5%

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

   if -10.199999999999999 < F < -2.00000000000000012e-295

   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. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 99.5

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

   7. Applied rewrites 99.5%

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

   8. Taylor expanded in F around 0

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

   10. Applied rewrites 99.5%

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

   11. Taylor expanded in B around 0

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

       1. lower-/.f64 76.2

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

   13. Applied rewrites 76.2%

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

   if -2.00000000000000012e-295 < F

   1. Initial program 77.0%

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

   3. Taylor expanded in B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 64.4

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

   5. Applied rewrites 64.4%

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

   6. Taylor expanded in F around inf

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

   8. Applied rewrites 68.9%

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

4. Final simplification 73.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -10.2:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2 \cdot 10^{-295}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} + \frac{-1}{\tan B} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 62.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -16000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -6.3 \cdot 10^{-209}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} + \frac{-1}{\tan B} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -16000.0)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -6.3e-209)
     (-
      (* (/ F B) (sqrt (/ 1.0 (+ (fma 2.0 x (* F F)) 2.0))))
      (*
       (/
        (fma
         (fma -0.022222222222222223 (* B B) -0.3333333333333333)
         (* B B)
         1.0)
        B)
       x))
     (+ (/ 1.0 B) (* (/ -1.0 (tan B)) x)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -16000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -6.3e-209) {
		tmp = ((F / B) * sqrt((1.0 / (fma(2.0, x, (F * F)) + 2.0)))) - ((fma(fma(-0.022222222222222223, (B * B), -0.3333333333333333), (B * B), 1.0) / B) * x);
	} else {
		tmp = (1.0 / B) + ((-1.0 / tan(B)) * x);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -16000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -6.3e-209)
		tmp = Float64(Float64(Float64(F / B) * sqrt(Float64(1.0 / Float64(fma(2.0, x, Float64(F * F)) + 2.0)))) - Float64(Float64(fma(fma(-0.022222222222222223, Float64(B * B), -0.3333333333333333), Float64(B * B), 1.0) / B) * x));
	else
		tmp = Float64(Float64(1.0 / B) + Float64(Float64(-1.0 / tan(B)) * x));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -16000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -6.3e-209], N[(N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(N[(2.0 * x + N[(F * F), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(-0.022222222222222223 * N[(B * B), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] + N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -16000

   1. Initial program 54.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. un-div-inv N/A

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

      4. lower-/.f64 54.7

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

   4. Applied rewrites 54.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. lower-/.f64 N/A

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

   6. Applied rewrites 66.2%

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

   7. Taylor expanded in B around 0

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

      1. lower-/.f64 48.0

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

   9. Applied rewrites 48.0%

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

   10. Taylor expanded in F around -inf

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

   12. Applied rewrites 82.1%

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

   if -16000 < F < -6.30000000000000014e-209

   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 B around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 77.0

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

   5. Applied rewrites 77.0%

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

   6. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      2. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{{B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) + 1}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      3. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) \cdot {B}^{2}} + 1}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}, {B}^{2}, 1\right)}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      5. sub-neg N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{45} \cdot {B}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      6. metadata-eval N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} + \color{blue}{\frac{-1}{3}}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{45}, {B}^{2}, \frac{-1}{3}\right)}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      10. unpow2 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, B \cdot B, \frac{-1}{3}\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      11. lower-*.f64 60.3

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   8. Applied rewrites 60.3%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   if -6.30000000000000014e-209 < F

   1. Initial program 78.8%

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

   3. Taylor expanded in B around 0

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

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, {F}^{2}\right)} + 2}} \cdot \frac{F}{B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      10. lower-/.f64 66.5

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \color{blue}{\frac{F}{B}} \]

   5. Applied rewrites 66.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B}} \]

   6. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 67.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{1}{\color{blue}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -16000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -6.3 \cdot 10^{-209}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} + \frac{-1}{\tan B} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 65.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -16000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 240:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot B} - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -16000.0)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 240.0)
     (-
      (/ F (* (sqrt (fma x 2.0 (fma F F 2.0))) B))
      (*
       (/
        (fma
         (fma -0.022222222222222223 (* B B) -0.3333333333333333)
         (* B B)
         1.0)
        B)
       x))
     (- (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -16000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 240.0) {
		tmp = (F / (sqrt(fma(x, 2.0, fma(F, F, 2.0))) * B)) - ((fma(fma(-0.022222222222222223, (B * B), -0.3333333333333333), (B * B), 1.0) / B) * x);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -16000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 240.0)
		tmp = Float64(Float64(F / Float64(sqrt(fma(x, 2.0, fma(F, F, 2.0))) * B)) - Float64(Float64(fma(fma(-0.022222222222222223, Float64(B * B), -0.3333333333333333), Float64(B * B), 1.0) / B) * x));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -16000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 240.0], N[(N[(F / N[(N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(-0.022222222222222223 * N[(B * B), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -16000

