VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.1% → 99.7%

Time: 21.8s

Alternatives: 22

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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: 77.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4 \cdot 10^{+47}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 20000000:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -4e+47)
     (- (/ -1.0 (sin B)) (/ (* x (cos B)) (sin B)))
     (if (<= F 20000000.0)
       (- (/ (/ F (sqrt (fma F F 2.0))) (sin B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -4e+47) {
		tmp = (-1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	} else if (F <= 20000000.0) {
		tmp = ((F / sqrt(fma(F, F, 2.0))) / sin(B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -4e+47)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(Float64(x * cos(B)) / sin(B)));
	elseif (F <= 20000000.0)
		tmp = Float64(Float64(Float64(F / sqrt(fma(F, F, 2.0))) / sin(B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4e+47], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 20000000.0], N[(N[(N[(F / N[Sqrt[N[(F * F + 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.0000000000000002e47

   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. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in x around 0 99.8%

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

   if -4.0000000000000002e47 < F < 2e7

   1. Initial program 98.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

      1. associate-*r/ 99.5%

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

      2. sqrt-div 99.6%

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

      3. metadata-eval 99.6%

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

      4. un-div-inv 99.6%

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

   9. Applied egg-rr 99.6%

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

   if 2e7 < F

   1. Initial program 63.3%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in x around 0 80.6%

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

      1. associate-*l/ 80.6%

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

      2. *-lft-identity 80.6%

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

      3. +-commutative 80.6%

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

      4. unpow2 80.6%

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

      5. fma-undefine 80.6%

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

   7. Simplified 80.6%

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

   8. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{+47}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 20000000:\\ \;\;\;\;\frac{\frac{F}{\sqrt{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4.4e+15)
   (- (/ -1.0 (sin B)) (/ (* x (cos B)) (sin B)))
   (if (<= F 100000000.0)
     (+
      (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
      (* x (/ -1.0 (tan B))))
     (- (/ 1.0 (sin B)) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.4e+15) {
		tmp = (-1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	} else if (F <= 100000000.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / tan(B)));
	} else {
		tmp = (1.0 / sin(B)) - (x / tan(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 <= (-4.4d+15)) then
        tmp = ((-1.0d0) / sin(b)) - ((x * cos(b)) / sin(b))
    else if (f <= 100000000.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) + (x * ((-1.0d0) / tan(b)))
    else
        tmp = (1.0d0 / sin(b)) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -4.4e+15) {
		tmp = (-1.0 / Math.sin(B)) - ((x * Math.cos(B)) / Math.sin(B));
	} else if (F <= 100000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / Math.tan(B)));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4.4e+15:
		tmp = (-1.0 / math.sin(B)) - ((x * math.cos(B)) / math.sin(B))
	elif F <= 100000000.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / math.tan(B)))
	else:
		tmp = (1.0 / math.sin(B)) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4.4e+15)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(Float64(x * cos(B)) / sin(B)));
	elseif (F <= 100000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) + Float64(x * Float64(-1.0 / tan(B))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -4.4e+15)
		tmp = (-1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	elseif (F <= 100000000.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) + (x * (-1.0 / tan(B)));
	else
		tmp = (1.0 / sin(B)) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4.4e+15], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 100000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] + N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.4e15

   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. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in x around 0 99.8%

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

   if -4.4e15 < F < 1e8

   1. Initial program 99.4%

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

   if 1e8 < F

   1. Initial program 63.3%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in x around 0 80.6%

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

      1. associate-*l/ 80.6%

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

      2. *-lft-identity 80.6%

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

      3. +-commutative 80.6%

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

      4. unpow2 80.6%

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

      5. fma-undefine 80.6%

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

   7. Simplified 80.6%

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

   8. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.4 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{+14}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 20000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -4e+14)
   (- (/ -1.0 (sin B)) (/ (* x (cos B)) (sin B)))
   (if (<= F 20000000.0)
     (-
      (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
      (/ 1.0 (/ (tan B) x)))
     (- (/ 1.0 (sin B)) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -4e+14) {
		tmp = (-1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	} else if (F <= 20000000.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (1.0 / (tan(B) / x));
	} else {
		tmp = (1.0 / sin(B)) - (x / tan(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 <= (-4d+14)) then
        tmp = ((-1.0d0) / sin(b)) - ((x * cos(b)) / sin(b))
    else if (f <= 20000000.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (1.0d0 / (tan(b) / x))
    else
        tmp = (1.0d0 / sin(b)) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -4e+14) {
		tmp = (-1.0 / Math.sin(B)) - ((x * Math.cos(B)) / Math.sin(B));
	} else if (F <= 20000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (1.0 / (Math.tan(B) / x));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -4e+14:
		tmp = (-1.0 / math.sin(B)) - ((x * math.cos(B)) / math.sin(B))
	elif F <= 20000000.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (1.0 / (math.tan(B) / x))
	else:
		tmp = (1.0 / math.sin(B)) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -4e+14)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(Float64(x * cos(B)) / sin(B)));
	elseif (F <= 20000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(1.0 / Float64(tan(B) / x)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -4e+14)
		tmp = (-1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	elseif (F <= 20000000.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (1.0 / (tan(B) / x));
	else
		tmp = (1.0 / sin(B)) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -4e+14], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 20000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4e14

   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. Taylor expanded in F around -inf 99.7%

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

   4. Taylor expanded in x around 0 99.8%

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

   if -4e14 < F < 2e7

   1. Initial program 99.4%

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

      1. div-inv 99.5%

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

      2. clear-num 99.4%

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

   4. Applied egg-rr 99.4%

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

   if 2e7 < F

   1. Initial program 63.3%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in x around 0 80.6%

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

      1. associate-*l/ 80.6%

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

      2. *-lft-identity 80.6%

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

      3. +-commutative 80.6%

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

      4. unpow2 80.6%

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

      5. fma-undefine 80.6%

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

   7. Simplified 80.6%

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

   8. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{+14}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 20000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.42:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.42)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.42)
       (- (* F (/ (sqrt 0.5) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.42) {
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.42d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.42d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.42) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.42:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.42:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.42)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.42)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.42)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.42)
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.42], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.42], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4199999999999999

   1. Initial program 77.1%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in x around 0 84.3%

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

      1. associate-*l/ 84.3%

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

      2. *-lft-identity 84.3%

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

      3. +-commutative 84.3%

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

      4. unpow2 84.3%

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

      5. fma-undefine 84.3%

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

   7. Simplified 84.3%

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

   8. Taylor expanded in F around -inf 97.7%

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

   if -1.4199999999999999 < F < 1.4199999999999999

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in F around 0 98.3%

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

   if 1.4199999999999999 < F

   1. Initial program 63.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in x around 0 80.8%

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

      1. associate-*l/ 80.9%

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

      2. *-lft-identity 80.9%

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

      3. +-commutative 80.9%

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

      4. unpow2 80.9%

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

      5. fma-undefine 80.9%

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

   7. Simplified 80.9%

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

   8. Taylor expanded in F around inf 99.1%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.42:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.42) {
		tmp = ((F / sqrt(2.0)) / sin(B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.42d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.42d0) then
        tmp = ((f / sqrt(2.0d0)) / sin(b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if F < -1.4199999999999999

   1. Initial program 77.1%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in x around 0 84.3%

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

      1. associate-*l/ 84.3%

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

      2. *-lft-identity 84.3%

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

      3. +-commutative 84.3%

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

      4. unpow2 84.3%

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

      5. fma-undefine 84.3%

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

   7. Simplified 84.3%

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

   8. Taylor expanded in F around -inf 97.7%

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

   if -1.4199999999999999 < F < 1.4199999999999999

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

      1. associate-*r/ 99.5%

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

      2. sqrt-div 99.5%

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

      3. metadata-eval 99.5%

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

      4. un-div-inv 99.6%

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in F around 0 98.3%

       \[\leadsto \frac{\frac{F}{\color{blue}{\sqrt{2}}}}{\sin B} - \frac{x}{\tan B} \]

   if 1.4199999999999999 < F

   1. Initial program 63.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in x around 0 80.8%

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

      1. associate-*l/ 80.9%

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

      2. *-lft-identity 80.9%

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

      3. +-commutative 80.9%

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

      4. unpow2 80.9%

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

      5. fma-undefine 80.9%

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

   7. Simplified 80.9%

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

   8. Taylor expanded in F around inf 99.1%

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

4. Final simplification 98.4%

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

Alternative 6: 99.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	} else if (F <= 1.42) {
		tmp = ((F / sqrt(2.0)) / sin(B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.42d0)) then
        tmp = ((-1.0d0) / sin(b)) - ((x * cos(b)) / sin(b))
    else if (f <= 1.42d0) then
        tmp = ((f / sqrt(2.0d0)) / sin(b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if F < -1.4199999999999999

   1. Initial program 77.1%

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

   3. Taylor expanded in F around -inf 97.7%

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

   4. Taylor expanded in x around 0 97.8%

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

   if -1.4199999999999999 < F < 1.4199999999999999

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

      1. associate-*r/ 99.5%

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

      2. sqrt-div 99.5%

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

      3. metadata-eval 99.5%

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

      4. un-div-inv 99.6%

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

   9. Applied egg-rr 99.6%

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

   10. Taylor expanded in F around 0 98.3%

       \[\leadsto \frac{\frac{F}{\color{blue}{\sqrt{2}}}}{\sin B} - \frac{x}{\tan B} \]

   if 1.4199999999999999 < F

   1. Initial program 63.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in x around 0 80.8%

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

      1. associate-*l/ 80.9%

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

      2. *-lft-identity 80.9%

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

      3. +-commutative 80.9%

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

      4. unpow2 80.9%

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

      5. fma-undefine 80.9%

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

   7. Simplified 80.9%

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

   8. Taylor expanded in F around inf 99.1%

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

4. Final simplification 98.4%

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

Alternative 7: 92.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -4 \cdot 10^{+14}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq -3.1 \cdot 10^{-70}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 1.62 \cdot 10^{-65}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_1\\ \mathbf{elif}\;F \leq 920000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0
         (-
          (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
          (/ x B)))
        (t_1 (/ x (tan B))))
   (if (<= F -4e+14)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F -3.1e-70)
       t_0
       (if (<= F 1.62e-65)
         (- (* F (/ (sqrt 0.5) B)) t_1)
         (if (<= F 920000.0) t_0 (- (/ 1.0 (sin B)) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -4e+14) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= -3.1e-70) {
		tmp = t_0;
	} else if (F <= 1.62e-65) {
		tmp = (F * (sqrt(0.5) / B)) - t_1;
	} else if (F <= 920000.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    t_1 = x / tan(b)
    if (f <= (-4d+14)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= (-3.1d-70)) then
        tmp = t_0
    else if (f <= 1.62d-65) then
        tmp = (f * (sqrt(0.5d0) / b)) - t_1
    else if (f <= 920000.0d0) then
        tmp = t_0
    else
        tmp = (1.0d0 / sin(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -4e+14) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= -3.1e-70) {
		tmp = t_0;
	} else if (F <= 1.62e-65) {
		tmp = (F * (Math.sqrt(0.5) / B)) - t_1;
	} else if (F <= 920000.0) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -4e+14:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= -3.1e-70:
		tmp = t_0
	elif F <= 1.62e-65:
		tmp = (F * (math.sqrt(0.5) / B)) - t_1
	elif F <= 920000.0:
		tmp = t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -4e+14)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= -3.1e-70)
		tmp = t_0;
	elseif (F <= 1.62e-65)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - t_1);
	elseif (F <= 920000.0)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (x / B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -4e+14)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= -3.1e-70)
		tmp = t_0;
	elseif (F <= 1.62e-65)
		tmp = (F * (sqrt(0.5) / B)) - t_1;
	elseif (F <= 920000.0)
		tmp = t_0;
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -4e+14], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, -3.1e-70], t$95$0, If[LessEqual[F, 1.62e-65], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 920000.0], t$95$0, N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -4e14

