VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.2% → 99.7%

Time: 23.8s

Alternatives: 30

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: 76.2% 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, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -5 \cdot 10^{+77}:\\ \;\;\;\;t_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -5e+77)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 100000000.0)
       (- (* F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B))) (/ x (tan B)))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -5e+77) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 100000000.0) {
		tmp = (F * (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B))) - (x / tan(B));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -5e+77)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 100000000.0)
		tmp = Float64(Float64(F * Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B))) - Float64(x / tan(B)));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -5e+77], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 100000000.0], N[(N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.00000000000000004e77

   1. Initial program 56.4%

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

   2. Taylor expanded in F around -inf 99.8%

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

   if -5.00000000000000004e77 < F < 1e8

   1. Initial program 98.8%

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

      1. +-commutative 98.8%

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

      2. unsub-neg 98.8%

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

      3. associate-*l/ 99.6%

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

      4. associate-*r/ 99.6%

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

      5. *-commutative 99.6%

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

   3. Simplified 99.7%

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

   if 1e8 < F

   1. Initial program 61.1%

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

   2. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+77}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 2: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -6 \cdot 10^{+37}:\\ \;\;\;\;t_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 200000000:\\ \;\;\;\;t_0 + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -6e+37)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 200000000.0)
       (+ t_0 (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -6e+37) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 200000000.0) {
		tmp = t_0 + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-6d+37)) then
        tmp = t_0 + ((-1.0d0) / sin(b))
    else if (f <= 200000000.0d0) then
        tmp = t_0 + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -6e+37) {
		tmp = t_0 + (-1.0 / Math.sin(B));
	} else if (F <= 200000000.0) {
		tmp = t_0 + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -6e+37:
		tmp = t_0 + (-1.0 / math.sin(B))
	elif F <= 200000000.0:
		tmp = t_0 + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -6e+37)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 200000000.0)
		tmp = Float64(t_0 + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -6e+37)
		tmp = t_0 + (-1.0 / sin(B));
	elseif (F <= 200000000.0)
		tmp = t_0 + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -6e+37], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 200000000.0], N[(t$95$0 + 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]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.00000000000000043e37

   1. Initial program 62.9%

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

   2. Taylor expanded in F around -inf 99.7%

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

   if -6.00000000000000043e37 < F < 2e8

   1. Initial program 99.6%

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

   if 2e8 < F

   1. Initial program 61.1%

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

   2. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6 \cdot 10^{+37}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 200000000:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 3: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;t_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 0.62:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -1.42)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 0.62)
       (- (* F (/ (sqrt 0.5) (sin B))) (/ x (tan B)))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -1.42) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 0.62) {
		tmp = (F * (sqrt(0.5) / sin(B))) - (x / tan(B));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-1.42d0)) then
        tmp = t_0 + ((-1.0d0) / sin(b))
    else if (f <= 0.62d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - (x / tan(b))
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -1.42) {
		tmp = t_0 + (-1.0 / Math.sin(B));
	} else if (F <= 0.62) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - (x / Math.tan(B));
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -1.42:
		tmp = t_0 + (-1.0 / math.sin(B))
	elif F <= 0.62:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - (x / math.tan(B))
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -1.42)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 0.62)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - Float64(x / tan(B)));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -1.42)
		tmp = t_0 + (-1.0 / sin(B));
	elseif (F <= 0.62)
		tmp = (F * (sqrt(0.5) / sin(B))) - (x / tan(B));
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.42], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.62], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4199999999999999

   1. Initial program 65.6%

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

   2. Taylor expanded in F around -inf 98.6%

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

   if -1.4199999999999999 < F < 0.619999999999999996

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.6%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   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. Taylor expanded in F around 0 99.0%

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

   5. Taylor expanded in x around 0 99.0%

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

   if 0.619999999999999996 < F

   1. Initial program 62.1%

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

   2. Taylor expanded in F around inf 99.2%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 0.62:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 4: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;t_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 0.62:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -1.42)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 0.62)
       (- (/ (sqrt 0.5) (/ (sin B) F)) (/ x (tan B)))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -1.42) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 0.62) {
		tmp = (sqrt(0.5) / (sin(B) / F)) - (x / tan(B));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-1.42d0)) then
        tmp = t_0 + ((-1.0d0) / sin(b))
    else if (f <= 0.62d0) then
        tmp = (sqrt(0.5d0) / (sin(b) / f)) - (x / tan(b))
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -1.42) {
		tmp = t_0 + (-1.0 / Math.sin(B));
	} else if (F <= 0.62) {
		tmp = (Math.sqrt(0.5) / (Math.sin(B) / F)) - (x / Math.tan(B));
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -1.42:
		tmp = t_0 + (-1.0 / math.sin(B))
	elif F <= 0.62:
		tmp = (math.sqrt(0.5) / (math.sin(B) / F)) - (x / math.tan(B))
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -1.42)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 0.62)
		tmp = Float64(Float64(sqrt(0.5) / Float64(sin(B) / F)) - Float64(x / tan(B)));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -1.42)
		tmp = t_0 + (-1.0 / sin(B));
	elseif (F <= 0.62)
		tmp = (sqrt(0.5) / (sin(B) / F)) - (x / tan(B));
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.42], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.62], N[(N[(N[Sqrt[0.5], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4199999999999999

   1. Initial program 65.6%

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

   2. Taylor expanded in F around -inf 98.6%

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

   if -1.4199999999999999 < F < 0.619999999999999996

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.6%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   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. Taylor expanded in F around 0 99.0%

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

   5. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{\frac{\sqrt{0.5} \cdot F}{\sin B}} - \frac{x}{\tan B} \]
   6. Step-by-step derivation

      1. associate-/l* 99.0%

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

   7. Simplified 99.0%

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

   if 0.619999999999999996 < F

   1. Initial program 62.1%

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

   2. Taylor expanded in F around inf 99.2%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 0.62:\\ \;\;\;\;\frac{\sqrt{0.5}}{\frac{\sin B}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 5: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;t_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 0.62:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -1.42)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 0.62)
       (- (/ (* F (sqrt 0.5)) (sin B)) (/ x (tan B)))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -1.42) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 0.62) {
		tmp = ((F * sqrt(0.5)) / sin(B)) - (x / tan(B));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-1.42d0)) then
        tmp = t_0 + ((-1.0d0) / sin(b))
    else if (f <= 0.62d0) then
        tmp = ((f * sqrt(0.5d0)) / sin(b)) - (x / tan(b))
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -1.42) {
		tmp = t_0 + (-1.0 / Math.sin(B));
	} else if (F <= 0.62) {
		tmp = ((F * Math.sqrt(0.5)) / Math.sin(B)) - (x / Math.tan(B));
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -1.42:
		tmp = t_0 + (-1.0 / math.sin(B))
	elif F <= 0.62:
		tmp = ((F * math.sqrt(0.5)) / math.sin(B)) - (x / math.tan(B))
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -1.42)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 0.62)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) / sin(B)) - Float64(x / tan(B)));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -1.42)
		tmp = t_0 + (-1.0 / sin(B));
	elseif (F <= 0.62)
		tmp = ((F * sqrt(0.5)) / sin(B)) - (x / tan(B));
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.42], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.62], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4199999999999999