   1. Initial program 54.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\color{blue}{x \cdot \frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. un-div-inv N/A

         \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. lower-/.f64 54.7

         \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 54.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\frac{x}{\tan B}\right) + \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-\frac{x}{\tan B}\right) + \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. associate-*l/ N/A

         \[\leadsto \left(-\frac{x}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-\frac{x}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} \]

   6. Applied rewrites 66.2%

      \[\leadsto \left(-\frac{x}{\tan B}\right) + \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}} \]

   7. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} \]
   8. Step-by-step derivation

      1. lower-/.f64 48.0

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} \]

   9. Applied rewrites 48.0%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} \]

   10. Taylor expanded in F around -inf

       \[\leadsto \left(-\frac{x}{B}\right) + \frac{\color{blue}{-1}}{\sin B} \]
   11. Step-by-step derivation

   12. Applied rewrites 82.1%

       \[\leadsto \left(-\frac{x}{B}\right) + \frac{\color{blue}{-1}}{\sin B} \]

   if -16000 < F < 240

   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 B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, {F}^{2}\right)} + 2}} \cdot \frac{F}{B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      10. lower-/.f64 81.9

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \color{blue}{\frac{F}{B}} \]

   5. Applied rewrites 81.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      2. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{{B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) + 1}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      3. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) \cdot {B}^{2}} + 1}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}, {B}^{2}, 1\right)}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      5. sub-neg N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{45} \cdot {B}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      6. metadata-eval N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} + \color{blue}{\frac{-1}{3}}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{45}, {B}^{2}, \frac{-1}{3}\right)}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      10. unpow2 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, B \cdot B, \frac{-1}{3}\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      11. lower-*.f64 47.2

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   8. Applied rewrites 47.2%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]
   9. Step-by-step derivation

   10. Applied rewrites 47.2%

       \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B}\right) + \frac{F}{\color{blue}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot B}} \]

   if 240 < F

   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. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\color{blue}{x \cdot \frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. un-div-inv N/A

         \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. lower-/.f64 55.3

         \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 55.3%

      \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\frac{x}{\tan B}\right) + \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-\frac{x}{\tan B}\right) + \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. associate-*l/ N/A

         \[\leadsto \left(-\frac{x}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-\frac{x}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} \]

   6. Applied rewrites 72.5%

      \[\leadsto \left(-\frac{x}{\tan B}\right) + \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}} \]

   7. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} \]
   8. Step-by-step derivation

      1. lower-/.f64 47.2

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} \]

   9. Applied rewrites 47.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} \]

   10. Taylor expanded in F around inf

       \[\leadsto \left(-\frac{x}{B}\right) + \frac{\color{blue}{1}}{\sin B} \]
   11. Step-by-step derivation

   12. Applied rewrites 74.2%

       \[\leadsto \left(-\frac{x}{B}\right) + \frac{\color{blue}{1}}{\sin B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -16000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 240:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot B} - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 55.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 0.052:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot B} - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} + \frac{-1}{\tan B} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 0.052)
   (-
    (/ F (* (sqrt (fma x 2.0 (fma F F 2.0))) B))
    (*
     (/
      (fma (fma -0.022222222222222223 (* B B) -0.3333333333333333) (* B B) 1.0)
      B)
     x))
   (+ (/ -1.0 B) (* (/ -1.0 (tan B)) x))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 0.052) {
		tmp = (F / (sqrt(fma(x, 2.0, fma(F, F, 2.0))) * B)) - ((fma(fma(-0.022222222222222223, (B * B), -0.3333333333333333), (B * B), 1.0) / B) * x);
	} else {
		tmp = (-1.0 / B) + ((-1.0 / tan(B)) * x);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 0.052)
		tmp = Float64(Float64(F / Float64(sqrt(fma(x, 2.0, fma(F, F, 2.0))) * B)) - Float64(Float64(fma(fma(-0.022222222222222223, Float64(B * B), -0.3333333333333333), Float64(B * B), 1.0) / B) * x));
	else
		tmp = Float64(Float64(-1.0 / B) + Float64(Float64(-1.0 / tan(B)) * x));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 0.052], N[(N[(F / N[(N[Sqrt[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(N[(-0.022222222222222223 * N[(B * B), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / B), $MachinePrecision] + N[(N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.0519999999999999976