   1. Initial program 75.4%

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

   3. Simplified 83.2%

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

   5. Taylor expanded in x around 0 83.2%

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

      1. associate-*l/ 83.2%

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

      2. *-lft-identity 83.2%

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

      3. +-commutative 83.2%

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

      4. unpow2 83.2%

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

      5. fma-undefine 83.2%

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

   7. Simplified 83.2%

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

   8. Taylor expanded in F around -inf 99.8%

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

   if -4e14 < F < -3.1e-70 or 1.6200000000000001e-65 < F < 9.2e5

   1. Initial program 99.2%

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

   3. Taylor expanded in B around 0 94.4%

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

   if -3.1e-70 < F < 1.6200000000000001e-65

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in B around 0 85.8%

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

      1. associate-*l/ 85.8%

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

      2. *-lft-identity 85.8%

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

      3. +-commutative 85.8%

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

      4. unpow2 85.8%

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

      5. fma-undefine 85.8%

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

      6. unpow-1 85.8%

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

      7. metadata-eval 85.8%

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

      8. pow-sqr 85.8%

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

      9. rem-sqrt-square 85.8%

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

      10. rem-square-sqrt 85.7%

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

      11. fabs-sqr 85.7%

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

      12. rem-square-sqrt 85.8%

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

   10. Simplified 85.8%

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

   11. Taylor expanded in F around 0 85.8%

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

   if 9.2e5 < F

   1. Initial program 63.3%

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

   3. Simplified 80.5%

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

   5. Taylor expanded in x around 0 80.6%

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

      1. associate-*l/ 80.6%

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

      2. *-lft-identity 80.6%

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

      3. +-commutative 80.6%

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

      4. unpow2 80.6%

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

      5. fma-undefine 80.6%

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

   7. Simplified 80.6%

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

   8. Taylor expanded in F around inf 99.9%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4 \cdot 10^{+14}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -3.1 \cdot 10^{-70}:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.62 \cdot 10^{-65}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 920000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9.5 \cdot 10^{+125}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -0.0065:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.3 \cdot 10^{-59}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9.5e+125)
   (- (/ -1.0 B) (/ x (tan B)))
   (if (<= F -0.0065)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -2.3e-59)
       (/ (- (* F (sqrt 0.5)) x) B)
       (if (<= F 8.5e-36)
         (/ (* x (cos B)) (- (sin B)))
         (- (/ 1.0 (sin B)) (/ x B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.5e+125) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= -0.0065) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -2.3e-59) {
		tmp = ((F * sqrt(0.5)) - x) / B;
	} else if (F <= 8.5e-36) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (1.0 / sin(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 <= (-9.5d+125)) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (f <= (-0.0065d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-2.3d-59)) then
        tmp = ((f * sqrt(0.5d0)) - x) / b
    else if (f <= 8.5d-36) then
        tmp = (x * cos(b)) / -sin(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.5e+125) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (F <= -0.0065) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -2.3e-59) {
		tmp = ((F * Math.sqrt(0.5)) - x) / B;
	} else if (F <= 8.5e-36) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -9.5e+125:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif F <= -0.0065:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -2.3e-59:
		tmp = ((F * math.sqrt(0.5)) - x) / B
	elif F <= 8.5e-36:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9.5e+125)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= -0.0065)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -2.3e-59)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B);
	elseif (F <= 8.5e-36)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -9.5e+125)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (F <= -0.0065)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -2.3e-59)
		tmp = ((F * sqrt(0.5)) - x) / B;
	elseif (F <= 8.5e-36)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9.5e+125], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -0.0065], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.3e-59], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 8.5e-36], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -9.50000000000000041e125

   1. Initial program 62.5%

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

   3. Simplified 72.9%

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

   5. Taylor expanded in x around 0 72.9%

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

      1. associate-*l/ 72.9%

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

      2. *-lft-identity 72.9%

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

      3. +-commutative 72.9%

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

      4. unpow2 72.9%

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

      5. fma-undefine 72.9%

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

   7. Simplified 72.9%

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

   8. Taylor expanded in B around 0 65.2%

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

      1. associate-*l/ 65.2%

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

      2. *-lft-identity 65.2%

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

      3. +-commutative 65.2%

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

      4. unpow2 65.2%

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

      5. fma-undefine 65.2%

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

      6. unpow-1 65.2%

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

      7. metadata-eval 65.2%

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

      8. pow-sqr 65.2%

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

      9. rem-sqrt-square 65.2%

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

      10. rem-square-sqrt 65.2%

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

      11. fabs-sqr 65.2%

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

      12. rem-square-sqrt 65.2%

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

   10. Simplified 65.2%

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

   11. Taylor expanded in F around -inf 81.6%

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

   if -9.50000000000000041e125 < F < -0.0064999999999999997

   1. Initial program 96.5%

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

   3. Taylor expanded in F around -inf 90.1%

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

   4. Taylor expanded in B around 0 71.6%

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

   5. Taylor expanded in x around 0 71.6%

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

      1. sub-neg 71.6%

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

      2. distribute-neg-frac 71.6%

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

      3. metadata-eval 71.6%

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

      4. +-commutative 71.6%

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

      5. mul-1-neg 71.6%

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

      6. sub-neg 71.6%

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

   7. Simplified 71.6%

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

   if -0.0064999999999999997 < F < -2.29999999999999979e-59

   1. Initial program 99.2%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.2%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in B around 0 83.0%

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

      1. associate-*l/ 83.4%

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

      2. *-lft-identity 83.4%

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

      3. +-commutative 83.4%

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

      4. unpow2 83.4%

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

      5. fma-undefine 83.4%

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

      6. unpow-1 83.4%

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

      7. metadata-eval 83.4%

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

      8. pow-sqr 83.4%

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

      9. rem-sqrt-square 83.4%

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

      10. rem-square-sqrt 82.8%

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

      11. fabs-sqr 82.8%

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

      12. rem-square-sqrt 83.4%

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

   10. Simplified 83.4%

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

   11. Taylor expanded in F around 0 81.9%

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

   12. Taylor expanded in B around 0 82.2%

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

   if -2.29999999999999979e-59 < F < 8.5000000000000007e-36

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in B around 0 81.0%

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

      1. associate-*l/ 81.1%

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

      2. *-lft-identity 81.1%

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

      3. +-commutative 81.1%

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

      4. unpow2 81.1%

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

      5. fma-undefine 81.1%

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

      6. unpow-1 81.1%

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

      7. metadata-eval 81.1%

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

      8. pow-sqr 81.1%

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

      9. rem-sqrt-square 81.1%

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

      10. rem-square-sqrt 80.9%

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

      11. fabs-sqr 80.9%

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

      12. rem-square-sqrt 81.1%

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

   10. Simplified 81.1%

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

   11. Taylor expanded in F around 0 81.1%

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

   12. Taylor expanded in F around 0 71.5%

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

       1. mul-1-neg 71.5%

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

   14. Simplified 71.5%

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

   if 8.5000000000000007e-36 < F

   1. Initial program 66.5%

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

   3. Taylor expanded in F around -inf 46.8%

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

   4. Taylor expanded in B around 0 31.5%

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

      1. *-un-lft-identity 31.5%

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

      2. *-commutative 31.5%

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

      3. add-sqr-sqrt 17.1%

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

      4. sqrt-unprod 40.4%

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

      5. frac-times 40.4%

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

      6. metadata-eval 40.4%

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

      7. metadata-eval 40.4%

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

      8. frac-times 40.4%

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

      9. rgt-mult-inverse 40.4%

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

      10. associate-*l/ 36.6%

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

      11. rgt-mult-inverse 36.6%

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

      12. associate-*l/ 36.5%

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

      13. sqrt-unprod 24.3%

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

      14. add-sqr-sqrt 52.9%

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

      15. associate-*l/ 75.5%

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

      16. rgt-mult-inverse 75.6%

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

   6. Applied egg-rr 75.6%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9.5 \cdot 10^{+125}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -0.0065:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.3 \cdot 10^{-59}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 84.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.0029:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -4.95 \cdot 10^{-54}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.0029)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -4.95e-54)
       (/ (- (* F (sqrt 0.5)) x) B)
       (if (<= F 8.5e-36)
         (/ (* x (cos B)) (- (sin B)))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.0029) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -4.95e-54) {
		tmp = ((F * sqrt(0.5)) - x) / B;
	} else if (F <= 8.5e-36) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.0029d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-4.95d-54)) then
        tmp = ((f * sqrt(0.5d0)) - x) / b
    else if (f <= 8.5d-36) then
        tmp = (x * cos(b)) / -sin(b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.0029) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -4.95e-54) {
		tmp = ((F * Math.sqrt(0.5)) - x) / B;
	} else if (F <= 8.5e-36) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.0029:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -4.95e-54:
		tmp = ((F * math.sqrt(0.5)) - x) / B
	elif F <= 8.5e-36:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.0029)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -4.95e-54)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B);
	elseif (F <= 8.5e-36)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.0029)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -4.95e-54)
		tmp = ((F * sqrt(0.5)) - x) / B;
	elseif (F <= 8.5e-36)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.0029], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -4.95e-54], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 8.5e-36], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.0029

   1. Initial program 77.7%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in x around 0 84.7%

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

      1. associate-*l/ 84.7%

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

      2. *-lft-identity 84.7%

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

      3. +-commutative 84.7%

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

      4. unpow2 84.7%

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

      5. fma-undefine 84.7%

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

   7. Simplified 84.7%

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

   8. Taylor expanded in F around -inf 95.5%

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

   if -0.0029 < F < -4.95000000000000018e-54

   1. Initial program 99.2%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.2%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in B around 0 83.0%

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

      1. associate-*l/ 83.4%

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

      2. *-lft-identity 83.4%

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

      3. +-commutative 83.4%

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

      4. unpow2 83.4%

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

      5. fma-undefine 83.4%

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

      6. unpow-1 83.4%

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

      7. metadata-eval 83.4%

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

      8. pow-sqr 83.4%

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

      9. rem-sqrt-square 83.4%

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

      10. rem-square-sqrt 82.8%

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

      11. fabs-sqr 82.8%

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

      12. rem-square-sqrt 83.4%

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

   10. Simplified 83.4%

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

   11. Taylor expanded in F around 0 81.9%

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

   12. Taylor expanded in B around 0 82.2%

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

   if -4.95000000000000018e-54 < F < 8.5000000000000007e-36

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in B around 0 81.0%

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

      1. associate-*l/ 81.1%

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

      2. *-lft-identity 81.1%

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

      3. +-commutative 81.1%

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

      4. unpow2 81.1%

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

      5. fma-undefine 81.1%

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

      6. unpow-1 81.1%

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

      7. metadata-eval 81.1%

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

      8. pow-sqr 81.1%

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

      9. rem-sqrt-square 81.1%

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

      10. rem-square-sqrt 80.9%

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

      11. fabs-sqr 80.9%

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

      12. rem-square-sqrt 81.1%

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

   10. Simplified 81.1%

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

   11. Taylor expanded in F around 0 81.1%

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

   12. Taylor expanded in F around 0 71.5%

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

       1. mul-1-neg 71.5%

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

   14. Simplified 71.5%

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

   if 8.5000000000000007e-36 < F

   1. Initial program 66.5%

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

   3. Simplified 82.2%

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

   5. Taylor expanded in x around 0 82.2%

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

      1. associate-*l/ 82.3%

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

      2. *-lft-identity 82.3%

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

      3. +-commutative 82.3%

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

      4. unpow2 82.3%

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

      5. fma-undefine 82.3%

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

   7. Simplified 82.3%

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

   8. Taylor expanded in F around inf 92.1%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0029:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -4.95 \cdot 10^{-54}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 77.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.0021:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.0021)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F -1.2e-54)
     (/ (- (* F (sqrt 0.5)) x) B)
     (if (<= F 8.5e-36)
       (/ (* x (cos B)) (- (sin B)))
       (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0021) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -1.2e-54) {
		tmp = ((F * sqrt(0.5)) - x) / B;
	} else if (F <= 8.5e-36) {
		tmp = (x * cos(B)) / -sin(B);
	} else {
		tmp = (1.0 / sin(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 <= (-0.0021d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-1.2d-54)) then
        tmp = ((f * sqrt(0.5d0)) - x) / b
    else if (f <= 8.5d-36) then
        tmp = (x * cos(b)) / -sin(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0021) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -1.2e-54) {
		tmp = ((F * Math.sqrt(0.5)) - x) / B;
	} else if (F <= 8.5e-36) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.0021:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -1.2e-54:
		tmp = ((F * math.sqrt(0.5)) - x) / B
	elif F <= 8.5e-36:
		tmp = (x * math.cos(B)) / -math.sin(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.0021)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -1.2e-54)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B);
	elseif (F <= 8.5e-36)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.0021)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -1.2e-54)
		tmp = ((F * sqrt(0.5)) - x) / B;
	elseif (F <= 8.5e-36)
		tmp = (x * cos(B)) / -sin(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.0021], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.2e-54], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 8.5e-36], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -0.00209999999999999987

   1. Initial program 77.7%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in x around 0 84.7%

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

      1. associate-*l/ 84.7%

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

      2. *-lft-identity 84.7%

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

      3. +-commutative 84.7%

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

      4. unpow2 84.7%

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

      5. fma-undefine 84.7%

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

   7. Simplified 84.7%

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

   8. Taylor expanded in F around -inf 95.5%

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

   if -0.00209999999999999987 < F < -1.20000000000000007e-54

   1. Initial program 99.2%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.2%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in B around 0 83.0%