   1. Initial program 65.6%

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

   2. Taylor expanded in F around -inf 98.6%

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

   if -1.4199999999999999 < F < 0.619999999999999996

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.6%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   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. Taylor expanded in F around 0 99.0%

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

   5. Taylor expanded in x around 0 99.2%

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

   if 0.619999999999999996 < F

   1. Initial program 62.1%

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

   2. Taylor expanded in F around inf 99.2%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 0.62:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 6: 91.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{F}{\sin B}\\ t_1 := \sqrt{\frac{1}{2 + x \cdot 2}}\\ t_2 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -1.02 \cdot 10^{-23}:\\ \;\;\;\;t_2 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -2.5 \cdot 10^{-165}:\\ \;\;\;\;t_0 \cdot t_1 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{-109}:\\ \;\;\;\;t_1 \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 950000:\\ \;\;\;\;t_0 \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t_2 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ F (sin B)))
        (t_1 (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))))
        (t_2 (* x (/ -1.0 (tan B)))))
   (if (<= F -1.02e-23)
     (+ t_2 (/ -1.0 (sin B)))
     (if (<= F -2.5e-165)
       (- (* t_0 t_1) (/ x B))
       (if (<= F 4.1e-109)
         (- (* t_1 (/ F B)) (/ x (tan B)))
         (if (<= F 950000.0)
           (- (* t_0 (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)) (/ x B))
           (+ t_2 (/ 1.0 (sin B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = F / sin(B);
	double t_1 = sqrt((1.0 / (2.0 + (x * 2.0))));
	double t_2 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -1.02e-23) {
		tmp = t_2 + (-1.0 / sin(B));
	} else if (F <= -2.5e-165) {
		tmp = (t_0 * t_1) - (x / B);
	} else if (F <= 4.1e-109) {
		tmp = (t_1 * (F / B)) - (x / tan(B));
	} else if (F <= 950000.0) {
		tmp = (t_0 * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else {
		tmp = t_2 + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = f / sin(b)
    t_1 = sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))
    t_2 = x * ((-1.0d0) / tan(b))
    if (f <= (-1.02d-23)) then
        tmp = t_2 + ((-1.0d0) / sin(b))
    else if (f <= (-2.5d-165)) then
        tmp = (t_0 * t_1) - (x / b)
    else if (f <= 4.1d-109) then
        tmp = (t_1 * (f / b)) - (x / tan(b))
    else if (f <= 950000.0d0) then
        tmp = (t_0 * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    else
        tmp = t_2 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = F / Math.sin(B);
	double t_1 = Math.sqrt((1.0 / (2.0 + (x * 2.0))));
	double t_2 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -1.02e-23) {
		tmp = t_2 + (-1.0 / Math.sin(B));
	} else if (F <= -2.5e-165) {
		tmp = (t_0 * t_1) - (x / B);
	} else if (F <= 4.1e-109) {
		tmp = (t_1 * (F / B)) - (x / Math.tan(B));
	} else if (F <= 950000.0) {
		tmp = (t_0 * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else {
		tmp = t_2 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = F / math.sin(B)
	t_1 = math.sqrt((1.0 / (2.0 + (x * 2.0))))
	t_2 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -1.02e-23:
		tmp = t_2 + (-1.0 / math.sin(B))
	elif F <= -2.5e-165:
		tmp = (t_0 * t_1) - (x / B)
	elif F <= 4.1e-109:
		tmp = (t_1 * (F / B)) - (x / math.tan(B))
	elif F <= 950000.0:
		tmp = (t_0 * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B)
	else:
		tmp = t_2 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(F / sin(B))
	t_1 = sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))
	t_2 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -1.02e-23)
		tmp = Float64(t_2 + Float64(-1.0 / sin(B)));
	elseif (F <= -2.5e-165)
		tmp = Float64(Float64(t_0 * t_1) - Float64(x / B));
	elseif (F <= 4.1e-109)
		tmp = Float64(Float64(t_1 * Float64(F / B)) - Float64(x / tan(B)));
	elseif (F <= 950000.0)
		tmp = Float64(Float64(t_0 * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B));
	else
		tmp = Float64(t_2 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = F / sin(B);
	t_1 = sqrt((1.0 / (2.0 + (x * 2.0))));
	t_2 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -1.02e-23)
		tmp = t_2 + (-1.0 / sin(B));
	elseif (F <= -2.5e-165)
		tmp = (t_0 * t_1) - (x / B);
	elseif (F <= 4.1e-109)
		tmp = (t_1 * (F / B)) - (x / tan(B));
	elseif (F <= 950000.0)
		tmp = (t_0 * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (x / B);
	else
		tmp = t_2 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.02e-23], N[(t$95$2 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.5e-165], N[(N[(t$95$0 * t$95$1), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.1e-109], N[(N[(t$95$1 * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 950000.0], N[(N[(t$95$0 * 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], N[(t$95$2 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.02000000000000005e-23

   1. Initial program 67.0%

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

   2. Taylor expanded in F around -inf 97.4%

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

   if -1.02000000000000005e-23 < F < -2.4999999999999999e-165

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0 89.9%

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

   3. Taylor expanded in F around 0 89.9%

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

   if -2.4999999999999999e-165 < F < 4.1000000000000002e-109

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*l/ 99.6%

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

      4. associate-*r/ 99.6%

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

      5. *-commutative 99.6%

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

   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. Taylor expanded in F around 0 99.6%

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

   5. Taylor expanded in B around 0 87.4%

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

   if 4.1000000000000002e-109 < F < 9.5e5

   1. Initial program 99.6%

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

   2. Taylor expanded in B around 0 76.8%

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

   if 9.5e5 < F

   1. Initial program 61.1%

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

   2. Taylor expanded in F around inf 99.8%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.02 \cdot 10^{-23}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -2.5 \cdot 10^{-165}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.1 \cdot 10^{-109}:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 950000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 7: 90.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -1.02 \cdot 10^{-23}:\\ \;\;\;\;t_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-165}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.62:\\ \;\;\;\;t_0 + {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -1.02e-23)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F -2.1e-165)
       (- (* (/ F (sin B)) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) (/ x B))
       (if (<= F 0.62)
         (+ t_0 (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)))
         (+ t_0 (/ 1.0 (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -1.02e-23) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= -2.1e-165) {
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else if (F <= 0.62) {
		tmp = t_0 + (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-1.02d-23)) then
        tmp = t_0 + ((-1.0d0) / sin(b))
    else if (f <= (-2.1d-165)) then
        tmp = ((f / sin(b)) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - (x / b)
    else if (f <= 0.62d0) then
        tmp = t_0 + ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b))
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -1.02e-23) {
		tmp = t_0 + (-1.0 / Math.sin(B));
	} else if (F <= -2.1e-165) {
		tmp = ((F / Math.sin(B)) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else if (F <= 0.62) {
		tmp = t_0 + (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B));
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -1.02e-23:
		tmp = t_0 + (-1.0 / math.sin(B))
	elif F <= -2.1e-165:
		tmp = ((F / math.sin(B)) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B)
	elif F <= 0.62:
		tmp = t_0 + (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B))
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -1.02e-23)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= -2.1e-165)
		tmp = Float64(Float64(Float64(F / sin(B)) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - Float64(x / B));
	elseif (F <= 0.62)
		tmp = Float64(t_0 + Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -1.02e-23)
		tmp = t_0 + (-1.0 / sin(B));
	elseif (F <= -2.1e-165)
		tmp = ((F / sin(B)) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	elseif (F <= 0.62)
		tmp = t_0 + ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B));
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.02e-23], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.1e-165], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.62], N[(t$95$0 + N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.02000000000000005e-23