   1. Initial program 74.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, {F}^{2}\right)} + 2}} \cdot \frac{F}{B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      10. lower-/.f64 62.1

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \color{blue}{\frac{F}{B}} \]

   5. Applied rewrites 62.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      2. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{{B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) + 1}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      3. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) \cdot {B}^{2}} + 1}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}, {B}^{2}, 1\right)}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      5. sub-neg N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{45} \cdot {B}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      6. metadata-eval N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} + \color{blue}{\frac{-1}{3}}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{45}, {B}^{2}, \frac{-1}{3}\right)}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      10. unpow2 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, B \cdot B, \frac{-1}{3}\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      11. lower-*.f64 43.7

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   8. Applied rewrites 43.7%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]
   9. Step-by-step derivation

   10. Applied rewrites 53.7%

       \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B}\right) + \frac{F}{\color{blue}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot B}} \]

   if 0.0519999999999999976 < B

   1. Initial program 83.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, {F}^{2}\right)} + 2}} \cdot \frac{F}{B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      10. lower-/.f64 57.6

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \color{blue}{\frac{F}{B}} \]

   5. Applied rewrites 57.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B}} \]

   6. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\color{blue}{B}} \]
   7. Step-by-step derivation

   8. Applied rewrites 56.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{-1}{\color{blue}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.052:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot B} - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} + \frac{-1}{\tan B} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -16000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+173}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{B} - \frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right)}{B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -16000.0)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 6e+173)
     (-
      (/ (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) B)
      (* (/ (fma -0.3333333333333333 (* B B) 1.0) B) x))
     (-
      (/ 1.0 B)
      (*
       (/
        (fma
         (fma -0.022222222222222223 (* B B) -0.3333333333333333)
         (* B B)
         1.0)
        B)
       x)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -16000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 6e+173) {
		tmp = ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / B) - ((fma(-0.3333333333333333, (B * B), 1.0) / B) * x);
	} else {
		tmp = (1.0 / B) - ((fma(fma(-0.022222222222222223, (B * B), -0.3333333333333333), (B * B), 1.0) / B) * x);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -16000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 6e+173)
		tmp = Float64(Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / B) - Float64(Float64(fma(-0.3333333333333333, Float64(B * B), 1.0) / B) * x));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(Float64(fma(fma(-0.022222222222222223, Float64(B * B), -0.3333333333333333), Float64(B * B), 1.0) / B) * x));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -16000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6e+173], N[(N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision] - N[(N[(N[(-0.3333333333333333 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(N[(N[(N[(-0.022222222222222223 * N[(B * B), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -16000

   1. Initial program 54.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\color{blue}{x \cdot \frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. un-div-inv N/A

         \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      4. lower-/.f64 54.7

         \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   4. Applied rewrites 54.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   5. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(-\frac{x}{\tan B}\right) + \color{blue}{\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(-\frac{x}{\tan B}\right) + \color{blue}{\frac{F}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

      3. associate-*l/ N/A

         \[\leadsto \left(-\frac{x}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-\frac{x}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)}}{\sin B}} \]

   6. Applied rewrites 66.2%

      \[\leadsto \left(-\frac{x}{\tan B}\right) + \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B}} \]

   7. Taylor expanded in B around 0

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} \]
   8. Step-by-step derivation

      1. lower-/.f64 48.0

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} \]

   9. Applied rewrites 48.0%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{\frac{F}{\sqrt{\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\sin B} \]

   10. Taylor expanded in F around -inf

       \[\leadsto \left(-\frac{x}{B}\right) + \frac{\color{blue}{-1}}{\sin B} \]
   11. Step-by-step derivation

   12. Applied rewrites 82.1%

       \[\leadsto \left(-\frac{x}{B}\right) + \frac{\color{blue}{-1}}{\sin B} \]

   if -16000 < F < 5.9999999999999995e173

   1. Initial program 95.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, {F}^{2}\right)} + 2}} \cdot \frac{F}{B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      10. lower-/.f64 77.4

