VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.0% → 99.5%

Time: 16.1s

Alternatives: 20

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: 77.0% 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: 99.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -8 \cdot 10^{+153}:\\ \;\;\;\;t\_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2000000000:\\ \;\;\;\;t\_0 + \frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -8e+153)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 2000000000.0)
       (+ t_0 (/ F (* (sin B) (sqrt (fma F F 2.0)))))
       (+ (/ -1.0 (/ (tan B) x)) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -8e+153) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 2000000000.0) {
		tmp = t_0 + (F / (sin(B) * sqrt(fma(F, F, 2.0))));
	} else {
		tmp = (-1.0 / (tan(B) / x)) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -8e+153)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 2000000000.0)
		tmp = Float64(t_0 + Float64(F / Float64(sin(B) * sqrt(fma(F, F, 2.0)))));
	else
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -8e+153], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2000000000.0], N[(t$95$0 + N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -8e153

   1. Initial program 35.3%

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

   3. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.6

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

   5. Simplified 99.6%

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

   if -8e153 < F < 2e9

   1. Initial program 97.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied egg-rr 99.6%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 99.7

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

   9. Simplified 99.7%

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

   if 2e9 < F

   1. Initial program 61.9%

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

      1. lift-tan.f64 N/A

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

      2. remove-double-neg N/A

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

      3. un-div-inv N/A

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

      4. remove-double-neg N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 61.9

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

   4. Applied egg-rr 61.9%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.7

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

   7. Simplified 99.7%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{+153}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 2000000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;t\_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.36:\\ \;\;\;\;t\_0 + \frac{F}{\sin B \cdot \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -1.42)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 1.36)
       (+ t_0 (/ F (* (sin B) (sqrt 2.0))))
       (+ (/ -1.0 (/ (tan B) x)) (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -1.42) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 1.36) {
		tmp = t_0 + (F / (sin(B) * sqrt(2.0)));
	} else {
		tmp = (-1.0 / (tan(B) / x)) + (1.0 / sin(B));
	}
	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 * ((-1.0d0) / tan(b))
    if (f <= (-1.42d0)) then
        tmp = t_0 + ((-1.0d0) / sin(b))
    else if (f <= 1.36d0) then
        tmp = t_0 + (f / (sin(b) * sqrt(2.0d0)))
    else
        tmp = ((-1.0d0) / (tan(b) / x)) + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -1.42) {
		tmp = t_0 + (-1.0 / Math.sin(B));
	} else if (F <= 1.36) {
		tmp = t_0 + (F / (Math.sin(B) * Math.sqrt(2.0)));
	} else {
		tmp = (-1.0 / (Math.tan(B) / x)) + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -1.42:
		tmp = t_0 + (-1.0 / math.sin(B))
	elif F <= 1.36:
		tmp = t_0 + (F / (math.sin(B) * math.sqrt(2.0)))
	else:
		tmp = (-1.0 / (math.tan(B) / x)) + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -1.42)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 1.36)
		tmp = Float64(t_0 + Float64(F / Float64(sin(B) * sqrt(2.0))));
	else
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -1.42)
		tmp = t_0 + (-1.0 / sin(B));
	elseif (F <= 1.36)
		tmp = t_0 + (F / (sin(B) * sqrt(2.0)));
	else
		tmp = (-1.0 / (tan(B) / x)) + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.42], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.36], N[(t$95$0 + N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4199999999999999

   1. Initial program 53.9%

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

   3. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 98.9

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

   5. Simplified 98.9%

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

   if -1.4199999999999999 < F < 1.3600000000000001

   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-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied egg-rr 99.5%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 99.6

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

   9. Simplified 99.6%

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

   10. Taylor expanded in F around 0

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

       1. lower-sqrt.f64 98.9

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

   12. Simplified 98.9%

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

   if 1.3600000000000001 < F

   1. Initial program 62.4%

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

      1. lift-tan.f64 N/A

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

      2. remove-double-neg N/A

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

      3. un-div-inv N/A

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

      4. remove-double-neg N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 62.4

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

   4. Applied egg-rr 62.4%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 99.4

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

   7. Simplified 99.4%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 1.36:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B \cdot \sqrt{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}\\ t_1 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -1.18 \cdot 10^{-33}:\\ \;\;\;\;t\_1 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -2 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{t\_0} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.55 \cdot 10^{-50}:\\ \;\;\;\;t\_1 + \frac{\frac{F}{B}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (fma F F (fma 2.0 x 2.0)))) (t_1 (* x (/ -1.0 (tan B)))))
   (if (<= F -1.18e-33)
     (+ t_1 (/ -1.0 (sin B)))
     (if (<= F -2e-80)
       (- (/ (/ F (sin B)) t_0) (/ x B))
       (if (<= F 3.55e-50)
         (+ t_1 (/ (/ F B) t_0))
         (+ (/ -1.0 (/ (tan B) x)) (/ 1.0 (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt(fma(F, F, fma(2.0, x, 2.0)));
	double t_1 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -1.18e-33) {
		tmp = t_1 + (-1.0 / sin(B));
	} else if (F <= -2e-80) {
		tmp = ((F / sin(B)) / t_0) - (x / B);
	} else if (F <= 3.55e-50) {
		tmp = t_1 + ((F / B) / t_0);
	} else {
		tmp = (-1.0 / (tan(B) / x)) + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = sqrt(fma(F, F, fma(2.0, x, 2.0)))
	t_1 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -1.18e-33)
		tmp = Float64(t_1 + Float64(-1.0 / sin(B)));
	elseif (F <= -2e-80)
		tmp = Float64(Float64(Float64(F / sin(B)) / t_0) - Float64(x / B));
	elseif (F <= 3.55e-50)
		tmp = Float64(t_1 + Float64(Float64(F / B) / t_0));
	else
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.18e-33], N[(t$95$1 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2e-80], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.55e-50], N[(t$95$1 + N[(N[(F / B), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.18e-33

   1. Initial program 58.7%

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

   3. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 97.8

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

   5. Simplified 97.8%

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

   if -1.18e-33 < F < -1.99999999999999992e-80

   1. Initial program 99.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 89.3

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

   7. Simplified 89.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. *-commutative N/A

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

   9. Applied egg-rr 89.6%

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

   if -1.99999999999999992e-80 < F < 3.5499999999999999e-50

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 85.1

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

   5. Simplified 85.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. un-div-inv N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 85.1

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

      13. lift-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-fma.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-+l+ N/A

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

      19. +-commutative N/A

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

   7. Applied egg-rr 85.1%

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

   if 3.5499999999999999e-50 < F

   1. Initial program 64.6%

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

      1. lift-tan.f64 N/A

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

      2. remove-double-neg N/A

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

      3. un-div-inv N/A

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

      4. remove-double-neg N/A

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

      5. clear-num N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 64.6

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

   4. Applied egg-rr 64.6%

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

   5. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 95.2

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

   7. Simplified 95.2%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.18 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -2 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.55 \cdot 10^{-50}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\frac{F}{B}}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}\\ t_1 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -1.18 \cdot 10^{-33}:\\ \;\;\;\;t\_1 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -2 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{t\_0} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.82 \cdot 10^{-63}:\\ \;\;\;\;t\_1 + \frac{\frac{F}{B}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (fma F F (fma 2.0 x 2.0)))) (t_1 (* x (/ -1.0 (tan B)))))
   (if (<= F -1.18e-33)
     (+ t_1 (/ -1.0 (sin B)))
     (if (<= F -2e-80)
       (- (/ (/ F (sin B)) t_0) (/ x B))
       (if (<= F 1.82e-63) (+ t_1 (/ (/ F B) t_0)) (+ t_1 (/ 1.0 (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt(fma(F, F, fma(2.0, x, 2.0)));
	double t_1 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -1.18e-33) {
		tmp = t_1 + (-1.0 / sin(B));
	} else if (F <= -2e-80) {
		tmp = ((F / sin(B)) / t_0) - (x / B);
	} else if (F <= 1.82e-63) {
		tmp = t_1 + ((F / B) / t_0);
	} else {
		tmp = t_1 + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = sqrt(fma(F, F, fma(2.0, x, 2.0)))
	t_1 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -1.18e-33)
		tmp = Float64(t_1 + Float64(-1.0 / sin(B)));
	elseif (F <= -2e-80)
		tmp = Float64(Float64(Float64(F / sin(B)) / t_0) - Float64(x / B));
	elseif (F <= 1.82e-63)
		tmp = Float64(t_1 + Float64(Float64(F / B) / t_0));
	else
		tmp = Float64(t_1 + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.18e-33], N[(t$95$1 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2e-80], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.82e-63], N[(t$95$1 + N[(N[(F / B), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.18e-33

   1. Initial program 58.7%

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

   3. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 97.8

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

   5. Simplified 97.8%

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

   if -1.18e-33 < F < -1.99999999999999992e-80

   1. Initial program 99.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 89.3

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

   7. Simplified 89.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. *-commutative N/A

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

   9. Applied egg-rr 89.6%

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

   if -1.99999999999999992e-80 < F < 1.81999999999999994e-63

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 84.9

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

   5. Simplified 84.9%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. un-div-inv N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 85.0

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

      13. lift-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-fma.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-+l+ N/A