   1. Initial program 67.0%

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

   2. Taylor expanded in F around -inf 97.4%

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

   if -1.02000000000000005e-23 < F < -2.09999999999999995e-165

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0 89.9%

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

   3. Taylor expanded in F around 0 89.9%

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

   if -2.09999999999999995e-165 < F < 0.619999999999999996

   1. Initial program 99.6%

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

   2. Taylor expanded in B around 0 78.8%

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

   if 0.619999999999999996 < F

   1. Initial program 62.1%

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

   2. Taylor expanded in F around inf 99.2%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.02 \cdot 10^{-23}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-165}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.62:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 8: 90.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{2 + x \cdot 2}}\\ t_1 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -1.02 \cdot 10^{-23}:\\ \;\;\;\;t_1 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -3.4 \cdot 10^{-165}:\\ \;\;\;\;\frac{F}{\sin B} \cdot t_0 - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.12:\\ \;\;\;\;t_0 \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;t_1 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) (t_1 (* x (/ -1.0 (tan B)))))
   (if (<= F -1.02e-23)
     (+ t_1 (/ -1.0 (sin B)))
     (if (<= F -3.4e-165)
       (- (* (/ F (sin B)) t_0) (/ x B))
       (if (<= F 0.12)
         (- (* t_0 (/ F B)) (/ x (tan B)))
         (+ t_1 (/ 1.0 (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = sqrt((1.0 / (2.0 + (x * 2.0))));
	double t_1 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -1.02e-23) {
		tmp = t_1 + (-1.0 / sin(B));
	} else if (F <= -3.4e-165) {
		tmp = ((F / sin(B)) * t_0) - (x / B);
	} else if (F <= 0.12) {
		tmp = (t_0 * (F / B)) - (x / tan(B));
	} else {
		tmp = t_1 + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))
    t_1 = x * ((-1.0d0) / tan(b))
    if (f <= (-1.02d-23)) then
        tmp = t_1 + ((-1.0d0) / sin(b))
    else if (f <= (-3.4d-165)) then
        tmp = ((f / sin(b)) * t_0) - (x / b)
    else if (f <= 0.12d0) then
        tmp = (t_0 * (f / b)) - (x / tan(b))
    else
        tmp = t_1 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.sqrt((1.0 / (2.0 + (x * 2.0))));
	double t_1 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -1.02e-23) {
		tmp = t_1 + (-1.0 / Math.sin(B));
	} else if (F <= -3.4e-165) {
		tmp = ((F / Math.sin(B)) * t_0) - (x / B);
	} else if (F <= 0.12) {
		tmp = (t_0 * (F / B)) - (x / Math.tan(B));
	} else {
		tmp = t_1 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.sqrt((1.0 / (2.0 + (x * 2.0))))
	t_1 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -1.02e-23:
		tmp = t_1 + (-1.0 / math.sin(B))
	elif F <= -3.4e-165:
		tmp = ((F / math.sin(B)) * t_0) - (x / B)
	elif F <= 0.12:
		tmp = (t_0 * (F / B)) - (x / math.tan(B))
	else:
		tmp = t_1 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))
	t_1 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -1.02e-23)
		tmp = Float64(t_1 + Float64(-1.0 / sin(B)));
	elseif (F <= -3.4e-165)
		tmp = Float64(Float64(Float64(F / sin(B)) * t_0) - Float64(x / B));
	elseif (F <= 0.12)
		tmp = Float64(Float64(t_0 * Float64(F / B)) - Float64(x / tan(B)));
	else
		tmp = Float64(t_1 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = sqrt((1.0 / (2.0 + (x * 2.0))));
	t_1 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -1.02e-23)
		tmp = t_1 + (-1.0 / sin(B));
	elseif (F <= -3.4e-165)
		tmp = ((F / sin(B)) * t_0) - (x / B);
	elseif (F <= 0.12)
		tmp = (t_0 * (F / B)) - (x / tan(B));
	else
		tmp = t_1 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.02e-23], N[(t$95$1 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.4e-165], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.12], N[(N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.02000000000000005e-23

   1. Initial program 67.0%

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

   2. Taylor expanded in F around -inf 97.4%

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

   if -1.02000000000000005e-23 < F < -3.4e-165

   1. Initial program 99.4%

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

   2. Taylor expanded in B around 0 89.9%

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

   3. Taylor expanded in F around 0 89.9%

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

   if -3.4e-165 < F < 0.12

   1. Initial program 99.6%

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

      1. +-commutative 99.6%

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

      2. unsub-neg 99.6%

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

      3. associate-*l/ 99.6%

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

      4. associate-*r/ 99.6%

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

      5. *-commutative 99.6%

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

   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. Taylor expanded in F around 0 98.9%

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

   5. Taylor expanded in B around 0 78.2%

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

   if 0.12 < F

   1. Initial program 62.1%

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

   2. Taylor expanded in F around inf 99.2%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.02 \cdot 10^{-23}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -3.4 \cdot 10^{-165}:\\ \;\;\;\;\frac{F}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.12:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 9: 90.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -1.1 \cdot 10^{-40}:\\ \;\;\;\;t_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 0.062:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -1.1e-40)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F 0.062)
       (- (* (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (/ F B)) (/ x (tan B)))
       (+ t_0 (/ 1.0 (sin B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -1.1e-40) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= 0.062) {
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - (x / tan(B));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-1.1d-40)) then
        tmp = t_0 + ((-1.0d0) / sin(b))
    else if (f <= 0.062d0) then
        tmp = (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) * (f / b)) - (x / tan(b))
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -1.1e-40) {
		tmp = t_0 + (-1.0 / Math.sin(B));
	} else if (F <= 0.062) {
		tmp = (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - (x / Math.tan(B));
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -1.1e-40:
		tmp = t_0 + (-1.0 / math.sin(B))
	elif F <= 0.062:
		tmp = (math.sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - (x / math.tan(B))
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -1.1e-40)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= 0.062)
		tmp = Float64(Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) * Float64(F / B)) - Float64(x / tan(B)));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -1.1e-40)
		tmp = t_0 + (-1.0 / sin(B));
	elseif (F <= 0.062)
		tmp = (sqrt((1.0 / (2.0 + (x * 2.0)))) * (F / B)) - (x / tan(B));
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.1e-40], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.062], N[(N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.10000000000000004e-40

   1. Initial program 67.9%

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

   2. Taylor expanded in F around -inf 95.0%

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

   if -1.10000000000000004e-40 < F < 0.062

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. unsub-neg 99.5%

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

      3. associate-*l/ 99.6%

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

      4. associate-*r/ 99.5%

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

      5. *-commutative 99.5%

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

   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. Taylor expanded in F around 0 99.1%