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

      1. associate-*l/ 83.4%

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

      2. *-lft-identity 83.4%

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

      3. +-commutative 83.4%

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

      4. unpow2 83.4%

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

      5. fma-undefine 83.4%

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

      6. unpow-1 83.4%

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

      7. metadata-eval 83.4%

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

      8. pow-sqr 83.4%

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

      9. rem-sqrt-square 83.4%

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

      10. rem-square-sqrt 82.8%

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

      11. fabs-sqr 82.8%

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

      12. rem-square-sqrt 83.4%

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

   10. Simplified 83.4%

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

   11. Taylor expanded in F around 0 81.9%

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

   12. Taylor expanded in B around 0 82.2%

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

   if -1.20000000000000007e-54 < F < 8.5000000000000007e-36

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.6%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in B around 0 81.0%

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

      1. associate-*l/ 81.1%

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

      2. *-lft-identity 81.1%

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

      3. +-commutative 81.1%

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

      4. unpow2 81.1%

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

      5. fma-undefine 81.1%

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

      6. unpow-1 81.1%

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

      7. metadata-eval 81.1%

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

      8. pow-sqr 81.1%

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

      9. rem-sqrt-square 81.1%

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

      10. rem-square-sqrt 80.9%

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

      11. fabs-sqr 80.9%

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

      12. rem-square-sqrt 81.1%

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

   10. Simplified 81.1%

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

   11. Taylor expanded in F around 0 81.1%

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

   12. Taylor expanded in F around 0 71.5%

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

       1. mul-1-neg 71.5%

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

   14. Simplified 71.5%

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

   if 8.5000000000000007e-36 < F

   1. Initial program 66.5%

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

   3. Taylor expanded in F around -inf 46.8%

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

   4. Taylor expanded in B around 0 31.5%

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

      1. *-un-lft-identity 31.5%

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

      2. *-commutative 31.5%

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

      3. add-sqr-sqrt 17.1%

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

      4. sqrt-unprod 40.4%

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

      5. frac-times 40.4%

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

      6. metadata-eval 40.4%

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

      7. metadata-eval 40.4%

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

      8. frac-times 40.4%

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

      9. rgt-mult-inverse 40.4%

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

      10. associate-*l/ 36.6%

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

      11. rgt-mult-inverse 36.6%

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

      12. associate-*l/ 36.5%

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

      13. sqrt-unprod 24.3%

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

      14. add-sqr-sqrt 52.9%

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

      15. associate-*l/ 75.5%

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

      16. rgt-mult-inverse 75.6%

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

   6. Applied egg-rr 75.6%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0021:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.2 \cdot 10^{-54}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 91.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -0.0065:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.085:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -0.0065)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.085)
       (- (* F (/ (sqrt 0.5) B)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -0.0065) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.085) {
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-0.0065d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.085d0) then
        tmp = (f * (sqrt(0.5d0) / b)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -0.0065) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.085) {
		tmp = (F * (Math.sqrt(0.5) / B)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -0.0065:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.085:
		tmp = (F * (math.sqrt(0.5) / B)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -0.0065)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.085)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -0.0065)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.085)
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -0.0065], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.085], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.0064999999999999997

   1. Initial program 77.7%

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

   3. Simplified 84.7%

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

   5. Taylor expanded in x around 0 84.7%

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

      1. associate-*l/ 84.7%

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

      2. *-lft-identity 84.7%

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

      3. +-commutative 84.7%

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

      4. unpow2 84.7%

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

      5. fma-undefine 84.7%

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

   7. Simplified 84.7%

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

   8. Taylor expanded in F around -inf 95.5%

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

   if -0.0064999999999999997 < F < 0.0850000000000000061

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in B around 0 80.5%

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

      1. associate-*l/ 80.5%

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

      2. *-lft-identity 80.5%

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

      3. +-commutative 80.5%

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

      4. unpow2 80.5%

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

      5. fma-undefine 80.5%

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

      6. unpow-1 80.5%

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

      7. metadata-eval 80.5%

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

      8. pow-sqr 80.5%

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

      9. rem-sqrt-square 80.5%

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

      10. rem-square-sqrt 80.3%

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

      11. fabs-sqr 80.3%

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

      12. rem-square-sqrt 80.5%

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

   10. Simplified 80.5%

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

   11. Taylor expanded in F around 0 80.4%

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

   if 0.0850000000000000061 < F

   1. Initial program 63.8%

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

   3. Simplified 80.8%

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

   5. Taylor expanded in x around 0 80.8%

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

      1. associate-*l/ 80.9%

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

      2. *-lft-identity 80.9%

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

      3. +-commutative 80.9%

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

      4. unpow2 80.9%

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

      5. fma-undefine 80.9%

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

   7. Simplified 80.9%

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

   8. Taylor expanded in F around inf 99.1%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0065:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.085:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 65.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.6 \cdot 10^{+126}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{elif}\;F \leq -0.006:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -5.6 \cdot 10^{-246}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{-202}:\\ \;\;\;\;\frac{\frac{F}{B}}{F} - t\_0\\ \mathbf{elif}\;F \leq 0.0019:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.55 \cdot 10^{+77} \lor \neg \left(F \leq 3.9 \cdot 10^{+125}\right):\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.6e+126)
     (- (/ -1.0 B) t_0)
     (if (<= F -0.006)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F -5.6e-246)
         (/ (- (* F (sqrt 0.5)) x) B)
         (if (<= F 6e-202)
           (- (/ (/ F B) F) t_0)
           (if (<= F 0.0019)
             (- (* F (/ (sqrt 0.5) B)) (/ x B))
             (if (or (<= F 4.55e+77) (not (<= F 3.9e+125)))
               (- (/ 1.0 (sin B)) (/ x B))
               (- (* F (/ (/ 1.0 F) B)) t_0)))))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.6e+126) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -0.006) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -5.6e-246) {
		tmp = ((F * sqrt(0.5)) - x) / B;
	} else if (F <= 6e-202) {
		tmp = ((F / B) / F) - t_0;
	} else if (F <= 0.0019) {
		tmp = (F * (sqrt(0.5) / B)) - (x / B);
	} else if ((F <= 4.55e+77) || !(F <= 3.9e+125)) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = (F * ((1.0 / F) / B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.6d+126)) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= (-0.006d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-5.6d-246)) then
        tmp = ((f * sqrt(0.5d0)) - x) / b
    else if (f <= 6d-202) then
        tmp = ((f / b) / f) - t_0
    else if (f <= 0.0019d0) then
        tmp = (f * (sqrt(0.5d0) / b)) - (x / b)
    else if ((f <= 4.55d+77) .or. (.not. (f <= 3.9d+125))) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = (f * ((1.0d0 / f) / b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.6e+126) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -0.006) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -5.6e-246) {
		tmp = ((F * Math.sqrt(0.5)) - x) / B;
	} else if (F <= 6e-202) {
		tmp = ((F / B) / F) - t_0;
	} else if (F <= 0.0019) {
		tmp = (F * (Math.sqrt(0.5) / B)) - (x / B);
	} else if ((F <= 4.55e+77) || !(F <= 3.9e+125)) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (F * ((1.0 / F) / B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.6e+126:
		tmp = (-1.0 / B) - t_0
	elif F <= -0.006:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -5.6e-246:
		tmp = ((F * math.sqrt(0.5)) - x) / B
	elif F <= 6e-202:
		tmp = ((F / B) / F) - t_0
	elif F <= 0.0019:
		tmp = (F * (math.sqrt(0.5) / B)) - (x / B)
	elif (F <= 4.55e+77) or not (F <= 3.9e+125):
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (F * ((1.0 / F) / B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.6e+126)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= -0.006)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -5.6e-246)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B);
	elseif (F <= 6e-202)
		tmp = Float64(Float64(Float64(F / B) / F) - t_0);
	elseif (F <= 0.0019)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - Float64(x / B));
	elseif ((F <= 4.55e+77) || !(F <= 3.9e+125))
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(F * Float64(Float64(1.0 / F) / B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.6e+126)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= -0.006)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -5.6e-246)
		tmp = ((F * sqrt(0.5)) - x) / B;
	elseif (F <= 6e-202)
		tmp = ((F / B) / F) - t_0;
	elseif (F <= 0.0019)
		tmp = (F * (sqrt(0.5) / B)) - (x / B);
	elseif ((F <= 4.55e+77) || ~((F <= 3.9e+125)))
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = (F * ((1.0 / F) / B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.6e+126], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -0.006], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -5.6e-246], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 6e-202], N[(N[(N[(F / B), $MachinePrecision] / F), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.0019], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 4.55e+77], N[Not[LessEqual[F, 3.9e+125]], $MachinePrecision]], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(F * N[(N[(1.0 / F), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 7 regimes

2. if F < -1.5999999999999999e126

   1. Initial program 62.5%

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

   3. Simplified 72.9%

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

   5. Taylor expanded in x around 0 72.9%

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

      1. associate-*l/ 72.9%

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

      2. *-lft-identity 72.9%

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

      3. +-commutative 72.9%

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

      4. unpow2 72.9%

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

      5. fma-undefine 72.9%

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

   7. Simplified 72.9%

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

   8. Taylor expanded in B around 0 65.2%

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

      1. associate-*l/ 65.2%

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

      2. *-lft-identity 65.2%

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

      3. +-commutative 65.2%

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

      4. unpow2 65.2%

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

      5. fma-undefine 65.2%

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

      6. unpow-1 65.2%

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

      7. metadata-eval 65.2%

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

      8. pow-sqr 65.2%

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

      9. rem-sqrt-square 65.2%

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

      10. rem-square-sqrt 65.2%

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

      11. fabs-sqr 65.2%

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

      12. rem-square-sqrt 65.2%

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

   10. Simplified 65.2%

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

   11. Taylor expanded in F around -inf 81.6%

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

   if -1.5999999999999999e126 < F < -0.0060000000000000001

   1. Initial program 96.5%

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

   3. Taylor expanded in F around -inf 90.1%

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

   4. Taylor expanded in B around 0 71.6%

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

   5. Taylor expanded in x around 0 71.6%

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

      1. sub-neg 71.6%

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

      2. distribute-neg-frac 71.6%

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

      3. metadata-eval 71.6%

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

      4. +-commutative 71.6%

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

      5. mul-1-neg 71.6%

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

      6. sub-neg 71.6%

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

   7. Simplified 71.6%

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

   if -0.0060000000000000001 < F < -5.5999999999999999e-246

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in B around 0 81.9%

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

      1. associate-*l/ 82.0%

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

      2. *-lft-identity 82.0%

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

      3. +-commutative 82.0%

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

      4. unpow2 82.0%

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

      5. fma-undefine 82.0%

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

      6. unpow-1 82.0%

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

      7. metadata-eval 82.0%

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

      8. pow-sqr 82.0%

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

      9. rem-sqrt-square 82.0%

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

      10. rem-square-sqrt 81.7%

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

      11. fabs-sqr 81.7%

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

      12. rem-square-sqrt 82.0%

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

   10. Simplified 82.0%

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

   11. Taylor expanded in F around 0 81.6%

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

   12. Taylor expanded in B around 0 62.0%

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

   if -5.5999999999999999e-246 < F < 6.00000000000000022e-202

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in B around 0 93.6%

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

      1. associate-*l/ 93.6%

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

      2. *-lft-identity 93.6%

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

      3. +-commutative 93.6%

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

      4. unpow2 93.6%

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

      5. fma-undefine 93.6%

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

      6. unpow-1 93.6%

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

      7. metadata-eval 93.6%

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

      8. pow-sqr 93.6%

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

      9. rem-sqrt-square 93.6%

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

      10. rem-square-sqrt 93.6%

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

      11. fabs-sqr 93.6%

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

      12. rem-square-sqrt 93.6%

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

   10. Simplified 93.6%

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

   11. Taylor expanded in F around inf 45.0%

       \[\leadsto F \cdot \color{blue}{\frac{1}{B \cdot F}} - \frac{x}{\tan B} \]
   12. Step-by-step derivation

       1. *-commutative 45.0%

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

   13. Simplified 45.0%

       \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot B}} - \frac{x}{\tan B} \]
   14. Step-by-step derivation

       1. un-div-inv 48.5%

          \[\leadsto \color{blue}{\frac{F}{F \cdot B}} - \frac{x}{\tan B} \]

       2. *-commutative 48.5%

          \[\leadsto \frac{F}{\color{blue}{B \cdot F}} - \frac{x}{\tan B} \]

       3. associate-/r* 74.3%

          \[\leadsto \color{blue}{\frac{\frac{F}{B}}{F}} - \frac{x}{\tan B} \]

   15. Applied egg-rr 74.3%

       \[\leadsto \color{blue}{\frac{\frac{F}{B}}{F}} - \frac{x}{\tan B} \]

   if 6.00000000000000022e-202 < F < 0.0019

   1. Initial program 99.2%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. associate-*l/ 99.5%

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

      2. *-lft-identity 99.5%

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

      3. +-commutative 99.5%

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

      4. unpow2 99.5%

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

      5. fma-undefine 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in B around 0 67.3%