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

      19. +-commutative N/A

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

   7. Applied egg-rr 85.0%

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

   if 1.81999999999999994e-63 < F

   1. Initial program 65.0%

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

   3. Taylor expanded in F around inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 94.0

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

   5. Simplified 94.0%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.18 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -2 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.82 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\frac{F}{B}}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}\\ t_1 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -1.18 \cdot 10^{-33}:\\ \;\;\;\;t\_1 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -2 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{t\_0} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{-58}:\\ \;\;\;\;t\_1 + \frac{\frac{F}{B}}{t\_0}\\ \mathbf{elif}\;F \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (fma F F (fma 2.0 x 2.0)))) (t_1 (* x (/ -1.0 (tan B)))))
   (if (<= F -1.18e-33)
     (+ t_1 (/ -1.0 (sin B)))
     (if (<= F -2e-80)
       (- (/ (/ F (sin B)) t_0) (/ x B))
       (if (<= F 2.9e-58)
         (+ t_1 (/ (/ F B) t_0))
         (if (<= F 7.6e+153)
           (- (/ (* F (sqrt (/ 1.0 (fma F F 2.0)))) (sin B)) (/ x B))
           (- (/ 1.0 B) (/ x (tan B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt(fma(F, F, fma(2.0, x, 2.0)));
	double t_1 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -1.18e-33) {
		tmp = t_1 + (-1.0 / sin(B));
	} else if (F <= -2e-80) {
		tmp = ((F / sin(B)) / t_0) - (x / B);
	} else if (F <= 2.9e-58) {
		tmp = t_1 + ((F / B) / t_0);
	} else if (F <= 7.6e+153) {
		tmp = ((F * sqrt((1.0 / fma(F, F, 2.0)))) / sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = sqrt(fma(F, F, fma(2.0, x, 2.0)))
	t_1 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -1.18e-33)
		tmp = Float64(t_1 + Float64(-1.0 / sin(B)));
	elseif (F <= -2e-80)
		tmp = Float64(Float64(Float64(F / sin(B)) / t_0) - Float64(x / B));
	elseif (F <= 2.9e-58)
		tmp = Float64(t_1 + Float64(Float64(F / B) / t_0));
	elseif (F <= 7.6e+153)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / fma(F, F, 2.0)))) / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.18e-33], N[(t$95$1 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2e-80], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.9e-58], N[(t$95$1 + N[(N[(F / B), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.6e+153], N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.18e-33

   1. Initial program 58.7%

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

   3. Taylor expanded in F around -inf

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

      1. lower-/.f64 N/A

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

      2. lower-sin.f64 97.8

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

   5. Simplified 97.8%

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

   if -1.18e-33 < F < -1.99999999999999992e-80

   1. Initial program 99.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 89.3

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

   7. Simplified 89.3%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. lift-sin.f64 N/A

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

      5. *-commutative N/A

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

   9. Applied egg-rr 89.6%

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

   if -1.99999999999999992e-80 < F < 2.8999999999999999e-58

   1. Initial program 99.6%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 85.1

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

   5. Simplified 85.1%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. frac-2neg N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

      6. frac-2neg N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

      9. lift-/.f64 N/A

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

      10. un-div-inv N/A

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

      11. lower-/.f64 N/A

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

      12. lower-sqrt.f64 85.1

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

      13. lift-fma.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lift-fma.f64 N/A

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

      17. lift-*.f64 N/A

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

      18. associate-+l+ N/A

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

      19. +-commutative N/A

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

   7. Applied egg-rr 85.1%

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

   if 2.8999999999999999e-58 < F < 7.59999999999999933e153

   1. Initial program 92.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 79.2

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

   7. Simplified 79.2%

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

   8. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 79.3

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

   10. Simplified 79.3%

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

   if 7.59999999999999933e153 < F

   1. Initial program 32.7%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 32.7

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

   5. Simplified 32.7%

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

      1. lift-tan.f64 N/A

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

      2. remove-double-neg N/A

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

      3. un-div-inv N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 32.8

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

   7. Applied egg-rr 32.8%

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

   8. Taylor expanded in F around inf

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

      1. lower-/.f64 82.7

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

   10. Simplified 82.7%

       \[\leadsto \left(-\frac{x}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.18 \cdot 10^{-33}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -2 \cdot 10^{-80}:\\ \;\;\;\;\frac{\frac{F}{\sin B}}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{-58}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\frac{F}{B}}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}\\ \mathbf{elif}\;F \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4.9 \cdot 10^{+200}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{-58}:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - t\_0\\ \mathbf{elif}\;F \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -4.9e+200)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F 2.9e-58)
       (- (/ F (* B (sqrt (fma F F (fma 2.0 x 2.0))))) t_0)
       (if (<= F 7.6e+153)
         (- (/ (* F (sqrt (/ 1.0 (fma F F 2.0)))) (sin B)) (/ x B))
         (- (/ 1.0 B) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -4.9e+200) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 2.9e-58) {
		tmp = (F / (B * sqrt(fma(F, F, fma(2.0, x, 2.0))))) - t_0;
	} else if (F <= 7.6e+153) {
		tmp = ((F * sqrt((1.0 / fma(F, F, 2.0)))) / sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -4.9e+200)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 2.9e-58)
		tmp = Float64(Float64(F / Float64(B * sqrt(fma(F, F, fma(2.0, x, 2.0))))) - t_0);
	elseif (F <= 7.6e+153)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / fma(F, F, 2.0)))) / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / 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, -4.9e+200], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.9e-58], N[(N[(F / N[(B * N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 7.6e+153], N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.89999999999999982e200

   1. Initial program 30.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied egg-rr 42.2%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 16.3

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

   7. Simplified 16.3%

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

   8. Taylor expanded in F around -inf

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. neg-mul-1 N/A

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

      5. lower--.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-/.f64 72.5

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

   10. Simplified 72.5%

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

   if -4.89999999999999982e200 < F < 2.8999999999999999e-58

   1. Initial program 94.0%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 77.8

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

   5. Simplified 77.8%

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

      1. lift-tan.f64 N/A

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

      2. remove-double-neg N/A

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

      3. un-div-inv N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 77.8

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

   7. Applied egg-rr 77.8%

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

      1. lift-fma.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. sqrt-div N/A

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

      4. metadata-eval N/A

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

      5. frac-times N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-sqrt.f64 81.3

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

      10. lift-fma.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-fma.f64 N/A

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

      15. lift-fma.f64 N/A

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

      16. lift-fma.f64 81.3

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

   9. Applied egg-rr 81.3%

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

   if 2.8999999999999999e-58 < F < 7.59999999999999933e153

   1. Initial program 92.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 79.2

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

   7. Simplified 79.2%

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

   8. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 79.3

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

   10. Simplified 79.3%

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

   if 7.59999999999999933e153 < F

   1. Initial program 32.7%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 32.7

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

   5. Simplified 32.7%

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

      1. lift-tan.f64 N/A

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

      2. remove-double-neg N/A

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

      3. un-div-inv N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 32.8

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

   7. Applied egg-rr 32.8%

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

   8. Taylor expanded in F around inf

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

      1. lower-/.f64 82.7

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

   10. Simplified 82.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.9 \cdot 10^{+200}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{-58}:\\ \;\;\;\;\frac{F}{B \cdot \sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.9 \cdot 10^{+200}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}\\ \mathbf{elif}\;F \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.9e+200)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 6e-63)
     (+ (* x (/ -1.0 (tan B))) (/ F (* B (sqrt (fma F F 2.0)))))
     (if (<= F 7.6e+153)
       (- (/ (* F (sqrt (/ 1.0 (fma F F 2.0)))) (sin B)) (/ x B))
       (- (/ 1.0 B) (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.9e+200) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 6e-63) {
		tmp = (x * (-1.0 / tan(B))) + (F / (B * sqrt(fma(F, F, 2.0))));
	} else if (F <= 7.6e+153) {
		tmp = ((F * sqrt((1.0 / fma(F, F, 2.0)))) / sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.9e+200)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 6e-63)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(F / Float64(B * sqrt(fma(F, F, 2.0)))));
	elseif (F <= 7.6e+153)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / fma(F, F, 2.0)))) / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.9e+200], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6e-63], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(B * N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.6e+153], N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.89999999999999982e200

   1. Initial program 30.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied egg-rr 42.2%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 16.3

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

   7. Simplified 16.3%

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

   8. Taylor expanded in F around -inf

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. neg-mul-1 N/A

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

      5. lower--.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-/.f64 72.5

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

   10. Simplified 72.5%

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

   if -4.89999999999999982e200 < F < 5.99999999999999959e-63

   1. Initial program 94.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied egg-rr 97.5%

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

   6. Applied egg-rr 97.6%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 97.6

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

   9. Simplified 97.6%

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

   10. Taylor expanded in B around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. +-commutative N/A

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

       4. unpow2 N/A

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

       5. lower-fma.f64 81.3

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

   12. Simplified 81.3%

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

   if 5.99999999999999959e-63 < F < 7.59999999999999933e153

   1. Initial program 92.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 79.2

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

   7. Simplified 79.2%

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

   8. Taylor expanded in x around 0

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

      1. lower-sqrt.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. lower-fma.f64 79.3

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

   10. Simplified 79.3%

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

   if 7.59999999999999933e153 < F

   1. Initial program 32.7%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 32.7

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

   5. Simplified 32.7%

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

      1. lift-tan.f64 N/A

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

      2. remove-double-neg N/A

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

      3. un-div-inv N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 32.8

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

   7. Applied egg-rr 32.8%

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

   8. Taylor expanded in F around inf

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

      1. lower-/.f64 82.7

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

   10. Simplified 82.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.9 \cdot 10^{+200}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{-63}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}\\ \mathbf{elif}\;F \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\mathsf{fma}\left(F, F, 2\right)}\\ \mathbf{if}\;F \leq -4.9 \cdot 10^{+200}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{-58}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot t\_0}\\ \mathbf{elif}\;F \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{F}{\sin B \cdot t\_0} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (fma F F 2.0))))
   (if (<= F -4.9e+200)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F 2.9e-58)
       (+ (* x (/ -1.0 (tan B))) (/ F (* B t_0)))
       (if (<= F 7.6e+153)
         (- (/ F (* (sin B) t_0)) (/ x B))
         (- (/ 1.0 B) (/ x (tan B))))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt(fma(F, F, 2.0));
	double tmp;
	if (F <= -4.9e+200) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 2.9e-58) {
		tmp = (x * (-1.0 / tan(B))) + (F / (B * t_0));
	} else if (F <= 7.6e+153) {
		tmp = (F / (sin(B) * t_0)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = sqrt(fma(F, F, 2.0))
	tmp = 0.0
	if (F <= -4.9e+200)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 2.9e-58)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(F / Float64(B * t_0)));
	elseif (F <= 7.6e+153)
		tmp = Float64(Float64(F / Float64(sin(B) * t_0)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[F, -4.9e+200], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.9e-58], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F / N[(B * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.6e+153], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4.89999999999999982e200