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

   5. Taylor expanded in B around 0 72.9%

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

   if 0.062 < F

   1. Initial program 62.1%

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

   2. Taylor expanded in F around inf 99.2%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.1 \cdot 10^{-40}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 0.062:\\ \;\;\;\;\sqrt{\frac{1}{2 + x \cdot 2}} \cdot \frac{F}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 10: 84.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{-41}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.15 \cdot 10^{-108}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \left(\frac{F}{B} + 0.16666666666666666 \cdot \left(F \cdot B\right)\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.75 \cdot 10^{-58}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8e-41)
   (+ (* x (/ -1.0 (tan B))) (/ -1.0 (sin B)))
   (if (<= F -1.15e-108)
     (-
      (*
       (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)
       (+ (/ F B) (* 0.16666666666666666 (* F B))))
      (/ x B))
     (if (<= F 2.75e-58)
       (* (cos B) (/ (- x) (sin B)))
       (- (/ F (* F (sin B))) (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8e-41) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / sin(B));
	} else if (F <= -1.15e-108) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * ((F / B) + (0.16666666666666666 * (F * B)))) - (x / B);
	} else if (F <= 2.75e-58) {
		tmp = cos(B) * (-x / sin(B));
	} else {
		tmp = (F / (F * 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 <= (-8d-41)) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / sin(b))
    else if (f <= (-1.15d-108)) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * ((f / b) + (0.16666666666666666d0 * (f * b)))) - (x / b)
    else if (f <= 2.75d-58) then
        tmp = cos(b) * (-x / sin(b))
    else
        tmp = (f / (f * 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 <= -8e-41) {
		tmp = (x * (-1.0 / Math.tan(B))) + (-1.0 / Math.sin(B));
	} else if (F <= -1.15e-108) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * ((F / B) + (0.16666666666666666 * (F * B)))) - (x / B);
	} else if (F <= 2.75e-58) {
		tmp = Math.cos(B) * (-x / Math.sin(B));
	} else {
		tmp = (F / (F * Math.sin(B))) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -8e-41:
		tmp = (x * (-1.0 / math.tan(B))) + (-1.0 / math.sin(B))
	elif F <= -1.15e-108:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * ((F / B) + (0.16666666666666666 * (F * B)))) - (x / B)
	elif F <= 2.75e-58:
		tmp = math.cos(B) * (-x / math.sin(B))
	else:
		tmp = (F / (F * math.sin(B))) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8e-41)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / sin(B)));
	elseif (F <= -1.15e-108)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(Float64(F / B) + Float64(0.16666666666666666 * Float64(F * B)))) - Float64(x / B));
	elseif (F <= 2.75e-58)
		tmp = Float64(cos(B) * Float64(Float64(-x) / sin(B)));
	else
		tmp = Float64(Float64(F / Float64(F * sin(B))) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -8e-41)
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / sin(B));
	elseif (F <= -1.15e-108)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * ((F / B) + (0.16666666666666666 * (F * B)))) - (x / B);
	elseif (F <= 2.75e-58)
		tmp = cos(B) * (-x / sin(B));
	else
		tmp = (F / (F * sin(B))) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8e-41], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.15e-108], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(N[(F / B), $MachinePrecision] + N[(0.16666666666666666 * N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.75e-58], N[(N[Cos[B], $MachinePrecision] * N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(F / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -8.00000000000000005e-41

   1. Initial program 67.9%

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

   2. Taylor expanded in F around -inf 95.0%

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

   if -8.00000000000000005e-41 < F < -1.14999999999999998e-108

   1. Initial program 99.6%

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

   2. Taylor expanded in B around 0 99.7%

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

   3. Taylor expanded in B around 0 63.9%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\left(0.16666666666666666 \cdot \left(F \cdot B\right) + \frac{F}{B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if -1.14999999999999998e-108 < F < 2.74999999999999998e-58

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. fma-def 99.6%

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

      3. +-commutative 99.6%

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

      4. *-commutative 99.6%

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

      5. fma-def 99.6%

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

      6. fma-def 99.6%

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

      7. metadata-eval 99.6%

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

      8. metadata-eval 99.6%

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

      9. distribute-lft-neg-in 99.6%

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

      10. associate-*r/ 99.7%

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

      11. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 67.6%

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

      1. mul-1-neg 67.6%

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

      2. *-commutative 67.6%

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

      3. associate-*l/ 67.7%

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

      4. *-commutative 67.7%

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

   6. Simplified 67.7%

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

   if 2.74999999999999998e-58 < F

   1. Initial program 67.5%

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

      1. +-commutative 67.5%

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

      2. unsub-neg 67.5%

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

      3. associate-*l/ 77.0%

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

      4. associate-*r/ 77.0%

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

      5. *-commutative 77.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in F around inf 91.2%

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

      1. un-div-inv 91.3%

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

      2. *-commutative 91.3%

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

   6. Applied egg-rr 91.3%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{-41}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.15 \cdot 10^{-108}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \left(\frac{F}{B} + 0.16666666666666666 \cdot \left(F \cdot B\right)\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.75 \cdot 10^{-58}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 11: 84.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{-1}{\tan B}\\ \mathbf{if}\;F \leq -1.1 \cdot 10^{-40}:\\ \;\;\;\;t_0 + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.05 \cdot 10^{-106}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \left(\frac{F}{B} + 0.16666666666666666 \cdot \left(F \cdot B\right)\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.75 \cdot 10^{-58}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;t_0 + \frac{1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (* x (/ -1.0 (tan B)))))
   (if (<= F -1.1e-40)
     (+ t_0 (/ -1.0 (sin B)))
     (if (<= F -1.05e-106)
       (-
        (*
         (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)
         (+ (/ F B) (* 0.16666666666666666 (* F B))))
        (/ x B))
       (if (<= F 2.75e-58)
         (* (cos B) (/ (- x) (sin B)))
         (+ t_0 (/ 1.0 (sin B))))))))