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \color{blue}{\frac{F}{B}} \]

   5. Applied rewrites 77.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + \frac{-1}{3} \cdot {B}^{2}}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + \frac{-1}{3} \cdot {B}^{2}}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      2. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\frac{-1}{3} \cdot {B}^{2} + 1}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{3}, {B}^{2}, 1\right)}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      4. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      5. lower-*.f64 44.2

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   8. Applied rewrites 44.2%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]
   9. Step-by-step derivation

   10. Applied rewrites 46.7%

       \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right)}{B}\right) + \frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\color{blue}{B}} \]

   if 5.9999999999999995e173 < F

   1. Initial program 28.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, {F}^{2}\right)} + 2}} \cdot \frac{F}{B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      10. lower-/.f64 28.3

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \color{blue}{\frac{F}{B}} \]

   5. Applied rewrites 28.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      2. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{{B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) + 1}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      3. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) \cdot {B}^{2}} + 1}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}, {B}^{2}, 1\right)}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      5. sub-neg N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{45} \cdot {B}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      6. metadata-eval N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} + \color{blue}{\frac{-1}{3}}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{45}, {B}^{2}, \frac{-1}{3}\right)}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      10. unpow2 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, B \cdot B, \frac{-1}{3}\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      11. lower-*.f64 7.2

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   8. Applied rewrites 7.2%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   9. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, B \cdot B, \frac{-1}{3}\right), B \cdot B, 1\right)}{B}\right) + \frac{1}{\color{blue}{B}} \]
   10. Step-by-step derivation

   11. Applied rewrites 61.3%

       \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B}\right) + \frac{1}{\color{blue}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -16000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+173}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{B} - \frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right)}{B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 51.3% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B} \cdot x\\ \mathbf{if}\;F \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+173}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{B} - \frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right)}{B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (*
          (/
           (fma
            (fma -0.022222222222222223 (* B B) -0.3333333333333333)
            (* B B)
            1.0)
           B)
          x)))
   (if (<= F -1.32e+154)
     (- (/ -1.0 B) t_0)
     (if (<= F 6e+173)
       (-
        (/ (/ F (sqrt (fma 2.0 x (fma F F 2.0)))) B)
        (* (/ (fma -0.3333333333333333 (* B B) 1.0) B) x))
       (- (/ 1.0 B) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (fma(fma(-0.022222222222222223, (B * B), -0.3333333333333333), (B * B), 1.0) / B) * x;
	double tmp;
	if (F <= -1.32e+154) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 6e+173) {
		tmp = ((F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / B) - ((fma(-0.3333333333333333, (B * B), 1.0) / B) * x);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(fma(fma(-0.022222222222222223, Float64(B * B), -0.3333333333333333), Float64(B * B), 1.0) / B) * x)
	tmp = 0.0
	if (F <= -1.32e+154)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= 6e+173)
		tmp = Float64(Float64(Float64(F / sqrt(fma(2.0, x, fma(F, F, 2.0)))) / B) - Float64(Float64(fma(-0.3333333333333333, Float64(B * B), 1.0) / B) * x));
	else
		tmp = Float64(Float64(1.0 / B) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(N[(-0.022222222222222223 * N[(B * B), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -1.32e+154], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 6e+173], N[(N[(N[(F / N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision] - N[(N[(N[(-0.3333333333333333 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.31999999999999998e154

   1. Initial program 26.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, {F}^{2}\right)} + 2}} \cdot \frac{F}{B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      10. lower-/.f64 26.9

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \color{blue}{\frac{F}{B}} \]

   5. Applied rewrites 26.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      2. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{{B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) + 1}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      3. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) \cdot {B}^{2}} + 1}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}, {B}^{2}, 1\right)}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      5. sub-neg N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{45} \cdot {B}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      6. metadata-eval N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} + \color{blue}{\frac{-1}{3}}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{45}, {B}^{2}, \frac{-1}{3}\right)}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      10. unpow2 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, B \cdot B, \frac{-1}{3}\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      11. lower-*.f64 4.6

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   8. Applied rewrites 4.6%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   9. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, B \cdot B, \frac{-1}{3}\right), B \cdot B, 1\right)}{B}\right) + \frac{-1}{\color{blue}{B}} \]
   10. Step-by-step derivation