   1. Initial program 30.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied egg-rr 42.2%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 16.3

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

   7. Simplified 16.3%

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

   8. Taylor expanded in F around -inf

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. neg-mul-1 N/A

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

      5. lower--.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-/.f64 72.5

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

   10. Simplified 72.5%

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

   if -4.89999999999999982e200 < F < 2.8999999999999999e-58

   1. Initial program 94.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied egg-rr 97.5%

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

   6. Applied egg-rr 97.6%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 97.6

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

   9. Simplified 97.6%

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

   10. Taylor expanded in B around 0

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

       1. lower-*.f64 N/A

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

       2. lower-sqrt.f64 N/A

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

       3. +-commutative N/A

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

       4. unpow2 N/A

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

       5. lower-fma.f64 81.3

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

   12. Simplified 81.3%

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

   if 2.8999999999999999e-58 < F < 7.59999999999999933e153

   1. Initial program 92.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied egg-rr 99.6%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 99.5

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

   9. Simplified 99.5%

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

   10. Taylor expanded in B around 0

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

       1. lower-/.f64 79.1

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

   12. Simplified 79.1%

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

   if 7.59999999999999933e153 < F

   1. Initial program 32.7%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 32.7

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

   5. Simplified 32.7%

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

      1. lift-tan.f64 N/A

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

      2. remove-double-neg N/A

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

      3. un-div-inv N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 32.8

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

   7. Applied egg-rr 32.8%

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

   8. Taylor expanded in F around inf

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

      1. lower-/.f64 82.7

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

   10. Simplified 82.7%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.9 \cdot 10^{+200}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.9 \cdot 10^{-58}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}}\\ \mathbf{elif}\;F \leq 7.6 \cdot 10^{+153}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 76.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{-56}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (/ x (tan B)))))
   (if (<= x -5.8e-56)
     t_0
     (if (<= x 2.2e-24)
       (- (/ F (* (sin B) (sqrt (fma F F 2.0)))) (/ x B))
       t_0))))

C

double code(double F, double B, double x) {
	double t_0 = (1.0 / B) - (x / tan(B));
	double tmp;
	if (x <= -5.8e-56) {
		tmp = t_0;
	} else if (x <= 2.2e-24) {
		tmp = (F / (sin(B) * sqrt(fma(F, F, 2.0)))) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -5.8e-56)
		tmp = t_0;
	elseif (x <= 2.2e-24)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(fma(F, F, 2.0)))) - Float64(x / B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.8e-56], t$95$0, If[LessEqual[x, 2.2e-24], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5.79999999999999982e-56 or 2.20000000000000002e-24 < x

   1. Initial program 81.1%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 78.8

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

   5. Simplified 78.8%

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

      1. lift-tan.f64 N/A

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

      2. remove-double-neg N/A

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

      3. un-div-inv N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 78.8

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

   7. Applied egg-rr 78.8%

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

   8. Taylor expanded in F around inf

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

      1. lower-/.f64 92.8

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

   10. Simplified 92.8%

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

   if -5.79999999999999982e-56 < x < 2.20000000000000002e-24

   1. Initial program 71.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied egg-rr 73.4%

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

   6. Applied egg-rr 73.5%

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

   7. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. +-commutative N/A

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

      5. unpow2 N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-sin.f64 73.5

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

   9. Simplified 73.5%

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

   10. Taylor expanded in B around 0

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

       1. lower-/.f64 61.8

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

   12. Simplified 61.8%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-56}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-24}:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{\mathsf{fma}\left(F, F, 2\right)}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{if}\;x \leq -2.9 \cdot 10^{-82}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-133}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-97}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (/ 1.0 B) (/ x (tan B)))))
   (if (<= x -2.9e-82)
     t_0
     (if (<= x 3.2e-133)
       (* (/ F (sin B)) (sqrt (/ 1.0 (fma F F 2.0))))
       (if (<= x 2.4e-97)
         (- (/ -1.0 (sin B)) (/ x B))
         (if (<= x 7.5e-25) (- (/ 1.0 (sin B)) (/ x B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = (1.0 / B) - (x / tan(B));
	double tmp;
	if (x <= -2.9e-82) {
		tmp = t_0;
	} else if (x <= 3.2e-133) {
		tmp = (F / sin(B)) * sqrt((1.0 / fma(F, F, 2.0)));
	} else if (x <= 2.4e-97) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (x <= 7.5e-25) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(Float64(1.0 / B) - Float64(x / tan(B)))
	tmp = 0.0
	if (x <= -2.9e-82)
		tmp = t_0;
	elseif (x <= 3.2e-133)
		tmp = Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / fma(F, F, 2.0))));
	elseif (x <= 2.4e-97)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (x <= 7.5e-25)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.9e-82], t$95$0, If[LessEqual[x, 3.2e-133], N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.4e-97], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.5e-25], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.89999999999999977e-82 or 7.49999999999999989e-25 < x

   1. Initial program 81.3%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 78.4

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

   5. Simplified 78.4%

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

      1. lift-tan.f64 N/A

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

      2. remove-double-neg N/A

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

      3. un-div-inv N/A

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

      4. remove-double-neg N/A

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

      5. lower-/.f64 78.4

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

   7. Applied egg-rr 78.4%

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

   8. Taylor expanded in F around inf

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

      1. lower-/.f64 89.4

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

   10. Simplified 89.4%

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

   if -2.89999999999999977e-82 < x < 3.20000000000000013e-133

   1. Initial program 75.4%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied egg-rr 76.7%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 69.2

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

   7. Simplified 69.2%

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

   8. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. +-commutative N/A

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

      6. unpow2 N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 62.0

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

   10. Simplified 62.0%

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

   if 3.20000000000000013e-133 < x < 2.4e-97

   1. Initial program 34.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied egg-rr 35.7%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 19.8

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

   7. Simplified 19.8%

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

   8. Taylor expanded in F around -inf

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

      1. distribute-lft-in N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. neg-mul-1 N/A

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

      5. lower--.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-/.f64 67.2

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

   10. Simplified 67.2%

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

   if 2.4e-97 < x < 7.49999999999999989e-25

   1. Initial program 71.3%

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

      1. lift-sin.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-+.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift-neg.f64 N/A

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

      8. lift-pow.f64 N/A

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

      9. associate-*l/ N/A

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

      10. lower-/.f64 N/A

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

   4. Applied egg-rr 75.8%

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

   5. Taylor expanded in B around 0

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

      1. lower-/.f64 51.3

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

   7. Simplified 51.3%

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

   8. Taylor expanded in F around inf

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sin.f64 N/A

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

      4. lower-/.f64 46.6

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

   10. Simplified 46.6%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.9 \cdot 10^{-82}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{-133}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{-97}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.35 \cdot 10^{-161}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.82 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.35e-161)
   (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))
   (if (<= F 1.82e-63)
     (/ (- (/ F (sqrt (fma F F (fma 2.0 x 2.0)))) x) B)
     (- (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.35e-161) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	} else if (F <= 1.82e-63) {
		tmp = ((F / sqrt(fma(F, F, fma(2.0, x, 2.0)))) - x) / B;
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.35e-161)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	elseif (F <= 1.82e-63)
		tmp = Float64(Float64(Float64(F / sqrt(fma(F, F, fma(2.0, x, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.35e-161], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.82e-63], N[(N[(N[(F / N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $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 < -2.3500000000000002e-161

   1. Initial program 68.1%

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

   3. Taylor expanded in B around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-sqrt.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. associate-+r+ N/A

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

      6. +-commutative N/A

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

      7. associate-+l+ N/A

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

      8. lower-fma.f64 N/A

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

      9. +-commutative N/A

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

      10. unpow2 N/A

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

      11. lower-fma.f64 55.2

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

   5. Simplified 55.2%

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

   6. Taylor expanded in F around -inf

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

      1. lower-/.f64 69.6

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

   8. Simplified 69.6%

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

   if -2.3500000000000002e-161 < F < 1.81999999999999994e-63

   1. Initial program 99.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B}} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      5. associate-+r+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + 2\right)} + {F}^{2}}} \]

      7. associate-+l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left(2 + {F}^{2}\right)}}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, 2 + {F}^{2}\right)}}} \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{{F}^{2} + 2}\right)}} \]

      10. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)}} \]

      11. lower-fma.f64 86.2

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}} \]

   5. Simplified 86.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 55.5

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   8. Simplified 55.5%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   10. Applied egg-rr 55.6%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - x}{B}} \]

   if 1.81999999999999994e-63 < F

   1. Initial program 65.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\color{blue}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(\color{blue}{F \cdot F} + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\color{blue}{\left(F \cdot F + 2\right)} + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + \color{blue}{2 \cdot x}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \]

      7. lift-neg.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      9. associate-*l/ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}} \]

   4. Applied egg-rr 78.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-0.5}}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) + \frac{F \cdot {\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B} \]
   6. Step-by-step derivation

      1. lower-/.f64 56.3

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-0.5}}{\sin B} \]

   7. Simplified 56.3%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-0.5}}{\sin B} \]

   8. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\sin B}} - \frac{x}{B} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} - \frac{x}{B} \]

      4. lower-/.f64 72.9

         \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x}{B}} \]

   10. Simplified 72.9%

       \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.35 \cdot 10^{-161}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.82 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.4 \cdot 10^{+25}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.82 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{F}{B}}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.4e+25)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 1.82e-63)
     (- (/ (/ F B) (sqrt (fma F F (fma 2.0 x 2.0)))) (/ x B))
     (- (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.4e+25) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 1.82e-63) {
		tmp = ((F / B) / sqrt(fma(F, F, fma(2.0, x, 2.0)))) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.4e+25)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 1.82e-63)
		tmp = Float64(Float64(Float64(F / B) / sqrt(fma(F, F, fma(2.0, x, 2.0)))) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.4e+25], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.82e-63], N[(N[(N[(F / B), $MachinePrecision] / N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $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 < -7.3999999999999998e25

   1. Initial program 51.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\color{blue}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(\color{blue}{F \cdot F} + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\color{blue}{\left(F \cdot F + 2\right)} + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + \color{blue}{2 \cdot x}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \]