C

double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / tan(B));
	double tmp;
	if (F <= -1.1e-40) {
		tmp = t_0 + (-1.0 / sin(B));
	} else if (F <= -1.05e-106) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * ((F / B) + (0.16666666666666666 * (F * B)))) - (x / B);
	} else if (F <= 2.75e-58) {
		tmp = cos(B) * (-x / sin(B));
	} else {
		tmp = t_0 + (1.0 / sin(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * ((-1.0d0) / tan(b))
    if (f <= (-1.1d-40)) then
        tmp = t_0 + ((-1.0d0) / sin(b))
    else if (f <= (-1.05d-106)) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * ((f / b) + (0.16666666666666666d0 * (f * b)))) - (x / b)
    else if (f <= 2.75d-58) then
        tmp = cos(b) * (-x / sin(b))
    else
        tmp = t_0 + (1.0d0 / sin(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x * (-1.0 / Math.tan(B));
	double tmp;
	if (F <= -1.1e-40) {
		tmp = t_0 + (-1.0 / Math.sin(B));
	} else if (F <= -1.05e-106) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * ((F / B) + (0.16666666666666666 * (F * B)))) - (x / B);
	} else if (F <= 2.75e-58) {
		tmp = Math.cos(B) * (-x / Math.sin(B));
	} else {
		tmp = t_0 + (1.0 / Math.sin(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x * (-1.0 / math.tan(B))
	tmp = 0
	if F <= -1.1e-40:
		tmp = t_0 + (-1.0 / math.sin(B))
	elif F <= -1.05e-106:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * ((F / B) + (0.16666666666666666 * (F * B)))) - (x / B)
	elif F <= 2.75e-58:
		tmp = math.cos(B) * (-x / math.sin(B))
	else:
		tmp = t_0 + (1.0 / math.sin(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x * Float64(-1.0 / tan(B)))
	tmp = 0.0
	if (F <= -1.1e-40)
		tmp = Float64(t_0 + Float64(-1.0 / sin(B)));
	elseif (F <= -1.05e-106)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(Float64(F / B) + Float64(0.16666666666666666 * Float64(F * B)))) - Float64(x / B));
	elseif (F <= 2.75e-58)
		tmp = Float64(cos(B) * Float64(Float64(-x) / sin(B)));
	else
		tmp = Float64(t_0 + Float64(1.0 / sin(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x * (-1.0 / tan(B));
	tmp = 0.0;
	if (F <= -1.1e-40)
		tmp = t_0 + (-1.0 / sin(B));
	elseif (F <= -1.05e-106)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * ((F / B) + (0.16666666666666666 * (F * B)))) - (x / B);
	elseif (F <= 2.75e-58)
		tmp = cos(B) * (-x / sin(B));
	else
		tmp = t_0 + (1.0 / sin(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.1e-40], N[(t$95$0 + N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.05e-106], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(N[(F / B), $MachinePrecision] + N[(0.16666666666666666 * N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.75e-58], N[(N[Cos[B], $MachinePrecision] * N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 + N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.10000000000000004e-40

   1. Initial program 67.9%

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

   2. Taylor expanded in F around -inf 95.0%

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

   if -1.10000000000000004e-40 < F < -1.05000000000000002e-106

   1. Initial program 99.6%

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

   2. Taylor expanded in B around 0 99.7%

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

   3. Taylor expanded in B around 0 63.9%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\left(0.16666666666666666 \cdot \left(F \cdot B\right) + \frac{F}{B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if -1.05000000000000002e-106 < F < 2.74999999999999998e-58

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. fma-def 99.6%

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

      3. +-commutative 99.6%

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

      4. *-commutative 99.6%

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

      5. fma-def 99.6%

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

      6. fma-def 99.6%

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

      7. metadata-eval 99.6%

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

      8. metadata-eval 99.6%

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

      9. distribute-lft-neg-in 99.6%

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

      10. associate-*r/ 99.7%

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

      11. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 67.6%

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

      1. mul-1-neg 67.6%

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

      2. *-commutative 67.6%

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

      3. associate-*l/ 67.7%

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

      4. *-commutative 67.7%

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

   6. Simplified 67.7%

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

   if 2.74999999999999998e-58 < F

   1. Initial program 67.5%

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

   2. Taylor expanded in F around inf 91.3%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.1 \cdot 10^{-40}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq -1.05 \cdot 10^{-106}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \left(\frac{F}{B} + 0.16666666666666666 \cdot \left(F \cdot B\right)\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.75 \cdot 10^{-58}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\sin B}\\ \end{array} \]

Alternative 12: 77.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.2 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{-59}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.2e+28)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 5.4e-59)
     (* (cos B) (/ (- x) (sin B)))
     (- (/ F (* F (sin B))) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.2e+28) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 5.4e-59) {
		tmp = cos(B) * (-x / sin(B));
	} else {
		tmp = (F / (F * 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 <= (-7.2d+28)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 5.4d-59) then
        tmp = cos(b) * (-x / sin(b))
    else
        tmp = (f / (f * 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 <= -7.2e+28) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 5.4e-59) {
		tmp = Math.cos(B) * (-x / Math.sin(B));
	} else {
		tmp = (F / (F * Math.sin(B))) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -7.2e+28:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 5.4e-59:
		tmp = math.cos(B) * (-x / math.sin(B))
	else:
		tmp = (F / (F * math.sin(B))) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.2e+28)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 5.4e-59)
		tmp = Float64(cos(B) * Float64(Float64(-x) / sin(B)));
	else
		tmp = Float64(Float64(F / Float64(F * sin(B))) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -7.2e+28)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 5.4e-59)
		tmp = cos(B) * (-x / sin(B));
	else
		tmp = (F / (F * sin(B))) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.2e+28], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.4e-59], N[(N[Cos[B], $MachinePrecision] * N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(F / N[(F * N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -7.1999999999999999e28

   1. Initial program 63.5%

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

   2. Taylor expanded in B around 0 43.3%

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

   3. Taylor expanded in F around -inf 79.1%

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

   if -7.1999999999999999e28 < F < 5.3999999999999998e-59

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. fma-def 99.6%

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

      3. +-commutative 99.6%

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

      4. *-commutative 99.6%

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

      5. fma-def 99.6%

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

      6. fma-def 99.6%

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

      7. metadata-eval 99.6%

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

      8. metadata-eval 99.6%

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

      9. distribute-lft-neg-in 99.6%

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

      10. associate-*r/ 99.7%

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

      11. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 61.7%

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

      1. mul-1-neg 61.7%

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

      2. *-commutative 61.7%

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

      3. associate-*l/ 61.8%

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

      4. *-commutative 61.8%

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

   6. Simplified 61.8%

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

   if 5.3999999999999998e-59 < F

   1. Initial program 67.5%

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

      1. +-commutative 67.5%

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

      2. unsub-neg 67.5%

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

      3. associate-*l/ 77.0%

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

      4. associate-*r/ 77.0%

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

      5. *-commutative 77.0%

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

   3. Simplified 77.0%

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

   4. Taylor expanded in F around inf 91.2%

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

      1. un-div-inv 91.3%

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

      2. *-commutative 91.3%

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

   6. Applied egg-rr 91.3%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.2 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.4 \cdot 10^{-59}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{F}{F \cdot \sin B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 13: 70.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.2 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{-14}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{elif}\;F \leq 10^{+181}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.2e+28)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 4.3e-14)
     (* (cos B) (/ (- x) (sin B)))
     (if (<= F 1e+181)
       (- (/ 1.0 (sin B)) (/ x B))
       (- (/ 1.0 B) (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.2e+28) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 4.3e-14) {
		tmp = cos(B) * (-x / sin(B));
	} else if (F <= 1e+181) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = (1.0 / 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 <= (-7.2d+28)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 4.3d-14) then
        tmp = cos(b) * (-x / sin(b))
    else if (f <= 1d+181) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = (1.0d0 / 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 <= -7.2e+28) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 4.3e-14) {
		tmp = Math.cos(B) * (-x / Math.sin(B));
	} else if (F <= 1e+181) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -7.2e+28:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 4.3e-14:
		tmp = math.cos(B) * (-x / math.sin(B))
	elif F <= 1e+181:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.2e+28)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 4.3e-14)
		tmp = Float64(cos(B) * Float64(Float64(-x) / sin(B)));
	elseif (F <= 1e+181)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -7.2e+28)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 4.3e-14)
		tmp = cos(B) * (-x / sin(B));
	elseif (F <= 1e+181)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.2e+28], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.3e-14], N[(N[Cos[B], $MachinePrecision] * N[((-x) / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e+181], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7.1999999999999999e28