   11. Applied rewrites 45.3%

       \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B}\right) + \frac{-1}{\color{blue}{B}} \]

   if -1.31999999999999998e154 < F < 5.9999999999999995e173

   1. Initial program 95.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, {F}^{2}\right)} + 2}} \cdot \frac{F}{B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      10. lower-/.f64 74.0

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \color{blue}{\frac{F}{B}} \]

   5. Applied rewrites 74.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + \frac{-1}{3} \cdot {B}^{2}}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + \frac{-1}{3} \cdot {B}^{2}}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      2. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\frac{-1}{3} \cdot {B}^{2} + 1}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{3}, {B}^{2}, 1\right)}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      4. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      5. lower-*.f64 44.2

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   8. Applied rewrites 44.2%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]
   9. Step-by-step derivation

   10. Applied rewrites 46.4%

       \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right)}{B}\right) + \frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{\color{blue}{B}} \]

   if 5.9999999999999995e173 < F

   1. Initial program 28.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, {F}^{2}\right)} + 2}} \cdot \frac{F}{B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      10. lower-/.f64 28.3

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \color{blue}{\frac{F}{B}} \]

   5. Applied rewrites 28.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      2. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{{B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) + 1}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      3. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) \cdot {B}^{2}} + 1}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}, {B}^{2}, 1\right)}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      5. sub-neg N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{45} \cdot {B}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      6. metadata-eval N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} + \color{blue}{\frac{-1}{3}}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{45}, {B}^{2}, \frac{-1}{3}\right)}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      10. unpow2 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, B \cdot B, \frac{-1}{3}\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      11. lower-*.f64 7.2

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   8. Applied rewrites 7.2%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   9. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, B \cdot B, \frac{-1}{3}\right), B \cdot B, 1\right)}{B}\right) + \frac{1}{\color{blue}{B}} \]
   10. Step-by-step derivation

   11. Applied rewrites 61.3%

       \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B}\right) + \frac{1}{\color{blue}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1}{B} - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B} \cdot x\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+173}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}}{B} - \frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right)}{B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 51.3% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B} \cdot x\\ \mathbf{if}\;F \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+173}:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot B} - \frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right)}{B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (*
          (/
           (fma
            (fma -0.022222222222222223 (* B B) -0.3333333333333333)
            (* B B)
            1.0)
           B)
          x)))
   (if (<= F -1.32e+154)
     (- (/ -1.0 B) t_0)
     (if (<= F 6e+173)
       (-
        (/ F (* (sqrt (fma 2.0 x (fma F F 2.0))) B))
        (* (/ (fma -0.3333333333333333 (* B B) 1.0) B) x))
       (- (/ 1.0 B) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (fma(fma(-0.022222222222222223, (B * B), -0.3333333333333333), (B * B), 1.0) / B) * x;
	double tmp;
	if (F <= -1.32e+154) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 6e+173) {
		tmp = (F / (sqrt(fma(2.0, x, fma(F, F, 2.0))) * B)) - ((fma(-0.3333333333333333, (B * B), 1.0) / B) * x);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(fma(fma(-0.022222222222222223, Float64(B * B), -0.3333333333333333), Float64(B * B), 1.0) / B) * x)
	tmp = 0.0
	if (F <= -1.32e+154)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= 6e+173)
		tmp = Float64(Float64(F / Float64(sqrt(fma(2.0, x, fma(F, F, 2.0))) * B)) - Float64(Float64(fma(-0.3333333333333333, Float64(B * B), 1.0) / B) * x));
	else
		tmp = Float64(Float64(1.0 / B) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(N[(-0.022222222222222223 * N[(B * B), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -1.32e+154], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 6e+173], N[(N[(F / N[(N[Sqrt[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * B), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(-0.3333333333333333 * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.31999999999999998e154

   1. Initial program 26.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, {F}^{2}\right)} + 2}} \cdot \frac{F}{B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      10. lower-/.f64 26.9

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \color{blue}{\frac{F}{B}} \]

   5. Applied rewrites 26.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      2. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{{B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) + 1}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      3. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) \cdot {B}^{2}} + 1}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}, {B}^{2}, 1\right)}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      5. sub-neg N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{45} \cdot {B}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      6. metadata-eval N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} + \color{blue}{\frac{-1}{3}}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{45}, {B}^{2}, \frac{-1}{3}\right)}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      10. unpow2 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, B \cdot B, \frac{-1}{3}\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      11. lower-*.f64 4.6