      7. lift-neg.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      9. associate-*l/ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}} \]

   4. Applied egg-rr 65.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-0.5}}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) + \frac{F \cdot {\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B} \]
   6. Step-by-step derivation

      1. lower-/.f64 34.1

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-0.5}}{\sin B} \]

   7. Simplified 34.1%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-0.5}}{\sin B} \]

   8. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x}{B}\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{1}{\sin B} + -1 \cdot \frac{x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{B}\right)\right)} \]

      3. unsub-neg N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{1}{\sin B} - \frac{x}{B}} \]

      4. neg-mul-1 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} - \frac{x}{B} \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) - \frac{x}{B}} \]

      6. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\sin B}} - \frac{x}{B} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\sin B} - \frac{x}{B} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\sin B}} - \frac{x}{B} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} - \frac{x}{B} \]

      10. lower-/.f64 67.9

          \[\leadsto \frac{-1}{\sin B} - \color{blue}{\frac{x}{B}} \]

   10. Simplified 67.9%

       \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   if -7.3999999999999998e25 < F < 1.81999999999999994e-63

   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

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B}} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      5. associate-+r+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + 2\right)} + {F}^{2}}} \]

      7. associate-+l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left(2 + {F}^{2}\right)}}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, 2 + {F}^{2}\right)}}} \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{{F}^{2} + 2}\right)}} \]

      10. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)}} \]

      11. lower-fma.f64 81.0

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}} \]

   5. Simplified 81.0%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 47.0

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   8. Simplified 47.0%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) + \color{blue}{\frac{F}{B}} \cdot \sqrt{\frac{1}{2 \cdot x + \left(F \cdot F + 2\right)}} \]

      2. lift-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{2 \cdot x + \color{blue}{\mathsf{fma}\left(F, F, 2\right)}}} \]

      3. lift-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

      4. sqrt-div N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) + \frac{F}{B} \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

      5. metadata-eval N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) + \frac{F}{B} \cdot \frac{\color{blue}{1}}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

      6. un-div-inv N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) + \color{blue}{\frac{\frac{F}{B}}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

      7. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) + \color{blue}{\frac{\frac{F}{B}}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

      8. lower-sqrt.f64 47.1

         \[\leadsto \left(-\frac{x}{B}\right) + \frac{\frac{F}{B}}{\color{blue}{\sqrt{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

      9. lift-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) + \frac{\frac{F}{B}}{\sqrt{\color{blue}{2 \cdot x + \mathsf{fma}\left(F, F, 2\right)}}} \]

      10. +-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) + \frac{\frac{F}{B}}{\sqrt{\color{blue}{\mathsf{fma}\left(F, F, 2\right) + 2 \cdot x}}} \]

      11. lift-fma.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) + \frac{\frac{F}{B}}{\sqrt{\color{blue}{\left(F \cdot F + 2\right)} + 2 \cdot x}} \]

      12. associate-+r+ N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) + \frac{\frac{F}{B}}{\sqrt{\color{blue}{F \cdot F + \left(2 + 2 \cdot x\right)}}} \]

      13. +-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) + \frac{\frac{F}{B}}{\sqrt{F \cdot F + \color{blue}{\left(2 \cdot x + 2\right)}}} \]

      14. lift-fma.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{x}{B}\right)\right) + \frac{\frac{F}{B}}{\sqrt{F \cdot F + \color{blue}{\mathsf{fma}\left(2, x, 2\right)}}} \]

      15. lift-fma.f64 47.1

          \[\leadsto \left(-\frac{x}{B}\right) + \frac{\frac{F}{B}}{\sqrt{\color{blue}{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}} \]

   10. Applied egg-rr 47.1%

       \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{\frac{F}{B}}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}}} \]

   if 1.81999999999999994e-63 < F

   1. Initial program 65.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\color{blue}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(\color{blue}{F \cdot F} + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\color{blue}{\left(F \cdot F + 2\right)} + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + \color{blue}{2 \cdot x}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \]

      7. lift-neg.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      9. associate-*l/ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}} \]

   4. Applied egg-rr 78.7%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-0.5}}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) + \frac{F \cdot {\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B} \]
   6. Step-by-step derivation

      1. lower-/.f64 56.3

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-0.5}}{\sin B} \]

   7. Simplified 56.3%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-0.5}}{\sin B} \]

   8. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]
   9. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]

      2. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{\sin B}} - \frac{x}{B} \]

      3. lower-sin.f64 N/A

         \[\leadsto \frac{1}{\color{blue}{\sin B}} - \frac{x}{B} \]

      4. lower-/.f64 72.9

         \[\leadsto \frac{1}{\sin B} - \color{blue}{\frac{x}{B}} \]

   10. Simplified 72.9%

       \[\leadsto \color{blue}{\frac{1}{\sin B} - \frac{x}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.4 \cdot 10^{+25}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.82 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{F}{B}}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 1.3 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 1.3e-12)
   (/ (- (/ F (sqrt (fma F F (fma 2.0 x 2.0)))) x) B)
   (- (/ 1.0 B) (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 1.3e-12) {
		tmp = ((F / sqrt(fma(F, F, fma(2.0, x, 2.0)))) - x) / B;
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 1.3e-12)
		tmp = Float64(Float64(Float64(F / sqrt(fma(F, F, fma(2.0, x, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 1.3e-12], N[(N[(N[(F / N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.29999999999999991e-12

   1. Initial program 73.3%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B}} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      5. associate-+r+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + 2\right)} + {F}^{2}}} \]

      7. associate-+l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left(2 + {F}^{2}\right)}}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, 2 + {F}^{2}\right)}}} \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{{F}^{2} + 2}\right)}} \]

      10. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)}} \]

      11. lower-fma.f64 62.1

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}} \]

   5. Simplified 62.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 42.1

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   8. Simplified 42.1%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   10. Applied egg-rr 53.1%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - x}{B}} \]

   if 1.29999999999999991e-12 < B

   1. Initial program 83.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B}} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      5. associate-+r+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + 2\right)} + {F}^{2}}} \]

      7. associate-+l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left(2 + {F}^{2}\right)}}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, 2 + {F}^{2}\right)}}} \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{{F}^{2} + 2}\right)}} \]

      10. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)}} \]

      11. lower-fma.f64 57.2

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}} \]

   5. Simplified 57.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]
   6. Step-by-step derivation

      1. lift-tan.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\color{blue}{\tan B}}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

      2. remove-double-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\color{blue}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\tan B\right)\right)\right)}}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

      3. un-div-inv N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{\mathsf{neg}\left(\left(\mathsf{neg}\left(\tan B\right)\right)\right)}}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

      4. remove-double-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{x}{\color{blue}{\tan B}}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

      5. lower-/.f64 57.3

         \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   7. Applied egg-rr 57.3%

      \[\leadsto \left(-\color{blue}{\frac{x}{\tan B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   8. Taylor expanded in F around inf

      \[\leadsto \left(\mathsf{neg}\left(\frac{x}{\tan B}\right)\right) + \color{blue}{\frac{1}{B}} \]
   9. Step-by-step derivation

      1. lower-/.f64 49.4

         \[\leadsto \left(-\frac{x}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]

   10. Simplified 49.4%

       \[\leadsto \left(-\frac{x}{\tan B}\right) + \color{blue}{\frac{1}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 52.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.3 \cdot 10^{-12}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.4 \cdot 10^{+25}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(B, B \cdot 0.16666666666666666, 1\right)}{B} - \frac{\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(B \cdot B\right), x\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.4e+25)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 3.4e+144)
     (/ (- (/ F (sqrt (fma F F (fma 2.0 x 2.0)))) x) B)
     (-
      (/ (fma B (* B 0.16666666666666666) 1.0) B)
      (/ (fma -0.3333333333333333 (* x (* B B)) x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.4e+25) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 3.4e+144) {
		tmp = ((F / sqrt(fma(F, F, fma(2.0, x, 2.0)))) - x) / B;
	} else {
		tmp = (fma(B, (B * 0.16666666666666666), 1.0) / B) - (fma(-0.3333333333333333, (x * (B * B)), x) / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.4e+25)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 3.4e+144)
		tmp = Float64(Float64(Float64(F / sqrt(fma(F, F, fma(2.0, x, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(fma(B, Float64(B * 0.16666666666666666), 1.0) / B) - Float64(fma(-0.3333333333333333, Float64(x * Float64(B * B)), x) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.4e+25], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.4e+144], N[(N[(N[(F / N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(B * N[(B * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision] - N[(N[(-0.3333333333333333 * N[(x * N[(B * B), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.3999999999999998e25

   1. Initial program 51.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\color{blue}{\sin B}} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(\color{blue}{F \cdot F} + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      3. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\color{blue}{\left(F \cdot F + 2\right)} + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + \color{blue}{2 \cdot x}\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      5. lift-+.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\color{blue}{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\color{blue}{\frac{1}{2}}\right)\right)} \]

      7. lift-neg.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      8. lift-pow.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot \color{blue}{{\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}} \]

      9. associate-*l/ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}} \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}}{\sin B}} \]

   4. Applied egg-rr 65.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F \cdot {\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-0.5}}{\sin B}} \]

   5. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) + \frac{F \cdot {\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{\frac{-1}{2}}}{\sin B} \]
   6. Step-by-step derivation

      1. lower-/.f64 34.1

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-0.5}}{\sin B} \]

   7. Simplified 34.1%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F \cdot {\left(\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)\right)}^{-0.5}}{\sin B} \]

   8. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{\sin B} + \frac{x}{B}\right)} \]
   9. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{1}{\sin B} + -1 \cdot \frac{x}{B}} \]

      2. mul-1-neg N/A

         \[\leadsto -1 \cdot \frac{1}{\sin B} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{B}\right)\right)} \]

      3. unsub-neg N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{1}{\sin B} - \frac{x}{B}} \]

      4. neg-mul-1 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right)} - \frac{x}{B} \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{\sin B}\right)\right) - \frac{x}{B}} \]

      6. distribute-neg-frac N/A

         \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{\sin B}} - \frac{x}{B} \]