   1. Initial program 63.5%

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

   2. Taylor expanded in B around 0 43.3%

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

   3. Taylor expanded in F around -inf 79.1%

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

   if -7.1999999999999999e28 < F < 4.29999999999999998e-14

   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. Step-by-step derivation

      1. +-commutative 99.5%

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

      2. fma-def 99.6%

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

      3. +-commutative 99.6%

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

      4. *-commutative 99.6%

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

      5. fma-def 99.6%

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

      6. fma-def 99.6%

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

      7. metadata-eval 99.6%

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

      8. metadata-eval 99.6%

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

      9. distribute-lft-neg-in 99.6%

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

      10. associate-*r/ 99.7%

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

      11. *-rgt-identity 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in F around 0 61.0%

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

      1. mul-1-neg 61.0%

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

      2. *-commutative 61.0%

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

      3. associate-*l/ 61.0%

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

      4. *-commutative 61.0%

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

   6. Simplified 61.0%

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

   if 4.29999999999999998e-14 < F < 9.9999999999999992e180

   1. Initial program 80.7%

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

   2. Taylor expanded in B around 0 65.6%

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

   3. Taylor expanded in F around inf 77.6%

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

   if 9.9999999999999992e180 < F

   1. Initial program 32.4%

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

      1. +-commutative 32.4%

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

      2. unsub-neg 32.4%

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

      3. associate-*l/ 50.1%

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

      4. associate-*r/ 50.1%

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

      5. *-commutative 50.1%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in F around inf 99.7%

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

   5. Taylor expanded in B around 0 93.0%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.2 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{-14}:\\ \;\;\;\;\cos B \cdot \frac{-x}{\sin B}\\ \mathbf{elif}\;F \leq 10^{+181}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 14: 64.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.2 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.62:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \left(\frac{F}{B} + 0.16666666666666666 \cdot \left(F \cdot B\right)\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{+180}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.2e+28)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 0.62)
     (-
      (*
       (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)
       (+ (/ F B) (* 0.16666666666666666 (* F B))))
      (/ x B))
     (if (<= F 9.5e+180)
       (- (/ 1.0 (sin B)) (/ x B))
       (- (/ 1.0 B) (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.2e+28) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 0.62) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * ((F / B) + (0.16666666666666666 * (F * B)))) - (x / B);
	} else if (F <= 9.5e+180) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = (1.0 / 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 <= (-7.2d+28)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 0.62d0) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * ((f / b) + (0.16666666666666666d0 * (f * b)))) - (x / b)
    else if (f <= 9.5d+180) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = (1.0d0 / 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 <= -7.2e+28) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 0.62) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * ((F / B) + (0.16666666666666666 * (F * B)))) - (x / B);
	} else if (F <= 9.5e+180) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -7.2e+28:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 0.62:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * ((F / B) + (0.16666666666666666 * (F * B)))) - (x / B)
	elif F <= 9.5e+180:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.2e+28)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 0.62)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(Float64(F / B) + Float64(0.16666666666666666 * Float64(F * B)))) - Float64(x / B));
	elseif (F <= 9.5e+180)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -7.2e+28)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 0.62)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * ((F / B) + (0.16666666666666666 * (F * B)))) - (x / B);
	elseif (F <= 9.5e+180)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.2e+28], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.62], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(N[(F / B), $MachinePrecision] + N[(0.16666666666666666 * N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.5e+180], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7.1999999999999999e28

   1. Initial program 63.5%

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

   2. Taylor expanded in B around 0 43.3%

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

   3. Taylor expanded in F around -inf 79.1%

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

   if -7.1999999999999999e28 < F < 0.619999999999999996

   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. Taylor expanded in B around 0 72.1%

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

   3. Taylor expanded in B around 0 46.6%

      \[\leadsto \left(-\frac{x}{B}\right) + \color{blue}{\left(0.16666666666666666 \cdot \left(F \cdot B\right) + \frac{F}{B}\right)} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{\left(-\frac{1}{2}\right)} \]

   if 0.619999999999999996 < F < 9.5000000000000003e180

   1. Initial program 78.8%

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

   2. Taylor expanded in B around 0 62.1%

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

   3. Taylor expanded in F around inf 82.5%

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

   if 9.5000000000000003e180 < F

   1. Initial program 32.4%

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

      1. +-commutative 32.4%

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

      2. unsub-neg 32.4%

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

      3. associate-*l/ 50.1%

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

      4. associate-*r/ 50.1%

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

      5. *-commutative 50.1%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in F around inf 99.7%

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

   5. Taylor expanded in B around 0 93.0%

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

4. Final simplification 66.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.2 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.62:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \left(\frac{F}{B} + 0.16666666666666666 \cdot \left(F \cdot B\right)\right) - \frac{x}{B}\\ \mathbf{elif}\;F \leq 9.5 \cdot 10^{+180}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 15: 64.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.2 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.62:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{1}{\frac{B}{F}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{+180}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.2e+28)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 0.62)
     (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ 1.0 (/ B F))) (/ x B))
     (if (<= F 5.2e+180)
       (- (/ 1.0 (sin B)) (/ x B))
       (- (/ 1.0 B) (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.2e+28) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 0.62) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (1.0 / (B / F))) - (x / B);
	} else if (F <= 5.2e+180) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = (1.0 / 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 <= (-7.2d+28)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 0.62d0) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (1.0d0 / (b / f))) - (x / b)
    else if (f <= 5.2d+180) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = (1.0d0 / 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 <= -7.2e+28) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 0.62) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (1.0 / (B / F))) - (x / B);
	} else if (F <= 5.2e+180) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -7.2e+28:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 0.62:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (1.0 / (B / F))) - (x / B)
	elif F <= 5.2e+180:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.2e+28)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 0.62)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(1.0 / Float64(B / F))) - Float64(x / B));
	elseif (F <= 5.2e+180)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -7.2e+28)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 0.62)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (1.0 / (B / F))) - (x / B);
	elseif (F <= 5.2e+180)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.2e+28], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.62], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(1.0 / N[(B / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.2e+180], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7.1999999999999999e28

   1. Initial program 63.5%

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

   2. Taylor expanded in B around 0 43.3%

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

   3. Taylor expanded in F around -inf 79.1%

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

   if -7.1999999999999999e28 < F < 0.619999999999999996

   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. Step-by-step derivation

      1. clear-num 99.5%

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

      2. inv-pow 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. unpow-1 99.5%

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

   5. Simplified 99.5%

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

   6. Taylor expanded in B around 0 72.0%

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

   7. Taylor expanded in B around 0 46.4%

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

   if 0.619999999999999996 < F < 5.20000000000000042e180

   1. Initial program 78.8%

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

   2. Taylor expanded in B around 0 62.1%

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

   3. Taylor expanded in F around inf 82.5%

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

   if 5.20000000000000042e180 < F

   1. Initial program 32.4%

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

      1. +-commutative 32.4%

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

      2. unsub-neg 32.4%

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

      3. associate-*l/ 50.1%

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

      4. associate-*r/ 50.1%

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

      5. *-commutative 50.1%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in F around inf 99.7%