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   8. Applied rewrites 4.6%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   9. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, B \cdot B, \frac{-1}{3}\right), B \cdot B, 1\right)}{B}\right) + \frac{-1}{\color{blue}{B}} \]
   10. Step-by-step derivation

   11. Applied rewrites 45.3%

       \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B}\right) + \frac{-1}{\color{blue}{B}} \]

   if -1.31999999999999998e154 < F < 5.9999999999999995e173

   1. Initial program 95.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, {F}^{2}\right)} + 2}} \cdot \frac{F}{B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      10. lower-/.f64 74.0

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \color{blue}{\frac{F}{B}} \]

   5. Applied rewrites 74.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + \frac{-1}{3} \cdot {B}^{2}}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + \frac{-1}{3} \cdot {B}^{2}}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      2. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\frac{-1}{3} \cdot {B}^{2} + 1}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{3}, {B}^{2}, 1\right)}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      4. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      5. lower-*.f64 44.2

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   8. Applied rewrites 44.2%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]
   9. Step-by-step derivation

   10. Applied rewrites 46.4%

       \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right)}{B}\right) + \frac{F}{\color{blue}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot B}} \]

   if 5.9999999999999995e173 < F

   1. Initial program 28.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, {F}^{2}\right)} + 2}} \cdot \frac{F}{B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      10. lower-/.f64 28.3

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \color{blue}{\frac{F}{B}} \]

   5. Applied rewrites 28.3%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      2. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{{B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) + 1}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      3. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) \cdot {B}^{2}} + 1}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}, {B}^{2}, 1\right)}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      5. sub-neg N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{45} \cdot {B}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      6. metadata-eval N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} + \color{blue}{\frac{-1}{3}}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{45}, {B}^{2}, \frac{-1}{3}\right)}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      10. unpow2 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, B \cdot B, \frac{-1}{3}\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      11. lower-*.f64 7.2

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   8. Applied rewrites 7.2%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   9. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, B \cdot B, \frac{-1}{3}\right), B \cdot B, 1\right)}{B}\right) + \frac{1}{\color{blue}{B}} \]
   10. Step-by-step derivation

   11. Applied rewrites 61.3%

       \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B}\right) + \frac{1}{\color{blue}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.32 \cdot 10^{+154}:\\ \;\;\;\;\frac{-1}{B} - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B} \cdot x\\ \mathbf{elif}\;F \leq 6 \cdot 10^{+173}:\\ \;\;\;\;\frac{F}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)} \cdot B} - \frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right)}{B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 37.2% accurate, 6.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B} \cdot x\\ \mathbf{if}\;F \leq -1.3 \cdot 10^{-257}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (*
          (/
           (fma
            (fma -0.022222222222222223 (* B B) -0.3333333333333333)
            (* B B)
            1.0)
           B)
          x)))
   (if (<= F -1.3e-257) (- (/ -1.0 B) t_0) (- (/ 1.0 B) t_0))))

C

double code(double F, double B, double x) {
	double t_0 = (fma(fma(-0.022222222222222223, (B * B), -0.3333333333333333), (B * B), 1.0) / B) * x;
	double tmp;
	if (F <= -1.3e-257) {
		tmp = (-1.0 / B) - t_0;
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(fma(fma(-0.022222222222222223, Float64(B * B), -0.3333333333333333), Float64(B * B), 1.0) / B) * x)
	tmp = 0.0
	if (F <= -1.3e-257)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	else
		tmp = Float64(Float64(1.0 / B) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(N[(-0.022222222222222223 * N[(B * B), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[F, -1.3e-257], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if F < -1.3e-257

   1. Initial program 75.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, {F}^{2}\right)} + 2}} \cdot \frac{F}{B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      10. lower-/.f64 55.5

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \color{blue}{\frac{F}{B}} \]

   5. Applied rewrites 55.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      2. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{{B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) + 1}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      3. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) \cdot {B}^{2}} + 1}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}, {B}^{2}, 1\right)}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      5. sub-neg N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{45} \cdot {B}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      6. metadata-eval N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} + \color{blue}{\frac{-1}{3}}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{45}, {B}^{2}, \frac{-1}{3}\right)}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      10. unpow2 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, B \cdot B, \frac{-1}{3}\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      11. lower-*.f64 37.3

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   8. Applied rewrites 37.3%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   9. Taylor expanded in F around -inf