      7. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{-1}}{\sin B} - \frac{x}{B} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{\sin B}} - \frac{x}{B} \]

      9. lower-sin.f64 N/A

         \[\leadsto \frac{-1}{\color{blue}{\sin B}} - \frac{x}{B} \]

      10. lower-/.f64 67.9

          \[\leadsto \frac{-1}{\sin B} - \color{blue}{\frac{x}{B}} \]

   10. Simplified 67.9%

       \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   if -7.3999999999999998e25 < F < 3.3999999999999999e144

   1. Initial program 97.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B}} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      5. associate-+r+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + 2\right)} + {F}^{2}}} \]

      7. associate-+l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left(2 + {F}^{2}\right)}}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, 2 + {F}^{2}\right)}}} \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{{F}^{2} + 2}\right)}} \]

      10. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)}} \]

      11. lower-fma.f64 73.9

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}} \]

   5. Simplified 73.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 43.5

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   8. Simplified 43.5%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   10. Applied egg-rr 45.6%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - x}{B}} \]

   if 3.3999999999999999e144 < F

   1. Initial program 35.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{F}} \]
   4. Step-by-step derivation

      1. lower-/.f64 66.5

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{F}} \]

   5. Simplified 66.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{F}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{1 + \frac{1}{6} \cdot {B}^{2}}{B}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{1 + \frac{1}{6} \cdot {B}^{2}}{B}} \]

      2. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{\color{blue}{\frac{1}{6} \cdot {B}^{2} + 1}}{B} \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{\color{blue}{{B}^{2} \cdot \frac{1}{6}} + 1}{B} \]

      4. unpow2 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{\color{blue}{\left(B \cdot B\right)} \cdot \frac{1}{6} + 1}{B} \]

      5. associate-*l* N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{\color{blue}{B \cdot \left(B \cdot \frac{1}{6}\right)} + 1}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{\color{blue}{\mathsf{fma}\left(B, B \cdot \frac{1}{6}, 1\right)}}{B} \]

      7. lower-*.f64 66.5

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(B, \color{blue}{B \cdot 0.16666666666666666}, 1\right)}{B} \]

   8. Simplified 66.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\mathsf{fma}\left(B, B \cdot 0.16666666666666666, 1\right)}{B}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x + \frac{-1}{3} \cdot \left({B}^{2} \cdot x\right)}{B}}\right)\right) + \frac{\mathsf{fma}\left(B, B \cdot \frac{1}{6}, 1\right)}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x + \frac{-1}{3} \cdot \left({B}^{2} \cdot x\right)}{B}}\right)\right) + \frac{\mathsf{fma}\left(B, B \cdot \frac{1}{6}, 1\right)}{B} \]

       2. +-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{-1}{3} \cdot \left({B}^{2} \cdot x\right) + x}}{B}\right)\right) + \frac{\mathsf{fma}\left(B, B \cdot \frac{1}{6}, 1\right)}{B} \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{3}, {B}^{2} \cdot x, x\right)}}{B}\right)\right) + \frac{\mathsf{fma}\left(B, B \cdot \frac{1}{6}, 1\right)}{B} \]

       4. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{x \cdot {B}^{2}}, x\right)}{B}\right)\right) + \frac{\mathsf{fma}\left(B, B \cdot \frac{1}{6}, 1\right)}{B} \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{x \cdot {B}^{2}}, x\right)}{B}\right)\right) + \frac{\mathsf{fma}\left(B, B \cdot \frac{1}{6}, 1\right)}{B} \]

       6. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, x \cdot \color{blue}{\left(B \cdot B\right)}, x\right)}{B}\right)\right) + \frac{\mathsf{fma}\left(B, B \cdot \frac{1}{6}, 1\right)}{B} \]

       7. lower-*.f64 59.6

          \[\leadsto \left(-\frac{\mathsf{fma}\left(-0.3333333333333333, x \cdot \color{blue}{\left(B \cdot B\right)}, x\right)}{B}\right) + \frac{\mathsf{fma}\left(B, B \cdot 0.16666666666666666, 1\right)}{B} \]

   11. Simplified 59.6%

       \[\leadsto \left(-\color{blue}{\frac{\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(B \cdot B\right), x\right)}{B}}\right) + \frac{\mathsf{fma}\left(B, B \cdot 0.16666666666666666, 1\right)}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 53.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.4 \cdot 10^{+25}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(B, B \cdot 0.16666666666666666, 1\right)}{B} - \frac{\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(B \cdot B\right), x\right)}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 51.4% accurate, 5.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{+159}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -1\right) - x}{B}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(B, B \cdot 0.16666666666666666, 1\right)}{B} - \frac{\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(B \cdot B\right), x\right)}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.8e+159)
   (/ (- (fma 0.5 (/ (fma 2.0 x 2.0) (* F F)) -1.0) x) B)
   (if (<= F 3.4e+144)
     (/ (- (/ F (sqrt (fma F F (fma 2.0 x 2.0)))) x) B)
     (-
      (/ (fma B (* B 0.16666666666666666) 1.0) B)
      (/ (fma -0.3333333333333333 (* x (* B B)) x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.8e+159) {
		tmp = (fma(0.5, (fma(2.0, x, 2.0) / (F * F)), -1.0) - x) / B;
	} else if (F <= 3.4e+144) {
		tmp = ((F / sqrt(fma(F, F, fma(2.0, x, 2.0)))) - x) / B;
	} else {
		tmp = (fma(B, (B * 0.16666666666666666), 1.0) / B) - (fma(-0.3333333333333333, (x * (B * B)), x) / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.8e+159)
		tmp = Float64(Float64(fma(0.5, Float64(fma(2.0, x, 2.0) / Float64(F * F)), -1.0) - x) / B);
	elseif (F <= 3.4e+144)
		tmp = Float64(Float64(Float64(F / sqrt(fma(F, F, fma(2.0, x, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(fma(B, Float64(B * 0.16666666666666666), 1.0) / B) - Float64(fma(-0.3333333333333333, Float64(x * Float64(B * B)), x) / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.8e+159], N[(N[(N[(0.5 * N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 3.4e+144], N[(N[(N[(F / N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(N[(B * N[(B * 0.16666666666666666), $MachinePrecision] + 1.0), $MachinePrecision] / B), $MachinePrecision] - N[(N[(-0.3333333333333333 * N[(x * N[(B * B), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.79999999999999965e159

   1. Initial program 35.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B}} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      5. associate-+r+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + 2\right)} + {F}^{2}}} \]

      7. associate-+l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left(2 + {F}^{2}\right)}}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, 2 + {F}^{2}\right)}}} \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{{F}^{2} + 2}\right)}} \]

      10. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)}} \]

      11. lower-fma.f64 35.4

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}} \]

   5. Simplified 35.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 6.9

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   8. Simplified 6.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   9. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{B \cdot {F}^{2}} - \left(\frac{1}{B} + \frac{x}{B}\right)} \]
   10. Step-by-step derivation

       1. associate--r+ N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{B \cdot {F}^{2}} - \frac{1}{B}\right) - \frac{x}{B}} \]

       2. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{B \cdot {F}^{2}} - \frac{1}{B}\right) - \frac{x}{B}} \]

       3. sub-neg N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{B \cdot {F}^{2}} + \left(\mathsf{neg}\left(\frac{1}{B}\right)\right)\right)} - \frac{x}{B} \]

       4. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{2 + 2 \cdot x}{B \cdot {F}^{2}}, \mathsf{neg}\left(\frac{1}{B}\right)\right)} - \frac{x}{B} \]

       5. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{2 + 2 \cdot x}{B \cdot {F}^{2}}}, \mathsf{neg}\left(\frac{1}{B}\right)\right) - \frac{x}{B} \]

       6. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{2 \cdot x + 2}}{B \cdot {F}^{2}}, \mathsf{neg}\left(\frac{1}{B}\right)\right) - \frac{x}{B} \]

       7. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{\mathsf{fma}\left(2, x, 2\right)}}{B \cdot {F}^{2}}, \mathsf{neg}\left(\frac{1}{B}\right)\right) - \frac{x}{B} \]

       8. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{{F}^{2} \cdot B}}, \mathsf{neg}\left(\frac{1}{B}\right)\right) - \frac{x}{B} \]

       9. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{{F}^{2} \cdot B}}, \mathsf{neg}\left(\frac{1}{B}\right)\right) - \frac{x}{B} \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{\left(F \cdot F\right)} \cdot B}, \mathsf{neg}\left(\frac{1}{B}\right)\right) - \frac{x}{B} \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{\left(F \cdot F\right)} \cdot B}, \mathsf{neg}\left(\frac{1}{B}\right)\right) - \frac{x}{B} \]

       12. distribute-neg-frac N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\left(F \cdot F\right) \cdot B}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{B}}\right) - \frac{x}{B} \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\left(F \cdot F\right) \cdot B}, \frac{\color{blue}{-1}}{B}\right) - \frac{x}{B} \]

       14. lower-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\left(F \cdot F\right) \cdot B}, \color{blue}{\frac{-1}{B}}\right) - \frac{x}{B} \]

       15. lower-/.f64 43.1

           \[\leadsto \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{\left(F \cdot F\right) \cdot B}, \frac{-1}{B}\right) - \color{blue}{\frac{x}{B}} \]

   11. Simplified 43.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{\left(F \cdot F\right) \cdot B}, \frac{-1}{B}\right) - \frac{x}{B}} \]

   12. Taylor expanded in F around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{B \cdot {F}^{2}} - \left(\frac{1}{B} + \frac{x}{B}\right)} \]
   13. Step-by-step derivation

       1. associate--r+ N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{B \cdot {F}^{2}} - \frac{1}{B}\right) - \frac{x}{B}} \]

       2. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{\color{blue}{{F}^{2} \cdot B}} - \frac{1}{B}\right) - \frac{x}{B} \]

       3. associate-/r* N/A

          \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\frac{\frac{2 + 2 \cdot x}{{F}^{2}}}{B}} - \frac{1}{B}\right) - \frac{x}{B} \]