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

   5. Taylor expanded in B around 0 93.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.2 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.62:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{1}{\frac{B}{F}} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{+180}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 16: 64.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.2 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.62:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{+180}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.2e+28)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 0.62)
     (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)) (/ x B))
     (if (<= F 5.6e+180)
       (- (/ 1.0 (sin B)) (/ x B))
       (- (/ 1.0 B) (/ x (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.2e+28) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 0.62) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 5.6e+180) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = (1.0 / 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 <= (-7.2d+28)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 0.62d0) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else if (f <= 5.6d+180) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = (1.0d0 / 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 <= -7.2e+28) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 0.62) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else if (F <= 5.6e+180) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -7.2e+28:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 0.62:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	elif F <= 5.6e+180:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.2e+28)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 0.62)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	elseif (F <= 5.6e+180)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -7.2e+28)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 0.62)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	elseif (F <= 5.6e+180)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.2e+28], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.62], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.6e+180], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7.1999999999999999e28

   1. Initial program 63.5%

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

   2. Taylor expanded in B around 0 43.3%

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

   3. Taylor expanded in F around -inf 79.1%

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

   if -7.1999999999999999e28 < F < 0.619999999999999996

   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. Taylor expanded in B around 0 72.1%

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

   3. Taylor expanded in B around 0 46.4%

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

   if 0.619999999999999996 < F < 5.60000000000000024e180

   1. Initial program 78.8%

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

   2. Taylor expanded in B around 0 62.1%

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

   3. Taylor expanded in F around inf 82.5%

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

   if 5.60000000000000024e180 < F

   1. Initial program 32.4%

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

      1. +-commutative 32.4%

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

      2. unsub-neg 32.4%

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

      3. associate-*l/ 50.1%

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

      4. associate-*r/ 50.1%

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

      5. *-commutative 50.1%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in F around inf 99.7%

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

   5. Taylor expanded in B around 0 93.0%

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

4. Final simplification 66.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.2 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 0.62:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{+180}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 17: 61.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -7.2 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -5.1 \cdot 10^{-241}:\\ \;\;\;\;\frac{F}{B} \cdot \frac{-1}{F} - t_0\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{-175}:\\ \;\;\;\;\cos B \cdot \left(0.16666666666666666 \cdot \left(-x \cdot B\right) - \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\frac{F}{B}}{-F}\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{+180}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -7.2e+28)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -5.1e-241)
       (- (* (/ F B) (/ -1.0 F)) t_0)
       (if (<= F 5.6e-175)
         (* (cos B) (- (* 0.16666666666666666 (- (* x B))) (/ x B)))
         (if (<= F 4.3e-14)
           (+ (* x (/ -1.0 (tan B))) (/ (/ F B) (- F)))
           (if (<= F 1.65e+180)
             (- (/ 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 <= -7.2e+28) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -5.1e-241) {
		tmp = ((F / B) * (-1.0 / F)) - t_0;
	} else if (F <= 5.6e-175) {
		tmp = cos(B) * ((0.16666666666666666 * -(x * B)) - (x / B));
	} else if (F <= 4.3e-14) {
		tmp = (x * (-1.0 / tan(B))) + ((F / B) / -F);
	} else if (F <= 1.65e+180) {
		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 <= (-7.2d+28)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-5.1d-241)) then
        tmp = ((f / b) * ((-1.0d0) / f)) - t_0
    else if (f <= 5.6d-175) then
        tmp = cos(b) * ((0.16666666666666666d0 * -(x * b)) - (x / b))
    else if (f <= 4.3d-14) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((f / b) / -f)
    else if (f <= 1.65d+180) 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 <= -7.2e+28) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -5.1e-241) {
		tmp = ((F / B) * (-1.0 / F)) - t_0;
	} else if (F <= 5.6e-175) {
		tmp = Math.cos(B) * ((0.16666666666666666 * -(x * B)) - (x / B));
	} else if (F <= 4.3e-14) {
		tmp = (x * (-1.0 / Math.tan(B))) + ((F / B) / -F);
	} else if (F <= 1.65e+180) {
		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 <= -7.2e+28:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -5.1e-241:
		tmp = ((F / B) * (-1.0 / F)) - t_0
	elif F <= 5.6e-175:
		tmp = math.cos(B) * ((0.16666666666666666 * -(x * B)) - (x / B))
	elif F <= 4.3e-14:
		tmp = (x * (-1.0 / math.tan(B))) + ((F / B) / -F)
	elif F <= 1.65e+180:
		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 <= -7.2e+28)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -5.1e-241)
		tmp = Float64(Float64(Float64(F / B) * Float64(-1.0 / F)) - t_0);
	elseif (F <= 5.6e-175)
		tmp = Float64(cos(B) * Float64(Float64(0.16666666666666666 * Float64(-Float64(x * B))) - Float64(x / B)));
	elseif (F <= 4.3e-14)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / B) / Float64(-F)));
	elseif (F <= 1.65e+180)
		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 <= -7.2e+28)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -5.1e-241)
		tmp = ((F / B) * (-1.0 / F)) - t_0;
	elseif (F <= 5.6e-175)
		tmp = cos(B) * ((0.16666666666666666 * -(x * B)) - (x / B));
	elseif (F <= 4.3e-14)
		tmp = (x * (-1.0 / tan(B))) + ((F / B) / -F);
	elseif (F <= 1.65e+180)
		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, -7.2e+28], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -5.1e-241], N[(N[(N[(F / B), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 5.6e-175], N[(N[Cos[B], $MachinePrecision] * N[(N[(0.16666666666666666 * (-N[(x * B), $MachinePrecision])), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.3e-14], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] / (-F)), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.65e+180], 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 6 regimes

2. if F < -7.1999999999999999e28

   1. Initial program 63.5%

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

   2. Taylor expanded in B around 0 43.3%

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

   3. Taylor expanded in F around -inf 79.1%

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

   if -7.1999999999999999e28 < F < -5.0999999999999998e-241

   1. Initial program 99.4%

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

   2. Taylor expanded in F around -inf 34.3%

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

   3. Taylor expanded in B around 0 40.4%

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

      1. div-inv 40.4%

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

      2. expm1-log1p-u 26.4%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{F}{B} \cdot \frac{-1}{F} \]

      3. expm1-udef 20.1%

         \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{F}{B} \cdot \frac{-1}{F} \]

   5. Applied egg-rr 20.1%

      \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{F}{B} \cdot \frac{-1}{F} \]
   6. Step-by-step derivation

      1. expm1-def 26.4%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{F}{B} \cdot \frac{-1}{F} \]