      \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, B \cdot B, \frac{-1}{3}\right), B \cdot B, 1\right)}{B}\right) + \frac{-1}{\color{blue}{B}} \]
   10. Step-by-step derivation

   11. Applied rewrites 34.3%

       \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B}\right) + \frac{-1}{\color{blue}{B}} \]

   if -1.3e-257 < F

   1. Initial program 77.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

      5. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      6. lower-+.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, {F}^{2}\right)} + 2}} \cdot \frac{F}{B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

      10. lower-/.f64 65.5

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \color{blue}{\frac{F}{B}} \]

   5. Applied rewrites 65.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      2. +-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{{B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) + 1}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      3. *-commutative N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) \cdot {B}^{2}} + 1}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      4. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}, {B}^{2}, 1\right)}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      5. sub-neg N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{45} \cdot {B}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      6. metadata-eval N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} + \color{blue}{\frac{-1}{3}}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      7. lower-fma.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{45}, {B}^{2}, \frac{-1}{3}\right)}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      8. unpow2 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      10. unpow2 N/A

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, B \cdot B, \frac{-1}{3}\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

      11. lower-*.f64 29.9

          \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   8. Applied rewrites 29.9%

      \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   9. Taylor expanded in F around inf

      \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, B \cdot B, \frac{-1}{3}\right), B \cdot B, 1\right)}{B}\right) + \frac{1}{\color{blue}{B}} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.4%

       \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B}\right) + \frac{1}{\color{blue}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{-257}:\\ \;\;\;\;\frac{-1}{B} - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 29.7% accurate, 6.9× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{B} - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B} \cdot x \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (-
  (/ -1.0 B)
  (*
   (/
    (fma (fma -0.022222222222222223 (* B B) -0.3333333333333333) (* B B) 1.0)
    B)
   x)))

C

double code(double F, double B, double x) {
	return (-1.0 / B) - ((fma(fma(-0.022222222222222223, (B * B), -0.3333333333333333), (B * B), 1.0) / B) * x);
}

Julia

function code(F, B, x)
	return Float64(Float64(-1.0 / B) - Float64(Float64(fma(fma(-0.022222222222222223, Float64(B * B), -0.3333333333333333), Float64(B * B), 1.0) / B) * x))
end

Wolfram

code[F_, B_, x_] := N[(N[(-1.0 / B), $MachinePrecision] - N[(N[(N[(N[(-0.022222222222222223 * N[(B * B), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(B * B), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in B around 0

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

   4. lower-/.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

   5. +-commutative N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

   6. lower-+.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

   7. lower-fma.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, {F}^{2}\right)} + 2}} \cdot \frac{F}{B} \]

   8. unpow2 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

   9. lower-*.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

   10. lower-/.f64 60.9

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \color{blue}{\frac{F}{B}} \]

5. Applied rewrites 60.9%

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B}} \]

6. Taylor expanded in B around 0

   \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + {B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   2. +-commutative N/A

      \[\leadsto \left(-x \cdot \frac{\color{blue}{{B}^{2} \cdot \left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) + 1}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   3. *-commutative N/A

      \[\leadsto \left(-x \cdot \frac{\color{blue}{\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}\right) \cdot {B}^{2}} + 1}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   4. lower-fma.f64 N/A

      \[\leadsto \left(-x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} - \frac{1}{3}, {B}^{2}, 1\right)}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   5. sub-neg N/A

      \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\frac{-1}{45} \cdot {B}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   6. metadata-eval N/A

      \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{45} \cdot {B}^{2} + \color{blue}{\frac{-1}{3}}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   7. lower-fma.f64 N/A

      \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{-1}{45}, {B}^{2}, \frac{-1}{3}\right)}, {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   8. unpow2 N/A

      \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   9. lower-*.f64 N/A

      \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, \color{blue}{B \cdot B}, \frac{-1}{3}\right), {B}^{2}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   10. unpow2 N/A

       \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, B \cdot B, \frac{-1}{3}\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   11. lower-*.f64 33.3

       \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

8. Applied rewrites 33.3%

   \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

9. Taylor expanded in F around -inf

   \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{45}, B \cdot B, \frac{-1}{3}\right), B \cdot B, 1\right)}{B}\right) + \frac{-1}{\color{blue}{B}} \]
10. Step-by-step derivation

11. Applied rewrites 26.9%

    \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B}\right) + \frac{-1}{\color{blue}{B}} \]