       4. associate-/l* N/A

          \[\leadsto \left(\color{blue}{\frac{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}}{B}} - \frac{1}{B}\right) - \frac{x}{B} \]

       5. div-sub N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - 1}{B}} - \frac{x}{B} \]

       6. div-sub N/A

          \[\leadsto \color{blue}{\frac{\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - 1\right) - x}{B}} \]

       7. associate--r+ N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - \left(1 + x\right)}}{B} \]

       8. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - \left(1 + x\right)}{B}} \]

   14. Simplified 43.2%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -1\right) - x}{B}} \]

   if -3.79999999999999965e159 < F < 3.3999999999999999e144

   1. Initial program 95.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B}} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      5. associate-+r+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + 2\right)} + {F}^{2}}} \]

      7. associate-+l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left(2 + {F}^{2}\right)}}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, 2 + {F}^{2}\right)}}} \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{{F}^{2} + 2}\right)}} \]

      10. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)}} \]

      11. lower-fma.f64 73.1

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}} \]

   5. Simplified 73.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 42.5

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   8. Simplified 42.5%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   10. Applied egg-rr 46.4%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - x}{B}} \]

   if 3.3999999999999999e144 < F

   1. Initial program 35.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around inf

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{F}} \]
   4. Step-by-step derivation

      1. lower-/.f64 66.5

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{F}} \]

   5. Simplified 66.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot \color{blue}{\frac{1}{F}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{1 + \frac{1}{6} \cdot {B}^{2}}{B}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{1 + \frac{1}{6} \cdot {B}^{2}}{B}} \]

      2. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{\color{blue}{\frac{1}{6} \cdot {B}^{2} + 1}}{B} \]

      3. *-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{\color{blue}{{B}^{2} \cdot \frac{1}{6}} + 1}{B} \]

      4. unpow2 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{\color{blue}{\left(B \cdot B\right)} \cdot \frac{1}{6} + 1}{B} \]

      5. associate-*l* N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{\color{blue}{B \cdot \left(B \cdot \frac{1}{6}\right)} + 1}{B} \]

      6. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{\color{blue}{\mathsf{fma}\left(B, B \cdot \frac{1}{6}, 1\right)}}{B} \]

      7. lower-*.f64 66.5

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{\mathsf{fma}\left(B, \color{blue}{B \cdot 0.16666666666666666}, 1\right)}{B} \]

   8. Simplified 66.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{\mathsf{fma}\left(B, B \cdot 0.16666666666666666, 1\right)}{B}} \]

   9. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x + \frac{-1}{3} \cdot \left({B}^{2} \cdot x\right)}{B}}\right)\right) + \frac{\mathsf{fma}\left(B, B \cdot \frac{1}{6}, 1\right)}{B} \]
   10. Step-by-step derivation

       1. lower-/.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x + \frac{-1}{3} \cdot \left({B}^{2} \cdot x\right)}{B}}\right)\right) + \frac{\mathsf{fma}\left(B, B \cdot \frac{1}{6}, 1\right)}{B} \]

       2. +-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\frac{-1}{3} \cdot \left({B}^{2} \cdot x\right) + x}}{B}\right)\right) + \frac{\mathsf{fma}\left(B, B \cdot \frac{1}{6}, 1\right)}{B} \]

       3. lower-fma.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{\color{blue}{\mathsf{fma}\left(\frac{-1}{3}, {B}^{2} \cdot x, x\right)}}{B}\right)\right) + \frac{\mathsf{fma}\left(B, B \cdot \frac{1}{6}, 1\right)}{B} \]

       4. *-commutative N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{x \cdot {B}^{2}}, x\right)}{B}\right)\right) + \frac{\mathsf{fma}\left(B, B \cdot \frac{1}{6}, 1\right)}{B} \]

       5. lower-*.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, \color{blue}{x \cdot {B}^{2}}, x\right)}{B}\right)\right) + \frac{\mathsf{fma}\left(B, B \cdot \frac{1}{6}, 1\right)}{B} \]

       6. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(\frac{\mathsf{fma}\left(\frac{-1}{3}, x \cdot \color{blue}{\left(B \cdot B\right)}, x\right)}{B}\right)\right) + \frac{\mathsf{fma}\left(B, B \cdot \frac{1}{6}, 1\right)}{B} \]

       7. lower-*.f64 59.6

          \[\leadsto \left(-\frac{\mathsf{fma}\left(-0.3333333333333333, x \cdot \color{blue}{\left(B \cdot B\right)}, x\right)}{B}\right) + \frac{\mathsf{fma}\left(B, B \cdot 0.16666666666666666, 1\right)}{B} \]

   11. Simplified 59.6%

       \[\leadsto \left(-\color{blue}{\frac{\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(B \cdot B\right), x\right)}{B}}\right) + \frac{\mathsf{fma}\left(B, B \cdot 0.16666666666666666, 1\right)}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 48.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{+159}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -1\right) - x}{B}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{+144}:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(B, B \cdot 0.16666666666666666, 1\right)}{B} - \frac{\mathsf{fma}\left(-0.3333333333333333, x \cdot \left(B \cdot B\right), x\right)}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 51.4% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{+159}:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -1\right) - x}{B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.8e+159)
   (/ (- (fma 0.5 (/ (fma 2.0 x 2.0) (* F F)) -1.0) x) B)
   (if (<= F 100000000.0)
     (/ (- (/ F (sqrt (fma F F (fma 2.0 x 2.0)))) x) B)
     (- (/ 1.0 B) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.8e+159) {
		tmp = (fma(0.5, (fma(2.0, x, 2.0) / (F * F)), -1.0) - x) / B;
	} else if (F <= 100000000.0) {
		tmp = ((F / sqrt(fma(F, F, fma(2.0, x, 2.0)))) - x) / B;
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.8e+159)
		tmp = Float64(Float64(fma(0.5, Float64(fma(2.0, x, 2.0) / Float64(F * F)), -1.0) - x) / B);
	elseif (F <= 100000000.0)
		tmp = Float64(Float64(Float64(F / sqrt(fma(F, F, fma(2.0, x, 2.0)))) - x) / B);
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.8e+159], N[(N[(N[(0.5 * N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 100000000.0], N[(N[(N[(F / N[Sqrt[N[(F * F + N[(2.0 * x + 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.79999999999999965e159

   1. Initial program 35.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B}} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      5. associate-+r+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + 2\right)} + {F}^{2}}} \]

      7. associate-+l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left(2 + {F}^{2}\right)}}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, 2 + {F}^{2}\right)}}} \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{{F}^{2} + 2}\right)}} \]

      10. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)}} \]

      11. lower-fma.f64 35.4

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}} \]

   5. Simplified 35.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 6.9

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   8. Simplified 6.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   9. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{B \cdot {F}^{2}} - \left(\frac{1}{B} + \frac{x}{B}\right)} \]
   10. Step-by-step derivation

       1. associate--r+ N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{B \cdot {F}^{2}} - \frac{1}{B}\right) - \frac{x}{B}} \]

       2. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{B \cdot {F}^{2}} - \frac{1}{B}\right) - \frac{x}{B}} \]

       3. sub-neg N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{B \cdot {F}^{2}} + \left(\mathsf{neg}\left(\frac{1}{B}\right)\right)\right)} - \frac{x}{B} \]

       4. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{2 + 2 \cdot x}{B \cdot {F}^{2}}, \mathsf{neg}\left(\frac{1}{B}\right)\right)} - \frac{x}{B} \]

       5. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{2 + 2 \cdot x}{B \cdot {F}^{2}}}, \mathsf{neg}\left(\frac{1}{B}\right)\right) - \frac{x}{B} \]

       6. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{2 \cdot x + 2}}{B \cdot {F}^{2}}, \mathsf{neg}\left(\frac{1}{B}\right)\right) - \frac{x}{B} \]

       7. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{\mathsf{fma}\left(2, x, 2\right)}}{B \cdot {F}^{2}}, \mathsf{neg}\left(\frac{1}{B}\right)\right) - \frac{x}{B} \]

       8. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{{F}^{2} \cdot B}}, \mathsf{neg}\left(\frac{1}{B}\right)\right) - \frac{x}{B} \]

       9. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{{F}^{2} \cdot B}}, \mathsf{neg}\left(\frac{1}{B}\right)\right) - \frac{x}{B} \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{\left(F \cdot F\right)} \cdot B}, \mathsf{neg}\left(\frac{1}{B}\right)\right) - \frac{x}{B} \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{\left(F \cdot F\right)} \cdot B}, \mathsf{neg}\left(\frac{1}{B}\right)\right) - \frac{x}{B} \]

       12. distribute-neg-frac N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\left(F \cdot F\right) \cdot B}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{B}}\right) - \frac{x}{B} \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\left(F \cdot F\right) \cdot B}, \frac{\color{blue}{-1}}{B}\right) - \frac{x}{B} \]

       14. lower-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\left(F \cdot F\right) \cdot B}, \color{blue}{\frac{-1}{B}}\right) - \frac{x}{B} \]

       15. lower-/.f64 43.1

           \[\leadsto \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{\left(F \cdot F\right) \cdot B}, \frac{-1}{B}\right) - \color{blue}{\frac{x}{B}} \]

   11. Simplified 43.1%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{\left(F \cdot F\right) \cdot B}, \frac{-1}{B}\right) - \frac{x}{B}} \]

   12. Taylor expanded in F around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{B \cdot {F}^{2}} - \left(\frac{1}{B} + \frac{x}{B}\right)} \]
   13. Step-by-step derivation

       1. associate--r+ N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{B \cdot {F}^{2}} - \frac{1}{B}\right) - \frac{x}{B}} \]

       2. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{\color{blue}{{F}^{2} \cdot B}} - \frac{1}{B}\right) - \frac{x}{B} \]

       3. associate-/r* N/A

          \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\frac{\frac{2 + 2 \cdot x}{{F}^{2}}}{B}} - \frac{1}{B}\right) - \frac{x}{B} \]

       4. associate-/l* N/A

          \[\leadsto \left(\color{blue}{\frac{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}}{B}} - \frac{1}{B}\right) - \frac{x}{B} \]