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

      1. associate-*l/ 67.5%

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

      2. *-lft-identity 67.5%

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

      3. +-commutative 67.5%

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

      4. unpow2 67.5%

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

      5. fma-undefine 67.5%

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

      6. unpow-1 67.5%

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

      7. metadata-eval 67.5%

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

      8. pow-sqr 67.5%

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

      9. rem-sqrt-square 67.5%

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

      10. rem-square-sqrt 67.1%

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

      11. fabs-sqr 67.1%

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

      12. rem-square-sqrt 67.5%

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

   10. Simplified 67.5%

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

   11. Taylor expanded in F around 0 67.5%

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

   12. Taylor expanded in B around 0 48.2%

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

   if 0.0019 < F < 4.55000000000000007e77 or 3.9000000000000002e125 < F

   1. Initial program 59.5%

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

   3. Taylor expanded in F around -inf 50.6%

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

   4. Taylor expanded in B around 0 35.9%

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

      1. *-un-lft-identity 35.9%

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

      2. *-commutative 35.9%

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

      3. add-sqr-sqrt 17.5%

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

      4. sqrt-unprod 47.5%

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

      5. frac-times 47.4%

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

      6. metadata-eval 47.4%

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

      7. metadata-eval 47.4%

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

      8. frac-times 47.5%

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

      9. rgt-mult-inverse 47.5%

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

      10. associate-*l/ 42.0%

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

      11. rgt-mult-inverse 42.0%

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

      12. associate-*l/ 41.9%

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

      13. sqrt-unprod 28.2%

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

      14. add-sqr-sqrt 57.3%

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

      15. associate-*l/ 82.6%

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

      16. rgt-mult-inverse 82.6%

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

   6. Applied egg-rr 82.6%

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

   if 4.55000000000000007e77 < F < 3.9000000000000002e125

   1. Initial program 77.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. associate-*l/ 99.8%

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

      2. *-lft-identity 99.8%

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

      3. +-commutative 99.8%

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

      4. unpow2 99.8%

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

      5. fma-undefine 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in B around 0 83.3%

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

      1. associate-*l/ 83.4%

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

      2. *-lft-identity 83.4%

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

      3. +-commutative 83.4%

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

      4. unpow2 83.4%

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

      5. fma-undefine 83.4%

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

      6. unpow-1 83.4%

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

      7. metadata-eval 83.4%

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

      8. pow-sqr 83.3%

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

      9. rem-sqrt-square 83.3%

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

      10. rem-square-sqrt 82.9%

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

      11. fabs-sqr 82.9%

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

      12. rem-square-sqrt 83.3%

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

   10. Simplified 83.3%

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

   11. Taylor expanded in F around inf 83.4%

       \[\leadsto F \cdot \frac{\color{blue}{\frac{1}{F}}}{B} - \frac{x}{\tan B} \]
3. Recombined 7 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.6 \cdot 10^{+126}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -0.006:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -5.6 \cdot 10^{-246}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 6 \cdot 10^{-202}:\\ \;\;\;\;\frac{\frac{F}{B}}{F} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.0019:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.55 \cdot 10^{+77} \lor \neg \left(F \leq 3.9 \cdot 10^{+125}\right):\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{\frac{1}{F}}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 65.6% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.18 \cdot 10^{+126}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{elif}\;F \leq -0.0065:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.62 \cdot 10^{-245}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-202}:\\ \;\;\;\;\frac{\frac{F}{B}}{F} - t\_0\\ \mathbf{elif}\;F \leq 0.455:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{+76} \lor \neg \left(F \leq 9.2 \cdot 10^{+132}\right):\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.18e+126)
     (- (/ -1.0 B) t_0)
     (if (<= F -0.0065)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F -1.62e-245)
         (/ (- (* F (sqrt 0.5)) x) B)
         (if (<= F 5e-202)
           (- (/ (/ F B) F) t_0)
           (if (<= F 0.455)
             (- (* F (/ (sqrt 0.5) B)) (/ x B))
             (if (or (<= F 4.5e+76) (not (<= F 9.2e+132)))
               (- (/ 1.0 (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 <= -1.18e+126) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -0.0065) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -1.62e-245) {
		tmp = ((F * sqrt(0.5)) - x) / B;
	} else if (F <= 5e-202) {
		tmp = ((F / B) / F) - t_0;
	} else if (F <= 0.455) {
		tmp = (F * (sqrt(0.5) / B)) - (x / B);
	} else if ((F <= 4.5e+76) || !(F <= 9.2e+132)) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.18d+126)) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= (-0.0065d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-1.62d-245)) then
        tmp = ((f * sqrt(0.5d0)) - x) / b
    else if (f <= 5d-202) then
        tmp = ((f / b) / f) - t_0
    else if (f <= 0.455d0) then
        tmp = (f * (sqrt(0.5d0) / b)) - (x / b)
    else if ((f <= 4.5d+76) .or. (.not. (f <= 9.2d+132))) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = (1.0d0 / b) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.18e+126) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -0.0065) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -1.62e-245) {
		tmp = ((F * Math.sqrt(0.5)) - x) / B;
	} else if (F <= 5e-202) {
		tmp = ((F / B) / F) - t_0;
	} else if (F <= 0.455) {
		tmp = (F * (Math.sqrt(0.5) / B)) - (x / B);
	} else if ((F <= 4.5e+76) || !(F <= 9.2e+132)) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.18e+126:
		tmp = (-1.0 / B) - t_0
	elif F <= -0.0065:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -1.62e-245:
		tmp = ((F * math.sqrt(0.5)) - x) / B
	elif F <= 5e-202:
		tmp = ((F / B) / F) - t_0
	elif F <= 0.455:
		tmp = (F * (math.sqrt(0.5) / B)) - (x / B)
	elif (F <= 4.5e+76) or not (F <= 9.2e+132):
		tmp = (1.0 / math.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 <= -1.18e+126)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= -0.0065)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -1.62e-245)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B);
	elseif (F <= 5e-202)
		tmp = Float64(Float64(Float64(F / B) / F) - t_0);
	elseif (F <= 0.455)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - Float64(x / B));
	elseif ((F <= 4.5e+76) || !(F <= 9.2e+132))
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.18e+126)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= -0.0065)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -1.62e-245)
		tmp = ((F * sqrt(0.5)) - x) / B;
	elseif (F <= 5e-202)
		tmp = ((F / B) / F) - t_0;
	elseif (F <= 0.455)
		tmp = (F * (sqrt(0.5) / B)) - (x / B);
	elseif ((F <= 4.5e+76) || ~((F <= 9.2e+132)))
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = (1.0 / B) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.18e+126], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -0.0065], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.62e-245], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 5e-202], N[(N[(N[(F / B), $MachinePrecision] / F), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.455], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, 4.5e+76], N[Not[LessEqual[F, 9.2e+132]], $MachinePrecision]], N[(N[(1.0 / 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 7 regimes

2. if F < -1.18e126

   1. Initial program 62.5%

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

   3. Simplified 72.9%

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

   5. Taylor expanded in x around 0 72.9%

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

      1. associate-*l/ 72.9%

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

      2. *-lft-identity 72.9%

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

      3. +-commutative 72.9%

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

      4. unpow2 72.9%

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

      5. fma-undefine 72.9%

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

   7. Simplified 72.9%

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

   8. Taylor expanded in B around 0 65.2%

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

      1. associate-*l/ 65.2%

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

      2. *-lft-identity 65.2%

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

      3. +-commutative 65.2%

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

      4. unpow2 65.2%

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

      5. fma-undefine 65.2%

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

      6. unpow-1 65.2%

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

      7. metadata-eval 65.2%

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

      8. pow-sqr 65.2%

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

      9. rem-sqrt-square 65.2%

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

      10. rem-square-sqrt 65.2%

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

      11. fabs-sqr 65.2%

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

      12. rem-square-sqrt 65.2%

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

   10. Simplified 65.2%

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

   11. Taylor expanded in F around -inf 81.6%

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

   if -1.18e126 < F < -0.0064999999999999997

   1. Initial program 96.5%

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

   3. Taylor expanded in F around -inf 90.1%

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

   4. Taylor expanded in B around 0 71.6%

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

   5. Taylor expanded in x around 0 71.6%

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

      1. sub-neg 71.6%

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

      2. distribute-neg-frac 71.6%

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

      3. metadata-eval 71.6%

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

      4. +-commutative 71.6%

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

      5. mul-1-neg 71.6%

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

      6. sub-neg 71.6%

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

   7. Simplified 71.6%

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

   if -0.0064999999999999997 < F < -1.6199999999999999e-245

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in x around 0 99.5%

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

      1. associate-*l/ 99.6%

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

      2. *-lft-identity 99.6%

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

      3. +-commutative 99.6%

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

      4. unpow2 99.6%

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

      5. fma-undefine 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in B around 0 81.9%

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

      1. associate-*l/ 82.0%

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

      2. *-lft-identity 82.0%

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

      3. +-commutative 82.0%

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

      4. unpow2 82.0%

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

      5. fma-undefine 82.0%

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

      6. unpow-1 82.0%

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

      7. metadata-eval 82.0%

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

      8. pow-sqr 82.0%

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

      9. rem-sqrt-square 82.0%

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

      10. rem-square-sqrt 81.7%

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

      11. fabs-sqr 81.7%

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

      12. rem-square-sqrt 82.0%

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

   10. Simplified 82.0%

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

   11. Taylor expanded in F around 0 81.6%

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

   12. Taylor expanded in B around 0 62.0%

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

   if -1.6199999999999999e-245 < F < 4.99999999999999973e-202

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 99.7%

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

      1. associate-*l/ 99.7%

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

      2. *-lft-identity 99.7%

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

      3. +-commutative 99.7%

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

      4. unpow2 99.7%

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

      5. fma-undefine 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in B around 0 93.6%

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

      1. associate-*l/ 93.6%

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

      2. *-lft-identity 93.6%

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

      3. +-commutative 93.6%

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

      4. unpow2 93.6%

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

      5. fma-undefine 93.6%

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

      6. unpow-1 93.6%

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

      7. metadata-eval 93.6%

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

      8. pow-sqr 93.6%

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

      9. rem-sqrt-square 93.6%

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

      10. rem-square-sqrt 93.6%

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

      11. fabs-sqr 93.6%

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

      12. rem-square-sqrt 93.6%

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

   10. Simplified 93.6%

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

   11. Taylor expanded in F around inf 45.0%

       \[\leadsto F \cdot \color{blue}{\frac{1}{B \cdot F}} - \frac{x}{\tan B} \]
   12. Step-by-step derivation

       1. *-commutative 45.0%

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

   13. Simplified 45.0%

       \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot B}} - \frac{x}{\tan B} \]
   14. Step-by-step derivation

       1. un-div-inv 48.5%

          \[\leadsto \color{blue}{\frac{F}{F \cdot B}} - \frac{x}{\tan B} \]

       2. *-commutative 48.5%

          \[\leadsto \frac{F}{\color{blue}{B \cdot F}} - \frac{x}{\tan B} \]

       3. associate-/r* 74.3%

          \[\leadsto \color{blue}{\frac{\frac{F}{B}}{F}} - \frac{x}{\tan B} \]

   15. Applied egg-rr 74.3%

       \[\leadsto \color{blue}{\frac{\frac{F}{B}}{F}} - \frac{x}{\tan B} \]

   if 4.99999999999999973e-202 < F < 0.455000000000000016

   1. Initial program 99.2%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 99.4%

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

      1. associate-*l/ 99.5%

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

      2. *-lft-identity 99.5%

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

      3. +-commutative 99.5%

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

      4. unpow2 99.5%

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

      5. fma-undefine 99.5%

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

   7. Simplified 99.5%

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

   8. Taylor expanded in B around 0 67.3%

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

      1. associate-*l/ 67.5%

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

      2. *-lft-identity 67.5%

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

      3. +-commutative 67.5%

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

      4. unpow2 67.5%

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

      5. fma-undefine 67.5%

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

      6. unpow-1 67.5%

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

      7. metadata-eval 67.5%

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

      8. pow-sqr 67.5%

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

      9. rem-sqrt-square 67.5%

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

      10. rem-square-sqrt 67.1%

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

      11. fabs-sqr 67.1%

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

      12. rem-square-sqrt 67.5%

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

   10. Simplified 67.5%

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

   11. Taylor expanded in F around 0 67.5%

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

   12. Taylor expanded in B around 0 48.2%

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

   if 0.455000000000000016 < F < 4.4999999999999997e76 or 9.2000000000000006e132 < F