12. Final simplification 26.9%

    \[\leadsto \frac{-1}{B} - \frac{\mathsf{fma}\left(\mathsf{fma}\left(-0.022222222222222223, B \cdot B, -0.3333333333333333\right), B \cdot B, 1\right)}{B} \cdot x \]
13. Add Preprocessing

Alternative 21: 9.4% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{F} \cdot \frac{F}{B} - \left(-0.3333333333333333 \cdot B\right) \cdot x \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (- (* (/ -1.0 F) (/ F B)) (* (* -0.3333333333333333 B) x)))

C

double code(double F, double B, double x) {
	return ((-1.0 / F) * (F / B)) - ((-0.3333333333333333 * B) * x);
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (((-1.0d0) / f) * (f / b)) - (((-0.3333333333333333d0) * b) * x)
end function

Java

public static double code(double F, double B, double x) {
	return ((-1.0 / F) * (F / B)) - ((-0.3333333333333333 * B) * x);
}

Python

def code(F, B, x):
	return ((-1.0 / F) * (F / B)) - ((-0.3333333333333333 * B) * x)

Julia

function code(F, B, x)
	return Float64(Float64(Float64(-1.0 / F) * Float64(F / B)) - Float64(Float64(-0.3333333333333333 * B) * x))
end

MATLAB

function tmp = code(F, B, x)
	tmp = ((-1.0 / F) * (F / B)) - ((-0.3333333333333333 * B) * x);
end

Wolfram

code[F_, B_, x_] := N[(N[(N[(-1.0 / F), $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(N[(-0.3333333333333333 * B), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 76.7%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in B around 0

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \cdot \frac{F}{B}} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

   4. lower-/.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \cdot \frac{F}{B} \]

   5. +-commutative N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

   6. lower-+.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + {F}^{2}\right) + 2}}} \cdot \frac{F}{B} \]

   7. lower-fma.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, {F}^{2}\right)} + 2}} \cdot \frac{F}{B} \]

   8. unpow2 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

   9. lower-*.f64 N/A

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F}\right) + 2}} \cdot \frac{F}{B} \]

   10. lower-/.f64 60.9

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \color{blue}{\frac{F}{B}} \]

5. Applied rewrites 60.9%

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B}} \]

6. Taylor expanded in B around 0

   \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + \frac{-1}{3} \cdot {B}^{2}}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]
7. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \left(-x \cdot \color{blue}{\frac{1 + \frac{-1}{3} \cdot {B}^{2}}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   2. +-commutative N/A

      \[\leadsto \left(-x \cdot \frac{\color{blue}{\frac{-1}{3} \cdot {B}^{2} + 1}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   3. lower-fma.f64 N/A

      \[\leadsto \left(-x \cdot \frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{3}, {B}^{2}, 1\right)}}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   4. unpow2 N/A

      \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

   5. lower-*.f64 33.5

      \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, \color{blue}{B \cdot B}, 1\right)}{B}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

8. Applied rewrites 33.5%

   \[\leadsto \left(-x \cdot \color{blue}{\frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right)}{B}}\right) + \sqrt{\frac{1}{\mathsf{fma}\left(2, x, F \cdot F\right) + 2}} \cdot \frac{F}{B} \]

9. Taylor expanded in F around -inf

   \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(\frac{-1}{3}, B \cdot B, 1\right)}{B}\right) + \frac{-1}{F} \cdot \frac{\color{blue}{F}}{B} \]
10. Step-by-step derivation

11. Applied rewrites 19.7%

    \[\leadsto \left(-x \cdot \frac{\mathsf{fma}\left(-0.3333333333333333, B \cdot B, 1\right)}{B}\right) + \frac{-1}{F} \cdot \frac{\color{blue}{F}}{B} \]

12. Taylor expanded in B around inf

    \[\leadsto \left(-x \cdot \left(\frac{-1}{3} \cdot \color{blue}{B}\right)\right) + \frac{-1}{F} \cdot \frac{F}{B} \]
13. Step-by-step derivation

14. Applied rewrites 8.0%

    \[\leadsto \left(-x \cdot \left(-0.3333333333333333 \cdot \color{blue}{B}\right)\right) + \frac{-1}{F} \cdot \frac{F}{B} \]

15. Final simplification 8.0%

    \[\leadsto \frac{-1}{F} \cdot \frac{F}{B} - \left(-0.3333333333333333 \cdot B\right) \cdot x \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024312 
(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))))))