       5. div-sub N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - 1}{B}} - \frac{x}{B} \]

       6. div-sub N/A

          \[\leadsto \color{blue}{\frac{\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - 1\right) - x}{B}} \]

       7. associate--r+ N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - \left(1 + x\right)}}{B} \]

       8. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - \left(1 + x\right)}{B}} \]

   14. Simplified 43.2%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -1\right) - x}{B}} \]

   if -3.79999999999999965e159 < F < 1e8

   1. Initial program 96.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B}} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      5. associate-+r+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + 2\right)} + {F}^{2}}} \]

      7. associate-+l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left(2 + {F}^{2}\right)}}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, 2 + {F}^{2}\right)}}} \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{{F}^{2} + 2}\right)}} \]

      10. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)}} \]

      11. lower-fma.f64 76.9

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}} \]

   5. Simplified 76.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 43.6

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   8. Simplified 43.6%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   10. Applied egg-rr 46.4%

       \[\leadsto \color{blue}{\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, \mathsf{fma}\left(2, x, 2\right)\right)}} - x}{B}} \]

   if 1e8 < F

   1. Initial program 61.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B}} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      5. associate-+r+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + 2\right)} + {F}^{2}}} \]

      7. associate-+l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left(2 + {F}^{2}\right)}}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, 2 + {F}^{2}\right)}}} \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{{F}^{2} + 2}\right)}} \]

      10. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)}} \]

      11. lower-fma.f64 45.2

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}} \]

   5. Simplified 45.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 24.2

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   8. Simplified 24.2%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   9. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{B} - \frac{x}{B}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{B} - \frac{x}{B}} \]

       2. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{B} \]

       3. lower-/.f64 53.2

          \[\leadsto \frac{1}{B} - \color{blue}{\frac{x}{B}} \]

   11. Simplified 53.2%

       \[\leadsto \color{blue}{\frac{1}{B} - \frac{x}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 44.4% accurate, 7.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.115:\\ \;\;\;\;\frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -1\right) - x}{B}\\ \mathbf{elif}\;F \leq 7.4 \cdot 10^{-64}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.115)
   (/ (- (fma 0.5 (/ (fma 2.0 x 2.0) (* F F)) -1.0) x) B)
   (if (<= F 7.4e-64) (- (/ x B)) (- (/ 1.0 B) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.115) {
		tmp = (fma(0.5, (fma(2.0, x, 2.0) / (F * F)), -1.0) - x) / B;
	} else if (F <= 7.4e-64) {
		tmp = -(x / B);
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	return tmp;
}

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.115)
		tmp = Float64(Float64(fma(0.5, Float64(fma(2.0, x, 2.0) / Float64(F * F)), -1.0) - x) / B);
	elseif (F <= 7.4e-64)
		tmp = Float64(-Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.115], N[(N[(N[(0.5 * N[(N[(2.0 * x + 2.0), $MachinePrecision] / N[(F * F), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 7.4e-64], (-N[(x / B), $MachinePrecision]), N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.115000000000000005

   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. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B}} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      5. associate-+r+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + 2\right)} + {F}^{2}}} \]

      7. associate-+l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left(2 + {F}^{2}\right)}}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, 2 + {F}^{2}\right)}}} \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{{F}^{2} + 2}\right)}} \]

      10. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)}} \]

      11. lower-fma.f64 49.8

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}} \]

   5. Simplified 49.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 19.5

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   8. Simplified 19.5%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   9. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{B \cdot {F}^{2}} - \left(\frac{1}{B} + \frac{x}{B}\right)} \]
   10. Step-by-step derivation

       1. associate--r+ N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{B \cdot {F}^{2}} - \frac{1}{B}\right) - \frac{x}{B}} \]

       2. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{B \cdot {F}^{2}} - \frac{1}{B}\right) - \frac{x}{B}} \]

       3. sub-neg N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{B \cdot {F}^{2}} + \left(\mathsf{neg}\left(\frac{1}{B}\right)\right)\right)} - \frac{x}{B} \]

       4. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{2}, \frac{2 + 2 \cdot x}{B \cdot {F}^{2}}, \mathsf{neg}\left(\frac{1}{B}\right)\right)} - \frac{x}{B} \]

       5. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \color{blue}{\frac{2 + 2 \cdot x}{B \cdot {F}^{2}}}, \mathsf{neg}\left(\frac{1}{B}\right)\right) - \frac{x}{B} \]

       6. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{2 \cdot x + 2}}{B \cdot {F}^{2}}, \mathsf{neg}\left(\frac{1}{B}\right)\right) - \frac{x}{B} \]

       7. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\color{blue}{\mathsf{fma}\left(2, x, 2\right)}}{B \cdot {F}^{2}}, \mathsf{neg}\left(\frac{1}{B}\right)\right) - \frac{x}{B} \]

       8. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{{F}^{2} \cdot B}}, \mathsf{neg}\left(\frac{1}{B}\right)\right) - \frac{x}{B} \]

       9. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{{F}^{2} \cdot B}}, \mathsf{neg}\left(\frac{1}{B}\right)\right) - \frac{x}{B} \]

       10. unpow2 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{\left(F \cdot F\right)} \cdot B}, \mathsf{neg}\left(\frac{1}{B}\right)\right) - \frac{x}{B} \]

       11. lower-*.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\color{blue}{\left(F \cdot F\right)} \cdot B}, \mathsf{neg}\left(\frac{1}{B}\right)\right) - \frac{x}{B} \]

       12. distribute-neg-frac N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\left(F \cdot F\right) \cdot B}, \color{blue}{\frac{\mathsf{neg}\left(1\right)}{B}}\right) - \frac{x}{B} \]

       13. metadata-eval N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\left(F \cdot F\right) \cdot B}, \frac{\color{blue}{-1}}{B}\right) - \frac{x}{B} \]

       14. lower-/.f64 N/A

           \[\leadsto \mathsf{fma}\left(\frac{1}{2}, \frac{\mathsf{fma}\left(2, x, 2\right)}{\left(F \cdot F\right) \cdot B}, \color{blue}{\frac{-1}{B}}\right) - \frac{x}{B} \]

       15. lower-/.f64 46.2

           \[\leadsto \mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{\left(F \cdot F\right) \cdot B}, \frac{-1}{B}\right) - \color{blue}{\frac{x}{B}} \]

   11. Simplified 46.2%

       \[\leadsto \color{blue}{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{\left(F \cdot F\right) \cdot B}, \frac{-1}{B}\right) - \frac{x}{B}} \]

   12. Taylor expanded in F around inf

       \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{B \cdot {F}^{2}} - \left(\frac{1}{B} + \frac{x}{B}\right)} \]
   13. Step-by-step derivation

       1. associate--r+ N/A

          \[\leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{B \cdot {F}^{2}} - \frac{1}{B}\right) - \frac{x}{B}} \]

       2. *-commutative N/A

          \[\leadsto \left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{\color{blue}{{F}^{2} \cdot B}} - \frac{1}{B}\right) - \frac{x}{B} \]

       3. associate-/r* N/A

          \[\leadsto \left(\frac{1}{2} \cdot \color{blue}{\frac{\frac{2 + 2 \cdot x}{{F}^{2}}}{B}} - \frac{1}{B}\right) - \frac{x}{B} \]

       4. associate-/l* N/A

          \[\leadsto \left(\color{blue}{\frac{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}}}{B}} - \frac{1}{B}\right) - \frac{x}{B} \]

       5. div-sub N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - 1}{B}} - \frac{x}{B} \]

       6. div-sub N/A

          \[\leadsto \color{blue}{\frac{\left(\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - 1\right) - x}{B}} \]

       7. associate--r+ N/A

          \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - \left(1 + x\right)}}{B} \]

       8. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{2 + 2 \cdot x}{{F}^{2}} - \left(1 + x\right)}{B}} \]

   14. Simplified 46.2%

       \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(0.5, \frac{\mathsf{fma}\left(2, x, 2\right)}{F \cdot F}, -1\right) - x}{B}} \]

   if -0.115000000000000005 < F < 7.39999999999999998e-64

   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

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B}} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      5. associate-+r+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + 2\right)} + {F}^{2}}} \]

      7. associate-+l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left(2 + {F}^{2}\right)}}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, 2 + {F}^{2}\right)}}} \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{{F}^{2} + 2}\right)}} \]

      10. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)}} \]

      11. lower-fma.f64 80.8

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}} \]

   5. Simplified 80.8%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 46.9

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   8. Simplified 46.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x}{B}\right)} \]

       2. lower-neg.f64 N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x}{B}\right)} \]

       3. lower-/.f64 30.0

          \[\leadsto -\color{blue}{\frac{x}{B}} \]

   11. Simplified 30.0%

       \[\leadsto \color{blue}{-\frac{x}{B}} \]

   if 7.39999999999999998e-64 < F

   1. Initial program 65.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B}} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      5. associate-+r+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + 2\right)} + {F}^{2}}} \]

      7. associate-+l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left(2 + {F}^{2}\right)}}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, 2 + {F}^{2}\right)}}} \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{{F}^{2} + 2}\right)}} \]

      10. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)}} \]

      11. lower-fma.f64 45.2

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}} \]

   5. Simplified 45.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 23.7

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   8. Simplified 23.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   9. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{B} - \frac{x}{B}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{B} - \frac{x}{B}} \]

       2. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{B} \]

       3. lower-/.f64 50.0

          \[\leadsto \frac{1}{B} - \color{blue}{\frac{x}{B}} \]