   1. Initial program 59.5%

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

   3. Taylor expanded in F around -inf 50.6%

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

   4. Taylor expanded in B around 0 35.9%

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

      1. *-un-lft-identity 35.9%

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

      2. *-commutative 35.9%

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

      3. add-sqr-sqrt 17.5%

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

      4. sqrt-unprod 47.5%

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

      5. frac-times 47.4%

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

      6. metadata-eval 47.4%

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

      7. metadata-eval 47.4%

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

      8. frac-times 47.5%

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

      9. rgt-mult-inverse 47.5%

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

      10. associate-*l/ 42.0%

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

      11. rgt-mult-inverse 42.0%

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

      12. associate-*l/ 41.9%

          \[\leadsto \left(-\frac{x}{B}\right) + \sqrt{\color{blue}{\left(\frac{F}{\sin B} \cdot \frac{1}{F}\right)} \cdot \left(\frac{F}{\sin B} \cdot \frac{1}{F}\right)} \cdot 1 \]

      13. sqrt-unprod 28.2%

          \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\left(\sqrt{\frac{F}{\sin B} \cdot \frac{1}{F}} \cdot \sqrt{\frac{F}{\sin B} \cdot \frac{1}{F}}\right)} \cdot 1 \]

      14. add-sqr-sqrt 57.3%

          \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\left(\frac{F}{\sin B} \cdot \frac{1}{F}\right)} \cdot 1 \]

      15. associate-*l/ 82.6%

          \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{F \cdot \frac{1}{F}}{\sin B}} \cdot 1 \]

      16. rgt-mult-inverse 82.6%

          \[\leadsto \left(-\frac{x}{B}\right) + \frac{\color{blue}{1}}{\sin B} \cdot 1 \]

   6. Applied egg-rr 82.6%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\frac{1}{\sin B} \cdot 1} \]

   if 4.4999999999999997e76 < F < 9.2000000000000006e132

   1. Initial program 77.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.5%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} - \frac{x}{\tan B} \]

      2. *-lft-identity 99.8%

         \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{\sin B} - \frac{x}{\tan B} \]

      3. +-commutative 99.8%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{\sin B} - \frac{x}{\tan B} \]

      4. unpow2 99.8%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{\sin B} - \frac{x}{\tan B} \]

      5. fma-undefine 99.8%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B} - \frac{x}{\tan B} \]

   7. Simplified 99.8%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 83.3%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   9. Step-by-step derivation

      1. associate-*l/ 83.4%

         \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{B}} - \frac{x}{\tan B} \]

      2. *-lft-identity 83.4%

         \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{B} - \frac{x}{\tan B} \]

      3. +-commutative 83.4%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{B} - \frac{x}{\tan B} \]

      4. unpow2 83.4%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{B} - \frac{x}{\tan B} \]

      5. fma-undefine 83.4%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{B} - \frac{x}{\tan B} \]

      6. unpow-1 83.4%

         \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}}}{B} - \frac{x}{\tan B} \]

      7. metadata-eval 83.4%

         \[\leadsto F \cdot \frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}}{B} - \frac{x}{\tan B} \]

      8. pow-sqr 83.3%

         \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

      9. rem-sqrt-square 83.3%

         \[\leadsto F \cdot \frac{\color{blue}{\left|{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}\right|}}{B} - \frac{x}{\tan B} \]

      10. rem-square-sqrt 82.9%

          \[\leadsto F \cdot \frac{\left|\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}\right|}{B} - \frac{x}{\tan B} \]

      11. fabs-sqr 82.9%

          \[\leadsto F \cdot \frac{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

      12. rem-square-sqrt 83.3%

          \[\leadsto F \cdot \frac{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}{B} - \frac{x}{\tan B} \]

   10. Simplified 83.3%

       \[\leadsto F \cdot \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}{B}} - \frac{x}{\tan B} \]

   11. Taylor expanded in F around inf 83.3%

       \[\leadsto F \cdot \color{blue}{\frac{1}{B \cdot F}} - \frac{x}{\tan B} \]
   12. Step-by-step derivation

       1. *-commutative 83.3%

          \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot B}} - \frac{x}{\tan B} \]

   13. Simplified 83.3%

       \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot B}} - \frac{x}{\tan B} \]

   14. Taylor expanded in F around 0 83.4%

       \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]
3. Recombined 7 regimes into one program.

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.18 \cdot 10^{+126}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -0.0065:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.62 \cdot 10^{-245}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-202}:\\ \;\;\;\;\frac{\frac{F}{B}}{F} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.455:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{+76} \lor \neg \left(F \leq 9.2 \cdot 10^{+132}\right):\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 64.3% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.65 \cdot 10^{+126}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{elif}\;F \leq -0.0065:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -7.5 \cdot 10^{-245}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-212}:\\ \;\;\;\;\frac{\frac{F}{B}}{F} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.65e+126)
     (- (/ -1.0 B) t_0)
     (if (<= F -0.0065)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F -7.5e-245)
         (/ (- (* F (sqrt 0.5)) x) B)
         (if (<= F 4.2e-212) (- (/ (/ F B) F) t_0) (- (/ 1.0 B) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.65e+126) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -0.0065) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -7.5e-245) {
		tmp = ((F * sqrt(0.5)) - x) / B;
	} else if (F <= 4.2e-212) {
		tmp = ((F / B) / F) - t_0;
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.65d+126)) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= (-0.0065d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-7.5d-245)) then
        tmp = ((f * sqrt(0.5d0)) - x) / b
    else if (f <= 4.2d-212) then
        tmp = ((f / b) / f) - t_0
    else
        tmp = (1.0d0 / b) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.65e+126) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -0.0065) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -7.5e-245) {
		tmp = ((F * Math.sqrt(0.5)) - x) / B;
	} else if (F <= 4.2e-212) {
		tmp = ((F / B) / F) - t_0;
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.65e+126:
		tmp = (-1.0 / B) - t_0
	elif F <= -0.0065:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -7.5e-245:
		tmp = ((F * math.sqrt(0.5)) - x) / B
	elif F <= 4.2e-212:
		tmp = ((F / B) / F) - t_0
	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 <= -1.65e+126)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= -0.0065)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -7.5e-245)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B);
	elseif (F <= 4.2e-212)
		tmp = Float64(Float64(Float64(F / B) / F) - t_0);
	else
		tmp = Float64(Float64(1.0 / B) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.65e+126)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= -0.0065)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -7.5e-245)
		tmp = ((F * sqrt(0.5)) - x) / B;
	elseif (F <= 4.2e-212)
		tmp = ((F / B) / F) - t_0;
	else
		tmp = (1.0 / B) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.65e+126], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -0.0065], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -7.5e-245], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4.2e-212], N[(N[(N[(F / B), $MachinePrecision] / F), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.65000000000000006e126

   1. Initial program 62.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 72.9%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 72.9%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-*l/ 72.9%

         \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} - \frac{x}{\tan B} \]

      2. *-lft-identity 72.9%

         \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{\sin B} - \frac{x}{\tan B} \]

      3. +-commutative 72.9%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{\sin B} - \frac{x}{\tan B} \]

      4. unpow2 72.9%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{\sin B} - \frac{x}{\tan B} \]

      5. fma-undefine 72.9%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B} - \frac{x}{\tan B} \]

   7. Simplified 72.9%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 65.2%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   9. Step-by-step derivation

      1. associate-*l/ 65.2%

         \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{B}} - \frac{x}{\tan B} \]

      2. *-lft-identity 65.2%

         \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{B} - \frac{x}{\tan B} \]

      3. +-commutative 65.2%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{B} - \frac{x}{\tan B} \]

      4. unpow2 65.2%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{B} - \frac{x}{\tan B} \]

      5. fma-undefine 65.2%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{B} - \frac{x}{\tan B} \]

      6. unpow-1 65.2%

         \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}}}{B} - \frac{x}{\tan B} \]

      7. metadata-eval 65.2%

         \[\leadsto F \cdot \frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}}{B} - \frac{x}{\tan B} \]

      8. pow-sqr 65.2%

         \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

      9. rem-sqrt-square 65.2%

         \[\leadsto F \cdot \frac{\color{blue}{\left|{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}\right|}}{B} - \frac{x}{\tan B} \]

      10. rem-square-sqrt 65.2%

          \[\leadsto F \cdot \frac{\left|\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}\right|}{B} - \frac{x}{\tan B} \]

      11. fabs-sqr 65.2%

          \[\leadsto F \cdot \frac{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

      12. rem-square-sqrt 65.2%

          \[\leadsto F \cdot \frac{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}{B} - \frac{x}{\tan B} \]

   10. Simplified 65.2%

       \[\leadsto F \cdot \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}{B}} - \frac{x}{\tan B} \]

   11. Taylor expanded in F around -inf 81.6%

       \[\leadsto \color{blue}{\frac{-1}{B}} - \frac{x}{\tan B} \]

   if -1.65000000000000006e126 < F < -0.0064999999999999997

   1. Initial program 96.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 90.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 71.6%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   5. Taylor expanded in x around 0 71.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} - \frac{1}{\sin B}} \]
   6. Step-by-step derivation

      1. sub-neg 71.6%

         \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} + \left(-\frac{1}{\sin B}\right)} \]

      2. distribute-neg-frac 71.6%

         \[\leadsto -1 \cdot \frac{x}{B} + \color{blue}{\frac{-1}{\sin B}} \]

      3. metadata-eval 71.6%

         \[\leadsto -1 \cdot \frac{x}{B} + \frac{\color{blue}{-1}}{\sin B} \]

      4. +-commutative 71.6%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} + -1 \cdot \frac{x}{B}} \]

      5. mul-1-neg 71.6%

         \[\leadsto \frac{-1}{\sin B} + \color{blue}{\left(-\frac{x}{B}\right)} \]

      6. sub-neg 71.6%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   7. Simplified 71.6%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   if -0.0064999999999999997 < F < -7.5000000000000003e-245

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.5%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-*l/ 99.6%

         \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} - \frac{x}{\tan B} \]

      2. *-lft-identity 99.6%

         \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{\sin B} - \frac{x}{\tan B} \]

      3. +-commutative 99.6%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{\sin B} - \frac{x}{\tan B} \]

      4. unpow2 99.6%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{\sin B} - \frac{x}{\tan B} \]

      5. fma-undefine 99.6%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B} - \frac{x}{\tan B} \]

   7. Simplified 99.6%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 81.9%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   9. Step-by-step derivation

      1. associate-*l/ 82.0%

         \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{B}} - \frac{x}{\tan B} \]

      2. *-lft-identity 82.0%

         \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{B} - \frac{x}{\tan B} \]

      3. +-commutative 82.0%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{B} - \frac{x}{\tan B} \]

      4. unpow2 82.0%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{B} - \frac{x}{\tan B} \]

      5. fma-undefine 82.0%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{B} - \frac{x}{\tan B} \]

      6. unpow-1 82.0%

         \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}}}{B} - \frac{x}{\tan B} \]

      7. metadata-eval 82.0%

         \[\leadsto F \cdot \frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}}{B} - \frac{x}{\tan B} \]

      8. pow-sqr 82.0%

         \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

      9. rem-sqrt-square 82.0%

         \[\leadsto F \cdot \frac{\color{blue}{\left|{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}\right|}}{B} - \frac{x}{\tan B} \]

      10. rem-square-sqrt 81.7%

          \[\leadsto F \cdot \frac{\left|\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}\right|}{B} - \frac{x}{\tan B} \]

      11. fabs-sqr 81.7%

          \[\leadsto F \cdot \frac{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

      12. rem-square-sqrt 82.0%

          \[\leadsto F \cdot \frac{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}{B} - \frac{x}{\tan B} \]

   10. Simplified 82.0%

       \[\leadsto F \cdot \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}{B}} - \frac{x}{\tan B} \]

   11. Taylor expanded in F around 0 81.6%

       \[\leadsto F \cdot \frac{\color{blue}{\sqrt{0.5}}}{B} - \frac{x}{\tan B} \]

   12. Taylor expanded in B around 0 62.0%

       \[\leadsto \color{blue}{\frac{F \cdot \sqrt{0.5} - x}{B}} \]

   if -7.5000000000000003e-245 < F < 4.1999999999999999e-212

   1. Initial program 99.6%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.7%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} - \frac{x}{\tan B} \]

      2. *-lft-identity 99.7%

         \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{\sin B} - \frac{x}{\tan B} \]

      3. +-commutative 99.7%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{\sin B} - \frac{x}{\tan B} \]

      4. unpow2 99.7%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{\sin B} - \frac{x}{\tan B} \]

      5. fma-undefine 99.7%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B} - \frac{x}{\tan B} \]

   7. Simplified 99.7%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 92.9%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   9. Step-by-step derivation

      1. associate-*l/ 92.9%

         \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{B}} - \frac{x}{\tan B} \]

      2. *-lft-identity 92.9%

         \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{B} - \frac{x}{\tan B} \]

      3. +-commutative 92.9%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{B} - \frac{x}{\tan B} \]