   11. Simplified 50.0%

       \[\leadsto \color{blue}{\frac{1}{B} - \frac{x}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 43.1% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.6 \cdot 10^{-173}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7.4 \cdot 10^{-64}:\\ \;\;\;\;-\frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.6e-173)
   (- (/ -1.0 B) (/ x B))
   (if (<= F 7.4e-64) (- (/ x B)) (- (/ 1.0 B) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.6e-173) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 7.4e-64) {
		tmp = -(x / B);
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	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 <= (-4.6d-173)) then
        tmp = ((-1.0d0) / b) - (x / b)
    else if (f <= 7.4d-64) then
        tmp = -(x / b)
    else
        tmp = (1.0d0 / b) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.6e-173) {
		tmp = (-1.0 / B) - (x / B);
	} else if (F <= 7.4e-64) {
		tmp = -(x / B);
	} else {
		tmp = (1.0 / B) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.6e-173:
		tmp = (-1.0 / B) - (x / B)
	elif F <= 7.4e-64:
		tmp = -(x / B)
	else:
		tmp = (1.0 / B) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.6e-173)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / B));
	elseif (F <= 7.4e-64)
		tmp = Float64(-Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -4.6e-173)
		tmp = (-1.0 / B) - (x / B);
	elseif (F <= 7.4e-64)
		tmp = -(x / B);
	else
		tmp = (1.0 / B) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.6e-173], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.4e-64], (-N[(x / B), $MachinePrecision]), N[(N[(1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.59999999999999976e-173

   1. Initial program 69.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B}} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      5. associate-+r+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + 2\right)} + {F}^{2}}} \]

      7. associate-+l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left(2 + {F}^{2}\right)}}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, 2 + {F}^{2}\right)}}} \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{{F}^{2} + 2}\right)}} \]

      10. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)}} \]

      11. lower-fma.f64 55.9

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}} \]

   5. Simplified 55.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 23.1

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   8. Simplified 23.1%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   9. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{B} + \frac{x}{B}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{1}{B} + -1 \cdot \frac{x}{B}} \]

       2. mul-1-neg N/A

          \[\leadsto -1 \cdot \frac{1}{B} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{B}\right)\right)} \]

       3. unsub-neg N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{1}{B} - \frac{x}{B}} \]

       4. neg-mul-1 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{B}\right)\right)} - \frac{x}{B} \]

       5. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{B}\right)\right) - \frac{x}{B}} \]

       6. distribute-neg-frac N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{B}} - \frac{x}{B} \]

       7. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{-1}}{B} - \frac{x}{B} \]

       8. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{B}} - \frac{x}{B} \]

       9. lower-/.f64 36.0

          \[\leadsto \frac{-1}{B} - \color{blue}{\frac{x}{B}} \]

   11. Simplified 36.0%

       \[\leadsto \color{blue}{\frac{-1}{B} - \frac{x}{B}} \]

   if -4.59999999999999976e-173 < F < 7.39999999999999998e-64

   1. Initial program 99.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B}} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      5. associate-+r+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + 2\right)} + {F}^{2}}} \]

      7. associate-+l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left(2 + {F}^{2}\right)}}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, 2 + {F}^{2}\right)}}} \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{{F}^{2} + 2}\right)}} \]

      10. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)}} \]

      11. lower-fma.f64 87.9

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}} \]

   5. Simplified 87.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 55.9

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   8. Simplified 55.9%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x}{B}\right)} \]

       2. lower-neg.f64 N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x}{B}\right)} \]

       3. lower-/.f64 37.0

          \[\leadsto -\color{blue}{\frac{x}{B}} \]

   11. Simplified 37.0%

       \[\leadsto \color{blue}{-\frac{x}{B}} \]

   if 7.39999999999999998e-64 < F

   1. Initial program 65.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B}} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      5. associate-+r+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + 2\right)} + {F}^{2}}} \]

      7. associate-+l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left(2 + {F}^{2}\right)}}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, 2 + {F}^{2}\right)}}} \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{{F}^{2} + 2}\right)}} \]

      10. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)}} \]

      11. lower-fma.f64 45.2

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}} \]

   5. Simplified 45.2%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 23.7

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   8. Simplified 23.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   9. Taylor expanded in F around inf

      \[\leadsto \color{blue}{\frac{1}{B} - \frac{x}{B}} \]
   10. Step-by-step derivation

       1. lower--.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{B} - \frac{x}{B}} \]

       2. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{B} \]

       3. lower-/.f64 50.0

          \[\leadsto \frac{1}{B} - \color{blue}{\frac{x}{B}} \]

   11. Simplified 50.0%

       \[\leadsto \color{blue}{\frac{1}{B} - \frac{x}{B}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 19: 35.9% accurate, 11.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.6 \cdot 10^{-173}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.6e-173) (- (/ -1.0 B) (/ x B)) (- (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.6e-173) {
		tmp = (-1.0 / B) - (x / B);
	} else {
		tmp = -(x / B);
	}
	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 <= (-4.6d-173)) then
        tmp = ((-1.0d0) / b) - (x / b)
    else
        tmp = -(x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.6e-173) {
		tmp = (-1.0 / B) - (x / B);
	} else {
		tmp = -(x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.6e-173:
		tmp = (-1.0 / B) - (x / B)
	else:
		tmp = -(x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.6e-173)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / B));
	else
		tmp = Float64(-Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -4.6e-173)
		tmp = (-1.0 / B) - (x / B);
	else
		tmp = -(x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.6e-173], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], (-N[(x / B), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if F < -4.59999999999999976e-173

   1. Initial program 69.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B}} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      5. associate-+r+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + 2\right)} + {F}^{2}}} \]

      7. associate-+l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left(2 + {F}^{2}\right)}}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, 2 + {F}^{2}\right)}}} \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{{F}^{2} + 2}\right)}} \]

      10. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)}} \]

      11. lower-fma.f64 55.9

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}} \]

   5. Simplified 55.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 23.1

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   8. Simplified 23.1%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   9. Taylor expanded in F around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{1}{B} + \frac{x}{B}\right)} \]
   10. Step-by-step derivation

       1. distribute-lft-in N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{1}{B} + -1 \cdot \frac{x}{B}} \]

       2. mul-1-neg N/A

          \[\leadsto -1 \cdot \frac{1}{B} + \color{blue}{\left(\mathsf{neg}\left(\frac{x}{B}\right)\right)} \]

       3. unsub-neg N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{1}{B} - \frac{x}{B}} \]

       4. neg-mul-1 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{B}\right)\right)} - \frac{x}{B} \]

       5. lower--.f64 N/A

          \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{1}{B}\right)\right) - \frac{x}{B}} \]

       6. distribute-neg-frac N/A

          \[\leadsto \color{blue}{\frac{\mathsf{neg}\left(1\right)}{B}} - \frac{x}{B} \]

       7. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{-1}}{B} - \frac{x}{B} \]

       8. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{-1}{B}} - \frac{x}{B} \]

       9. lower-/.f64 36.0

          \[\leadsto \frac{-1}{B} - \color{blue}{\frac{x}{B}} \]

   11. Simplified 36.0%

       \[\leadsto \color{blue}{\frac{-1}{B} - \frac{x}{B}} \]

   if -4.59999999999999976e-173 < F

   1. Initial program 80.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      2. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B}} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

      5. associate-+r+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \]

      6. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + 2\right)} + {F}^{2}}} \]

      7. associate-+l+ N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left(2 + {F}^{2}\right)}}} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, 2 + {F}^{2}\right)}}} \]

      9. +-commutative N/A

         \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{{F}^{2} + 2}\right)}} \]

      10. unpow2 N/A

          \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)}} \]

      11. lower-fma.f64 64.1

          \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}} \]

   5. Simplified 64.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

   6. Taylor expanded in B around 0

      \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]
   7. Step-by-step derivation

      1. lower-/.f64 38.0

         \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   8. Simplified 38.0%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

   9. Taylor expanded in x around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x}{B}\right)} \]

       2. lower-neg.f64 N/A

          \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x}{B}\right)} \]

       3. lower-/.f64 30.2

          \[\leadsto -\color{blue}{\frac{x}{B}} \]

   11. Simplified 30.2%

       \[\leadsto \color{blue}{-\frac{x}{B}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 20: 29.8% accurate, 26.3× speedup

Math

\[\begin{array}{l} \\ -\frac{x}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (- (/ x B)))

C

double code(double F, double B, double x) {
	return -(x / B);
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = -(x / b)
end function

Java

public static double code(double F, double B, double x) {
	return -(x / B);
}

Python

def code(F, B, x):
	return -(x / B)

Julia

function code(F, B, x)
	return Float64(-Float64(x / B))
end

MATLAB

function tmp = code(F, B, x)
	tmp = -(x / B);
end

Wolfram

code[F_, B_, x_] := (-N[(x / B), $MachinePrecision])

Derivation

1. Initial program 76.1%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
2. Add Preprocessing

3. Taylor expanded in B around 0

   \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]
4. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

   2. lower-/.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \color{blue}{\frac{F}{B}} \cdot \sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \color{blue}{\sqrt{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

   4. lower-/.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\color{blue}{\frac{1}{2 + \left(2 \cdot x + {F}^{2}\right)}}} \]

   5. associate-+r+ N/A

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 + 2 \cdot x\right) + {F}^{2}}}} \]

   6. +-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\left(2 \cdot x + 2\right)} + {F}^{2}}} \]

   7. associate-+l+ N/A

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{2 \cdot x + \left(2 + {F}^{2}\right)}}} \]

   8. lower-fma.f64 N/A

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(2, x, 2 + {F}^{2}\right)}}} \]

   9. +-commutative N/A

      \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{{F}^{2} + 2}\right)}} \]

   10. unpow2 N/A

       \[\leadsto \left(\mathsf{neg}\left(x \cdot \frac{1}{\tan B}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{F \cdot F} + 2\right)}} \]

   11. lower-fma.f64 60.7

       \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \color{blue}{\mathsf{fma}\left(F, F, 2\right)}\right)}} \]

5. Simplified 60.7%

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}}} \]

6. Taylor expanded in B around 0

   \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{x}{B}}\right)\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]
7. Step-by-step derivation

   1. lower-/.f64 31.9

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

8. Simplified 31.9%

   \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{F}{B} \cdot \sqrt{\frac{1}{\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)}} \]

9. Taylor expanded in x around inf

   \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
10. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x}{B}\right)} \]

    2. lower-neg.f64 N/A

       \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x}{B}\right)} \]

    3. lower-/.f64 26.1

       \[\leadsto -\color{blue}{\frac{x}{B}} \]

11. Simplified 26.1%

    \[\leadsto \color{blue}{-\frac{x}{B}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(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))))))