      4. unpow2 92.9%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{B} - \frac{x}{\tan B} \]

      5. fma-undefine 92.9%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{B} - \frac{x}{\tan B} \]

      6. unpow-1 92.9%

         \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}}}{B} - \frac{x}{\tan B} \]

      7. metadata-eval 92.9%

         \[\leadsto F \cdot \frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}}{B} - \frac{x}{\tan B} \]

      8. pow-sqr 92.9%

         \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

      9. rem-sqrt-square 92.9%

         \[\leadsto F \cdot \frac{\color{blue}{\left|{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}\right|}}{B} - \frac{x}{\tan B} \]

      10. rem-square-sqrt 92.9%

          \[\leadsto F \cdot \frac{\left|\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}\right|}{B} - \frac{x}{\tan B} \]

      11. fabs-sqr 92.9%

          \[\leadsto F \cdot \frac{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

      12. rem-square-sqrt 92.9%

          \[\leadsto F \cdot \frac{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}{B} - \frac{x}{\tan B} \]

   10. Simplified 92.9%

       \[\leadsto F \cdot \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}{B}} - \frac{x}{\tan B} \]

   11. Taylor expanded in F around inf 46.3%

       \[\leadsto F \cdot \color{blue}{\frac{1}{B \cdot F}} - \frac{x}{\tan B} \]
   12. Step-by-step derivation

       1. *-commutative 46.3%

          \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot B}} - \frac{x}{\tan B} \]

   13. Simplified 46.3%

       \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot B}} - \frac{x}{\tan B} \]
   14. Step-by-step derivation

       1. un-div-inv 50.2%

          \[\leadsto \color{blue}{\frac{F}{F \cdot B}} - \frac{x}{\tan B} \]

       2. *-commutative 50.2%

          \[\leadsto \frac{F}{\color{blue}{B \cdot F}} - \frac{x}{\tan B} \]

       3. associate-/r* 71.4%

          \[\leadsto \color{blue}{\frac{\frac{F}{B}}{F}} - \frac{x}{\tan B} \]

   15. Applied egg-rr 71.4%

       \[\leadsto \color{blue}{\frac{\frac{F}{B}}{F}} - \frac{x}{\tan B} \]

   if 4.1999999999999999e-212 < F

   1. Initial program 76.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 87.3%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 87.3%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-*l/ 87.4%

         \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} - \frac{x}{\tan B} \]

      2. *-lft-identity 87.4%

         \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{\sin B} - \frac{x}{\tan B} \]

      3. +-commutative 87.4%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{\sin B} - \frac{x}{\tan B} \]

      4. unpow2 87.4%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{\sin B} - \frac{x}{\tan B} \]

      5. fma-undefine 87.4%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B} - \frac{x}{\tan B} \]

   7. Simplified 87.4%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 67.2%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   9. Step-by-step derivation

      1. associate-*l/ 67.3%

         \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{B}} - \frac{x}{\tan B} \]

      2. *-lft-identity 67.3%

         \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{B} - \frac{x}{\tan B} \]

      3. +-commutative 67.3%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{B} - \frac{x}{\tan B} \]

      4. unpow2 67.3%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{B} - \frac{x}{\tan B} \]

      5. fma-undefine 67.3%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{B} - \frac{x}{\tan B} \]

      6. unpow-1 67.3%

         \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}}}{B} - \frac{x}{\tan B} \]

      7. metadata-eval 67.3%

         \[\leadsto F \cdot \frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}}{B} - \frac{x}{\tan B} \]

      8. pow-sqr 67.3%

         \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

      9. rem-sqrt-square 67.3%

         \[\leadsto F \cdot \frac{\color{blue}{\left|{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}\right|}}{B} - \frac{x}{\tan B} \]

      10. rem-square-sqrt 67.2%

          \[\leadsto F \cdot \frac{\left|\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}\right|}{B} - \frac{x}{\tan B} \]

      11. fabs-sqr 67.2%

          \[\leadsto F \cdot \frac{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

      12. rem-square-sqrt 67.3%

          \[\leadsto F \cdot \frac{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}{B} - \frac{x}{\tan B} \]

   10. Simplified 67.3%

       \[\leadsto F \cdot \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}{B}} - \frac{x}{\tan B} \]

   11. Taylor expanded in F around inf 57.7%

       \[\leadsto F \cdot \color{blue}{\frac{1}{B \cdot F}} - \frac{x}{\tan B} \]
   12. Step-by-step derivation

       1. *-commutative 57.7%

          \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot B}} - \frac{x}{\tan B} \]

   13. Simplified 57.7%

       \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot B}} - \frac{x}{\tan B} \]

   14. Taylor expanded in F around 0 63.2%

       \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]
3. Recombined 5 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.65 \cdot 10^{+126}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -0.0065:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -7.5 \cdot 10^{-245}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-212}:\\ \;\;\;\;\frac{\frac{F}{B}}{F} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 63.1% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.75 \cdot 10^{+126}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{elif}\;F \leq -0.0065:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-240}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.75e+126)
     (- (/ -1.0 B) t_0)
     (if (<= F -0.0065)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F -2.1e-240) (/ (- (* F (sqrt 0.5)) 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 <= -1.75e+126) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -0.0065) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -2.1e-240) {
		tmp = ((F * sqrt(0.5)) - x) / B;
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.75d+126)) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= (-0.0065d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-2.1d-240)) then
        tmp = ((f * sqrt(0.5d0)) - x) / b
    else
        tmp = (1.0d0 / b) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.75e+126) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -0.0065) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -2.1e-240) {
		tmp = ((F * Math.sqrt(0.5)) - x) / B;
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.75e+126:
		tmp = (-1.0 / B) - t_0
	elif F <= -0.0065:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -2.1e-240:
		tmp = ((F * math.sqrt(0.5)) - 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 <= -1.75e+126)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= -0.0065)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -2.1e-240)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B);
	else
		tmp = Float64(Float64(1.0 / B) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.75e+126)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= -0.0065)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -2.1e-240)
		tmp = ((F * sqrt(0.5)) - x) / B;
	else
		tmp = (1.0 / B) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.75e+126], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -0.0065], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.1e-240], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.7500000000000001e126

   1. Initial program 62.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 72.9%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 72.9%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-*l/ 72.9%

         \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} - \frac{x}{\tan B} \]

      2. *-lft-identity 72.9%

         \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{\sin B} - \frac{x}{\tan B} \]

      3. +-commutative 72.9%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{\sin B} - \frac{x}{\tan B} \]

      4. unpow2 72.9%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{\sin B} - \frac{x}{\tan B} \]

      5. fma-undefine 72.9%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B} - \frac{x}{\tan B} \]

   7. Simplified 72.9%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 65.2%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   9. Step-by-step derivation

      1. associate-*l/ 65.2%

         \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{B}} - \frac{x}{\tan B} \]

      2. *-lft-identity 65.2%

         \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{B} - \frac{x}{\tan B} \]

      3. +-commutative 65.2%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{B} - \frac{x}{\tan B} \]

      4. unpow2 65.2%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{B} - \frac{x}{\tan B} \]

      5. fma-undefine 65.2%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{B} - \frac{x}{\tan B} \]

      6. unpow-1 65.2%

         \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}}}{B} - \frac{x}{\tan B} \]

      7. metadata-eval 65.2%

         \[\leadsto F \cdot \frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}}{B} - \frac{x}{\tan B} \]

      8. pow-sqr 65.2%

         \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

      9. rem-sqrt-square 65.2%

         \[\leadsto F \cdot \frac{\color{blue}{\left|{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}\right|}}{B} - \frac{x}{\tan B} \]

      10. rem-square-sqrt 65.2%

          \[\leadsto F \cdot \frac{\left|\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}\right|}{B} - \frac{x}{\tan B} \]

      11. fabs-sqr 65.2%

          \[\leadsto F \cdot \frac{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

      12. rem-square-sqrt 65.2%

          \[\leadsto F \cdot \frac{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}{B} - \frac{x}{\tan B} \]

   10. Simplified 65.2%

       \[\leadsto F \cdot \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}{B}} - \frac{x}{\tan B} \]

   11. Taylor expanded in F around -inf 81.6%

       \[\leadsto \color{blue}{\frac{-1}{B}} - \frac{x}{\tan B} \]

   if -1.7500000000000001e126 < F < -0.0064999999999999997

   1. Initial program 96.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 90.1%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 71.6%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   5. Taylor expanded in x around 0 71.6%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} - \frac{1}{\sin B}} \]
   6. Step-by-step derivation

      1. sub-neg 71.6%

         \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} + \left(-\frac{1}{\sin B}\right)} \]

      2. distribute-neg-frac 71.6%

         \[\leadsto -1 \cdot \frac{x}{B} + \color{blue}{\frac{-1}{\sin B}} \]

      3. metadata-eval 71.6%

         \[\leadsto -1 \cdot \frac{x}{B} + \frac{\color{blue}{-1}}{\sin B} \]

      4. +-commutative 71.6%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} + -1 \cdot \frac{x}{B}} \]

      5. mul-1-neg 71.6%

         \[\leadsto \frac{-1}{\sin B} + \color{blue}{\left(-\frac{x}{B}\right)} \]

      6. sub-neg 71.6%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   7. Simplified 71.6%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   if -0.0064999999999999997 < F < -2.09999999999999994e-240

   1. Initial program 99.4%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.5%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-*l/ 99.6%

         \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} - \frac{x}{\tan B} \]

      2. *-lft-identity 99.6%

         \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{\sin B} - \frac{x}{\tan B} \]

      3. +-commutative 99.6%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{\sin B} - \frac{x}{\tan B} \]

      4. unpow2 99.6%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{\sin B} - \frac{x}{\tan B} \]

      5. fma-undefine 99.6%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B} - \frac{x}{\tan B} \]

   7. Simplified 99.6%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 81.9%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   9. Step-by-step derivation

      1. associate-*l/ 82.0%

         \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{B}} - \frac{x}{\tan B} \]

      2. *-lft-identity 82.0%

         \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{B} - \frac{x}{\tan B} \]

      3. +-commutative 82.0%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{B} - \frac{x}{\tan B} \]

      4. unpow2 82.0%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{B} - \frac{x}{\tan B} \]

      5. fma-undefine 82.0%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{B} - \frac{x}{\tan B} \]

      6. unpow-1 82.0%

         \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}}}{B} - \frac{x}{\tan B} \]

      7. metadata-eval 82.0%

         \[\leadsto F \cdot \frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}}{B} - \frac{x}{\tan B} \]

      8. pow-sqr 82.0%

         \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

      9. rem-sqrt-square 82.0%

         \[\leadsto F \cdot \frac{\color{blue}{\left|{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}\right|}}{B} - \frac{x}{\tan B} \]

      10. rem-square-sqrt 81.7%

          \[\leadsto F \cdot \frac{\left|\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}\right|}{B} - \frac{x}{\tan B} \]

      11. fabs-sqr 81.7%

          \[\leadsto F \cdot \frac{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

      12. rem-square-sqrt 82.0%

          \[\leadsto F \cdot \frac{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}{B} - \frac{x}{\tan B} \]

   10. Simplified 82.0%

       \[\leadsto F \cdot \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}{B}} - \frac{x}{\tan B} \]

   11. Taylor expanded in F around 0 81.6%

       \[\leadsto F \cdot \frac{\color{blue}{\sqrt{0.5}}}{B} - \frac{x}{\tan B} \]

   12. Taylor expanded in B around 0 62.0%

       \[\leadsto \color{blue}{\frac{F \cdot \sqrt{0.5} - x}{B}} \]

   if -2.09999999999999994e-240 < F

   1. Initial program 80.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 89.8%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 89.8%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-*l/ 89.8%

         \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} - \frac{x}{\tan B} \]

      2. *-lft-identity 89.8%

         \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{\sin B} - \frac{x}{\tan B} \]

      3. +-commutative 89.8%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{\sin B} - \frac{x}{\tan B} \]

      4. unpow2 89.8%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{\sin B} - \frac{x}{\tan B} \]

      5. fma-undefine 89.8%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B} - \frac{x}{\tan B} \]

   7. Simplified 89.8%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 72.3%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   9. Step-by-step derivation

      1. associate-*l/ 72.4%

         \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{B}} - \frac{x}{\tan B} \]

      2. *-lft-identity 72.4%

         \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{B} - \frac{x}{\tan B} \]

      3. +-commutative 72.4%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{B} - \frac{x}{\tan B} \]

      4. unpow2 72.4%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{B} - \frac{x}{\tan B} \]

      5. fma-undefine 72.4%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{B} - \frac{x}{\tan B} \]

      6. unpow-1 72.4%

         \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}}}{B} - \frac{x}{\tan B} \]

      7. metadata-eval 72.4%

         \[\leadsto F \cdot \frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}}{B} - \frac{x}{\tan B} \]

      8. pow-sqr 72.4%

         \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

      9. rem-sqrt-square 72.4%

         \[\leadsto F \cdot \frac{\color{blue}{\left|{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}\right|}}{B} - \frac{x}{\tan B} \]

      10. rem-square-sqrt 72.3%

          \[\leadsto F \cdot \frac{\left|\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}\right|}{B} - \frac{x}{\tan B} \]

      11. fabs-sqr 72.3%

          \[\leadsto F \cdot \frac{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

      12. rem-square-sqrt 72.4%

          \[\leadsto F \cdot \frac{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}{B} - \frac{x}{\tan B} \]

   10. Simplified 72.4%

       \[\leadsto F \cdot \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}{B}} - \frac{x}{\tan B} \]

   11. Taylor expanded in F around inf 55.4%

       \[\leadsto F \cdot \color{blue}{\frac{1}{B \cdot F}} - \frac{x}{\tan B} \]
   12. Step-by-step derivation

       1. *-commutative 55.4%

          \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot B}} - \frac{x}{\tan B} \]

   13. Simplified 55.4%

       \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot B}} - \frac{x}{\tan B} \]

   14. Taylor expanded in F around 0 61.4%

       \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]
3. Recombined 4 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.75 \cdot 10^{+126}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -0.0065:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-240}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 56.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-7} \lor \neg \left(x \leq 2.9 \cdot 10^{-120}\right):\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -1.15e-7) (not (<= x 2.9e-120)))
   (- (/ -1.0 B) (/ x (tan B)))
   (- (/ -1.0 (sin B)) (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -1.15e-7) || !(x <= 2.9e-120)) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else {
		tmp = (-1.0 / sin(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 ((x <= (-1.15d-7)) .or. (.not. (x <= 2.9d-120))) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -1.15e-7) || !(x <= 2.9e-120)) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -1.15e-7) or not (x <= 2.9e-120):
		tmp = (-1.0 / B) - (x / math.tan(B))
	else:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -1.15e-7) || !(x <= 2.9e-120))
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -1.15e-7) || ~((x <= 2.9e-120)))
		tmp = (-1.0 / B) - (x / tan(B));
	else
		tmp = (-1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -1.15e-7], N[Not[LessEqual[x, 2.9e-120]], $MachinePrecision]], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.14999999999999997e-7 or 2.9e-120 < x

   1. Initial program 84.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 94.0%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 94.0%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-*l/ 94.0%

         \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} - \frac{x}{\tan B} \]

      2. *-lft-identity 94.0%

         \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{\sin B} - \frac{x}{\tan B} \]

      3. +-commutative 94.0%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{\sin B} - \frac{x}{\tan B} \]

      4. unpow2 94.0%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{\sin B} - \frac{x}{\tan B} \]

      5. fma-undefine 94.0%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B} - \frac{x}{\tan B} \]

   7. Simplified 94.0%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 90.6%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   9. Step-by-step derivation

      1. associate-*l/ 90.6%

         \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{B}} - \frac{x}{\tan B} \]

      2. *-lft-identity 90.6%

         \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{B} - \frac{x}{\tan B} \]

      3. +-commutative 90.6%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{B} - \frac{x}{\tan B} \]

      4. unpow2 90.6%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{B} - \frac{x}{\tan B} \]

      5. fma-undefine 90.6%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{B} - \frac{x}{\tan B} \]

      6. unpow-1 90.6%

         \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}}}{B} - \frac{x}{\tan B} \]

      7. metadata-eval 90.6%

         \[\leadsto F \cdot \frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}}{B} - \frac{x}{\tan B} \]

      8. pow-sqr 90.6%

         \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

      9. rem-sqrt-square 90.6%

         \[\leadsto F \cdot \frac{\color{blue}{\left|{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}\right|}}{B} - \frac{x}{\tan B} \]

      10. rem-square-sqrt 90.6%

          \[\leadsto F \cdot \frac{\left|\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}\right|}{B} - \frac{x}{\tan B} \]

      11. fabs-sqr 90.6%

          \[\leadsto F \cdot \frac{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

      12. rem-square-sqrt 90.6%

          \[\leadsto F \cdot \frac{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}{B} - \frac{x}{\tan B} \]

   10. Simplified 90.6%

       \[\leadsto F \cdot \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}{B}} - \frac{x}{\tan B} \]

   11. Taylor expanded in F around -inf 88.0%

       \[\leadsto \color{blue}{\frac{-1}{B}} - \frac{x}{\tan B} \]

   if -1.14999999999999997e-7 < x < 2.9e-120

   1. Initial program 82.1%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 25.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 25.3%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   5. Taylor expanded in x around 0 25.3%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} - \frac{1}{\sin B}} \]
   6. Step-by-step derivation

      1. sub-neg 25.3%

         \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} + \left(-\frac{1}{\sin B}\right)} \]

      2. distribute-neg-frac 25.3%

         \[\leadsto -1 \cdot \frac{x}{B} + \color{blue}{\frac{-1}{\sin B}} \]

      3. metadata-eval 25.3%

         \[\leadsto -1 \cdot \frac{x}{B} + \frac{\color{blue}{-1}}{\sin B} \]

      4. +-commutative 25.3%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} + -1 \cdot \frac{x}{B}} \]

      5. mul-1-neg 25.3%

         \[\leadsto \frac{-1}{\sin B} + \color{blue}{\left(-\frac{x}{B}\right)} \]

      6. sub-neg 25.3%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   7. Simplified 25.3%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.15 \cdot 10^{-7} \lor \neg \left(x \leq 2.9 \cdot 10^{-120}\right):\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 61.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.18 \cdot 10^{+126}:\\ \;\;\;\;\frac{-1}{B} - t\_0\\ \mathbf{elif}\;F \leq -1.95 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.18e+126)
     (- (/ -1.0 B) t_0)
     (if (<= F -1.95e-52) (- (/ -1.0 (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 <= -1.18e+126) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -1.95e-52) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.18d+126)) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= (-1.95d-52)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else
        tmp = (1.0d0 / b) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.18e+126) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= -1.95e-52) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.18e+126:
		tmp = (-1.0 / B) - t_0
	elif F <= -1.95e-52:
		tmp = (-1.0 / math.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 <= -1.18e+126)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= -1.95e-52)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.18e+126)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= -1.95e-52)
		tmp = (-1.0 / sin(B)) - (x / B);
	else
		tmp = (1.0 / B) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.18e+126], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -1.95e-52], N[(N[(-1.0 / 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 3 regimes

2. if F < -1.18e126

   1. Initial program 62.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 72.9%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 72.9%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-*l/ 72.9%

         \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} - \frac{x}{\tan B} \]

      2. *-lft-identity 72.9%

         \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{\sin B} - \frac{x}{\tan B} \]

      3. +-commutative 72.9%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{\sin B} - \frac{x}{\tan B} \]

      4. unpow2 72.9%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{\sin B} - \frac{x}{\tan B} \]

      5. fma-undefine 72.9%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B} - \frac{x}{\tan B} \]

   7. Simplified 72.9%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 65.2%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   9. Step-by-step derivation

      1. associate-*l/ 65.2%

         \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{B}} - \frac{x}{\tan B} \]

      2. *-lft-identity 65.2%

         \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{B} - \frac{x}{\tan B} \]

      3. +-commutative 65.2%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{B} - \frac{x}{\tan B} \]

      4. unpow2 65.2%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{B} - \frac{x}{\tan B} \]

      5. fma-undefine 65.2%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{B} - \frac{x}{\tan B} \]

      6. unpow-1 65.2%

         \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}}}{B} - \frac{x}{\tan B} \]

      7. metadata-eval 65.2%

         \[\leadsto F \cdot \frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}}{B} - \frac{x}{\tan B} \]

      8. pow-sqr 65.2%

         \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

      9. rem-sqrt-square 65.2%

         \[\leadsto F \cdot \frac{\color{blue}{\left|{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}\right|}}{B} - \frac{x}{\tan B} \]

      10. rem-square-sqrt 65.2%

          \[\leadsto F \cdot \frac{\left|\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}\right|}{B} - \frac{x}{\tan B} \]

      11. fabs-sqr 65.2%

          \[\leadsto F \cdot \frac{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

      12. rem-square-sqrt 65.2%

          \[\leadsto F \cdot \frac{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}{B} - \frac{x}{\tan B} \]

   10. Simplified 65.2%

       \[\leadsto F \cdot \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}{B}} - \frac{x}{\tan B} \]

   11. Taylor expanded in F around -inf 81.6%

       \[\leadsto \color{blue}{\frac{-1}{B}} - \frac{x}{\tan B} \]

   if -1.18e126 < F < -1.95000000000000009e-52

   1. Initial program 97.2%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 76.6%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 63.1%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   5. Taylor expanded in x around 0 63.1%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} - \frac{1}{\sin B}} \]
   6. Step-by-step derivation

      1. sub-neg 63.1%

         \[\leadsto \color{blue}{-1 \cdot \frac{x}{B} + \left(-\frac{1}{\sin B}\right)} \]

      2. distribute-neg-frac 63.1%

         \[\leadsto -1 \cdot \frac{x}{B} + \color{blue}{\frac{-1}{\sin B}} \]

      3. metadata-eval 63.1%

         \[\leadsto -1 \cdot \frac{x}{B} + \frac{\color{blue}{-1}}{\sin B} \]

      4. +-commutative 63.1%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} + -1 \cdot \frac{x}{B}} \]

      5. mul-1-neg 63.1%

         \[\leadsto \frac{-1}{\sin B} + \color{blue}{\left(-\frac{x}{B}\right)} \]

      6. sub-neg 63.1%

         \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   7. Simplified 63.1%

      \[\leadsto \color{blue}{\frac{-1}{\sin B} - \frac{x}{B}} \]

   if -1.95000000000000009e-52 < F

   1. Initial program 84.5%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 91.7%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 91.7%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-*l/ 91.8%

         \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} - \frac{x}{\tan B} \]

      2. *-lft-identity 91.8%

         \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{\sin B} - \frac{x}{\tan B} \]

      3. +-commutative 91.8%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{\sin B} - \frac{x}{\tan B} \]

      4. unpow2 91.8%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{\sin B} - \frac{x}{\tan B} \]

      5. fma-undefine 91.8%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B} - \frac{x}{\tan B} \]

   7. Simplified 91.8%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 74.2%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   9. Step-by-step derivation

      1. associate-*l/ 74.2%

         \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{B}} - \frac{x}{\tan B} \]

      2. *-lft-identity 74.2%

         \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{B} - \frac{x}{\tan B} \]

      3. +-commutative 74.2%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{B} - \frac{x}{\tan B} \]

      4. unpow2 74.2%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{B} - \frac{x}{\tan B} \]

      5. fma-undefine 74.2%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{B} - \frac{x}{\tan B} \]

      6. unpow-1 74.2%

         \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}}}{B} - \frac{x}{\tan B} \]

      7. metadata-eval 74.2%

         \[\leadsto F \cdot \frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}}{B} - \frac{x}{\tan B} \]

      8. pow-sqr 74.2%

         \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

      9. rem-sqrt-square 74.2%

         \[\leadsto F \cdot \frac{\color{blue}{\left|{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}\right|}}{B} - \frac{x}{\tan B} \]

      10. rem-square-sqrt 74.1%

          \[\leadsto F \cdot \frac{\left|\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}\right|}{B} - \frac{x}{\tan B} \]

      11. fabs-sqr 74.1%

          \[\leadsto F \cdot \frac{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

      12. rem-square-sqrt 74.2%

          \[\leadsto F \cdot \frac{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}{B} - \frac{x}{\tan B} \]

   10. Simplified 74.2%

       \[\leadsto F \cdot \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}{B}} - \frac{x}{\tan B} \]

   11. Taylor expanded in F around inf 52.7%

       \[\leadsto F \cdot \color{blue}{\frac{1}{B \cdot F}} - \frac{x}{\tan B} \]
   12. Step-by-step derivation

       1. *-commutative 52.7%

          \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot B}} - \frac{x}{\tan B} \]

   13. Simplified 52.7%

       \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot B}} - \frac{x}{\tan B} \]

   14. Taylor expanded in F around 0 57.8%

       \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.18 \cdot 10^{+126}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.95 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 53.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{B} - \frac{x}{\tan B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (- (/ -1.0 B) (/ x (tan B))))

C

double code(double F, double B, double x) {
	return (-1.0 / B) - (x / tan(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 = ((-1.0d0) / b) - (x / tan(b))
end function

Java

public static double code(double F, double B, double x) {
	return (-1.0 / B) - (x / Math.tan(B));
}

Python

def code(F, B, x):
	return (-1.0 / B) - (x / math.tan(B))

Julia

function code(F, B, x)
	return Float64(Float64(-1.0 / B) - Float64(x / tan(B)))
end

MATLAB

function tmp = code(F, B, x)
	tmp = (-1.0 / B) - (x / tan(B));
end

Wolfram

code[F_, B_, x_] := N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 83.2%

   \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

3. Simplified 90.1%

   \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 90.0%

   \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
6. Step-by-step derivation

   1. associate-*l/ 90.1%

      \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} - \frac{x}{\tan B} \]

   2. *-lft-identity 90.1%

      \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{\sin B} - \frac{x}{\tan B} \]

   3. +-commutative 90.1%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{\sin B} - \frac{x}{\tan B} \]

   4. unpow2 90.1%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{\sin B} - \frac{x}{\tan B} \]

   5. fma-undefine 90.1%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B} - \frac{x}{\tan B} \]

7. Simplified 90.1%

   \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}} - \frac{x}{\tan B} \]

8. Taylor expanded in B around 0 71.0%

   \[\leadsto F \cdot \color{blue}{\left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
9. Step-by-step derivation

   1. associate-*l/ 71.1%

      \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{B}} - \frac{x}{\tan B} \]

   2. *-lft-identity 71.1%

      \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{B} - \frac{x}{\tan B} \]

   3. +-commutative 71.1%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{B} - \frac{x}{\tan B} \]

   4. unpow2 71.1%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{B} - \frac{x}{\tan B} \]

   5. fma-undefine 71.1%

      \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{B} - \frac{x}{\tan B} \]

   6. unpow-1 71.1%

      \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}}}{B} - \frac{x}{\tan B} \]

   7. metadata-eval 71.1%

      \[\leadsto F \cdot \frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}}{B} - \frac{x}{\tan B} \]

   8. pow-sqr 71.1%

      \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

   9. rem-sqrt-square 71.1%

      \[\leadsto F \cdot \frac{\color{blue}{\left|{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}\right|}}{B} - \frac{x}{\tan B} \]

   10. rem-square-sqrt 70.9%

       \[\leadsto F \cdot \frac{\left|\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}\right|}{B} - \frac{x}{\tan B} \]

   11. fabs-sqr 70.9%

       \[\leadsto F \cdot \frac{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

   12. rem-square-sqrt 71.1%

       \[\leadsto F \cdot \frac{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}{B} - \frac{x}{\tan B} \]

10. Simplified 71.1%

    \[\leadsto F \cdot \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}{B}} - \frac{x}{\tan B} \]

11. Taylor expanded in F around -inf 53.4%

    \[\leadsto \color{blue}{\frac{-1}{B}} - \frac{x}{\tan B} \]

12. Final simplification 53.4%

    \[\leadsto \frac{-1}{B} - \frac{x}{\tan B} \]
13. Add Preprocessing

Alternative 19: 43.8% accurate, 15.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{-30}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-67}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3e-30)
   (/ (- -1.0 x) B)
   (if (<= F 1.2e-67) (/ x (- B)) (- (* F (/ 1.0 (* F B))) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3e-30) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.2e-67) {
		tmp = x / -B;
	} else {
		tmp = (F * (1.0 / (F * 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 <= (-3d-30)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 1.2d-67) then
        tmp = x / -b
    else
        tmp = (f * (1.0d0 / (f * b))) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3e-30) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 1.2e-67) {
		tmp = x / -B;
	} else {
		tmp = (F * (1.0 / (F * B))) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3e-30:
		tmp = (-1.0 - x) / B
	elif F <= 1.2e-67:
		tmp = x / -B
	else:
		tmp = (F * (1.0 / (F * B))) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3e-30)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 1.2e-67)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * B))) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3e-30)
		tmp = (-1.0 - x) / B;
	elseif (F <= 1.2e-67)
		tmp = x / -B;
	else
		tmp = (F * (1.0 / (F * B))) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3e-30], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 1.2e-67], N[(x / (-B)), $MachinePrecision], N[(N[(F * N[(1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.9999999999999999e-30

   1. Initial program 80.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 89.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 63.5%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   5. Taylor expanded in B around 0 41.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   6. Step-by-step derivation

      1. mul-1-neg 41.9%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 41.9%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   7. Simplified 41.9%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   if -2.9999999999999999e-30 < F < 1.2e-67

   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 F around -inf 38.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 21.7%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   5. Taylor expanded in x around inf 39.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   6. Step-by-step derivation

      1. mul-1-neg 39.2%

         \[\leadsto \color{blue}{-\frac{x}{B}} \]

      2. distribute-neg-frac2 39.2%

         \[\leadsto \color{blue}{\frac{x}{-B}} \]

   7. Simplified 39.2%

      \[\leadsto \color{blue}{\frac{x}{-B}} \]

   if 1.2e-67 < F

   1. Initial program 68.9%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   3. Simplified 83.4%

      \[\leadsto \color{blue}{F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 83.5%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-*l/ 83.5%

         \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{\sin B}} - \frac{x}{\tan B} \]

      2. *-lft-identity 83.5%

         \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{\sin B} - \frac{x}{\tan B} \]

      3. +-commutative 83.5%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{\sin B} - \frac{x}{\tan B} \]

      4. unpow2 83.5%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{\sin B} - \frac{x}{\tan B} \]

      5. fma-undefine 83.5%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{\sin B} - \frac{x}{\tan B} \]

   7. Simplified 83.5%

      \[\leadsto F \cdot \color{blue}{\frac{\sqrt{\frac{1}{\mathsf{fma}\left(F, F, 2\right)}}}{\sin B}} - \frac{x}{\tan B} \]

   8. Taylor expanded in B around 0 63.9%

      \[\leadsto F \cdot \color{blue}{\left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + {F}^{2}}}\right)} - \frac{x}{\tan B} \]
   9. Step-by-step derivation

      1. associate-*l/ 64.0%

         \[\leadsto F \cdot \color{blue}{\frac{1 \cdot \sqrt{\frac{1}{2 + {F}^{2}}}}{B}} - \frac{x}{\tan B} \]

      2. *-lft-identity 64.0%

         \[\leadsto F \cdot \frac{\color{blue}{\sqrt{\frac{1}{2 + {F}^{2}}}}}{B} - \frac{x}{\tan B} \]

      3. +-commutative 64.0%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{{F}^{2} + 2}}}}{B} - \frac{x}{\tan B} \]

      4. unpow2 64.0%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{F \cdot F} + 2}}}{B} - \frac{x}{\tan B} \]

      5. fma-undefine 64.0%

         \[\leadsto F \cdot \frac{\sqrt{\frac{1}{\color{blue}{\mathsf{fma}\left(F, F, 2\right)}}}}{B} - \frac{x}{\tan B} \]

      6. unpow-1 64.0%

         \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-1}}}}{B} - \frac{x}{\tan B} \]

      7. metadata-eval 64.0%

         \[\leadsto F \cdot \frac{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{\color{blue}{\left(2 \cdot -0.5\right)}}}}{B} - \frac{x}{\tan B} \]

      8. pow-sqr 63.9%

         \[\leadsto F \cdot \frac{\sqrt{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5} \cdot {\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

      9. rem-sqrt-square 63.9%

         \[\leadsto F \cdot \frac{\color{blue}{\left|{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}\right|}}{B} - \frac{x}{\tan B} \]

      10. rem-square-sqrt 63.8%

          \[\leadsto F \cdot \frac{\left|\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}\right|}{B} - \frac{x}{\tan B} \]

      11. fabs-sqr 63.8%

          \[\leadsto F \cdot \frac{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}} \cdot \sqrt{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}}{B} - \frac{x}{\tan B} \]

      12. rem-square-sqrt 63.9%

          \[\leadsto F \cdot \frac{\color{blue}{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}}{B} - \frac{x}{\tan B} \]

   10. Simplified 63.9%

       \[\leadsto F \cdot \color{blue}{\frac{{\left(\mathsf{fma}\left(F, F, 2\right)\right)}^{-0.5}}{B}} - \frac{x}{\tan B} \]

   11. Taylor expanded in F around inf 67.0%

       \[\leadsto F \cdot \color{blue}{\frac{1}{B \cdot F}} - \frac{x}{\tan B} \]
   12. Step-by-step derivation

       1. *-commutative 67.0%

          \[\leadsto F \cdot \frac{1}{\color{blue}{F \cdot B}} - \frac{x}{\tan B} \]

   13. Simplified 67.0%

       \[\leadsto F \cdot \color{blue}{\frac{1}{F \cdot B}} - \frac{x}{\tan B} \]

   14. Taylor expanded in B around 0 50.1%

       \[\leadsto F \cdot \frac{1}{F \cdot B} - \frac{x}{\color{blue}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3 \cdot 10^{-30}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 1.2 \cdot 10^{-67}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 37.0% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -9.2 \cdot 10^{-30}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -9.2e-30) (/ (- -1.0 x) B) (/ x (- B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.2e-30) {
		tmp = (-1.0 - 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 <= (-9.2d-30)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = x / -b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -9.2e-30) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = x / -B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -9.2e-30:
		tmp = (-1.0 - x) / B
	else:
		tmp = x / -B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -9.2e-30)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(x / Float64(-B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -9.2e-30)
		tmp = (-1.0 - x) / B;
	else
		tmp = x / -B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -9.2e-30], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(x / (-B)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -9.19999999999999937e-30

   1. Initial program 80.0%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 89.4%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 63.5%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   5. Taylor expanded in B around 0 41.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{1 + x}{B}} \]
   6. Step-by-step derivation

      1. mul-1-neg 41.9%

         \[\leadsto \color{blue}{-\frac{1 + x}{B}} \]

      2. distribute-neg-frac2 41.9%

         \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   7. Simplified 41.9%

      \[\leadsto \color{blue}{\frac{1 + x}{-B}} \]

   if -9.19999999999999937e-30 < F

   1. Initial program 84.8%

      \[\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in F around -inf 41.5%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

   4. Taylor expanded in B around 0 25.3%

      \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

   5. Taylor expanded in x around inf 34.9%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
   6. Step-by-step derivation

      1. mul-1-neg 34.9%

         \[\leadsto \color{blue}{-\frac{x}{B}} \]

      2. distribute-neg-frac2 34.9%

         \[\leadsto \color{blue}{\frac{x}{-B}} \]

   7. Simplified 34.9%

      \[\leadsto \color{blue}{\frac{x}{-B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9.2 \cdot 10^{-30}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 29.4% accurate, 81.0× 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(x / Float64(-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 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 F around -inf 57.0%

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

4. Taylor expanded in B around 0 37.7%

   \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]

5. Taylor expanded in x around inf 31.1%

   \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} \]
6. Step-by-step derivation

   1. mul-1-neg 31.1%

      \[\leadsto \color{blue}{-\frac{x}{B}} \]

   2. distribute-neg-frac2 31.1%

      \[\leadsto \color{blue}{\frac{x}{-B}} \]

7. Simplified 31.1%

   \[\leadsto \color{blue}{\frac{x}{-B}} \]

8. Final simplification 31.1%

   \[\leadsto \frac{x}{-B} \]
9. Add Preprocessing

Alternative 22: 2.8% accurate, 108.0× 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(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 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 F around -inf 57.0%

   \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\frac{-1}{\sin B}} \]

4. Taylor expanded in B around 0 37.7%

   \[\leadsto \left(-\color{blue}{\frac{x}{B}}\right) + \frac{-1}{\sin B} \]
5. Step-by-step derivation

   1. log1p-expm1-u 21.3%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\left(-\frac{x}{B}\right) + \frac{-1}{\sin B}\right)\right)} \]

   2. add-sqr-sqrt 14.1%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sqrt{-\frac{x}{B}} \cdot \sqrt{-\frac{x}{B}}} + \frac{-1}{\sin B}\right)\right) \]

   3. sqrt-unprod 16.3%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sqrt{\left(-\frac{x}{B}\right) \cdot \left(-\frac{x}{B}\right)}} + \frac{-1}{\sin B}\right)\right) \]

   4. sqr-neg 16.3%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\sqrt{\color{blue}{\frac{x}{B} \cdot \frac{x}{B}}} + \frac{-1}{\sin B}\right)\right) \]

   5. sqrt-unprod 6.4%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\sqrt{\frac{x}{B}} \cdot \sqrt{\frac{x}{B}}} + \frac{-1}{\sin B}\right)\right) \]

   6. add-sqr-sqrt 9.4%

      \[\leadsto \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{\frac{x}{B}} + \frac{-1}{\sin B}\right)\right) \]

6. Applied egg-rr 9.4%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\frac{x}{B} + \frac{-1}{\sin B}\right)\right)} \]

7. Taylor expanded in x around inf 2.6%

   \[\leadsto \color{blue}{\frac{x}{B}} \]

8. Final simplification 2.6%

   \[\leadsto \frac{x}{B} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024081 
(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))))))