      2. expm1-log1p 40.4%

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

   7. Simplified 40.4%

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

   if -5.0999999999999998e-241 < F < 5.6e-175

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. fma-def 99.7%

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

      3. +-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. fma-def 99.7%

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

      6. fma-def 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

      9. distribute-lft-neg-in 99.7%

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

      10. associate-*r/ 99.9%

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

      11. *-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in F around 0 90.5%

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

      1. mul-1-neg 90.5%

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

      2. *-commutative 90.5%

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

      3. associate-*l/ 90.6%

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

      4. *-commutative 90.6%

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

   6. Simplified 90.6%

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

   7. Taylor expanded in B around 0 59.5%

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

   if 5.6e-175 < F < 4.29999999999999998e-14

   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. Taylor expanded in F around -inf 32.6%

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

      1. expm1-log1p-u 18.9%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{F}{\sin B} \cdot \frac{-1}{F}\right)\right)} \]

      2. expm1-udef 18.9%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{F}{\sin B} \cdot \frac{-1}{F}\right)} - 1\right)} \]

      3. *-commutative 18.9%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{-1}{F} \cdot \frac{F}{\sin B}}\right)} - 1\right) \]

      4. frac-2neg 18.9%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{--1}{-F}} \cdot \frac{F}{\sin B}\right)} - 1\right) \]

      5. metadata-eval 18.9%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{-F} \cdot \frac{F}{\sin B}\right)} - 1\right) \]

      6. frac-times 18.9%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot F}{\left(-F\right) \cdot \sin B}}\right)} - 1\right) \]

      7. *-un-lft-identity 18.9%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{F}}{\left(-F\right) \cdot \sin B}\right)} - 1\right) \]

   4. Applied egg-rr 18.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{F}{\left(-F\right) \cdot \sin B}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 18.9%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{F}{\left(-F\right) \cdot \sin B}\right)\right)} \]

      2. expm1-log1p 32.6%

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

      3. *-commutative 32.6%

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

      4. associate-/r* 32.6%

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

   6. Simplified 32.6%

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

   7. Taylor expanded in B around 0 44.7%

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

   if 4.29999999999999998e-14 < F < 1.64999999999999995e180

   1. Initial program 80.7%

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

   2. Taylor expanded in B around 0 65.6%

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

   3. Taylor expanded in F around inf 77.6%

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

   if 1.64999999999999995e180 < F

   1. Initial program 32.4%

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

      1. +-commutative 32.4%

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

      2. unsub-neg 32.4%

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

      3. associate-*l/ 50.1%

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

      4. associate-*r/ 50.1%

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

      5. *-commutative 50.1%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in F around inf 99.7%

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

   5. Taylor expanded in B around 0 93.0%

      \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]
3. Recombined 6 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.2 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -5.1 \cdot 10^{-241}:\\ \;\;\;\;\frac{F}{B} \cdot \frac{-1}{F} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 5.6 \cdot 10^{-175}:\\ \;\;\;\;\cos B \cdot \left(0.16666666666666666 \cdot \left(-x \cdot B\right) - \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\frac{F}{B}}{-F}\\ \mathbf{elif}\;F \leq 1.65 \cdot 10^{+180}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 18: 61.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.7 \cdot 10^{-241}:\\ \;\;\;\;\frac{F}{B} \cdot \frac{-1}{F} - \frac{1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-176}:\\ \;\;\;\;\cos B \cdot \left(0.16666666666666666 \cdot \left(-x \cdot B\right) - \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\frac{F}{B}}{-F}\\ \mathbf{elif}\;F \leq 10^{+181}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8e+28)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -3.7e-241)
     (- (* (/ F B) (/ -1.0 F)) (/ 1.0 (/ (tan B) x)))
     (if (<= F 6.5e-176)
       (* (cos B) (- (* 0.16666666666666666 (- (* x B))) (/ x B)))
       (if (<= F 3.5e-14)
         (+ (* x (/ -1.0 (tan B))) (/ (/ F B) (- F)))
         (if (<= F 1e+181)
           (- (/ 1.0 (sin B)) (/ x B))
           (- (/ 1.0 B) (/ x (tan B)))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8e+28) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -3.7e-241) {
		tmp = ((F / B) * (-1.0 / F)) - (1.0 / (tan(B) / x));
	} else if (F <= 6.5e-176) {
		tmp = cos(B) * ((0.16666666666666666 * -(x * B)) - (x / B));
	} else if (F <= 3.5e-14) {
		tmp = (x * (-1.0 / tan(B))) + ((F / B) / -F);
	} else if (F <= 1e+181) {
		tmp = (1.0 / sin(B)) - (x / B);
	} else {
		tmp = (1.0 / 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 <= (-8d+28)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-3.7d-241)) then
        tmp = ((f / b) * ((-1.0d0) / f)) - (1.0d0 / (tan(b) / x))
    else if (f <= 6.5d-176) then
        tmp = cos(b) * ((0.16666666666666666d0 * -(x * b)) - (x / b))
    else if (f <= 3.5d-14) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((f / b) / -f)
    else if (f <= 1d+181) then
        tmp = (1.0d0 / sin(b)) - (x / b)
    else
        tmp = (1.0d0 / 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 <= -8e+28) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -3.7e-241) {
		tmp = ((F / B) * (-1.0 / F)) - (1.0 / (Math.tan(B) / x));
	} else if (F <= 6.5e-176) {
		tmp = Math.cos(B) * ((0.16666666666666666 * -(x * B)) - (x / B));
	} else if (F <= 3.5e-14) {
		tmp = (x * (-1.0 / Math.tan(B))) + ((F / B) / -F);
	} else if (F <= 1e+181) {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -8e+28:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -3.7e-241:
		tmp = ((F / B) * (-1.0 / F)) - (1.0 / (math.tan(B) / x))
	elif F <= 6.5e-176:
		tmp = math.cos(B) * ((0.16666666666666666 * -(x * B)) - (x / B))
	elif F <= 3.5e-14:
		tmp = (x * (-1.0 / math.tan(B))) + ((F / B) / -F)
	elif F <= 1e+181:
		tmp = (1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8e+28)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -3.7e-241)
		tmp = Float64(Float64(Float64(F / B) * Float64(-1.0 / F)) - Float64(1.0 / Float64(tan(B) / x)));
	elseif (F <= 6.5e-176)
		tmp = Float64(cos(B) * Float64(Float64(0.16666666666666666 * Float64(-Float64(x * B))) - Float64(x / B)));
	elseif (F <= 3.5e-14)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(F / B) / Float64(-F)));
	elseif (F <= 1e+181)
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -8e+28)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -3.7e-241)
		tmp = ((F / B) * (-1.0 / F)) - (1.0 / (tan(B) / x));
	elseif (F <= 6.5e-176)
		tmp = cos(B) * ((0.16666666666666666 * -(x * B)) - (x / B));
	elseif (F <= 3.5e-14)
		tmp = (x * (-1.0 / tan(B))) + ((F / B) / -F);
	elseif (F <= 1e+181)
		tmp = (1.0 / sin(B)) - (x / B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8e+28], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -3.7e-241], N[(N[(N[(F / B), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 6.5e-176], N[(N[Cos[B], $MachinePrecision] * N[(N[(0.16666666666666666 * (-N[(x * B), $MachinePrecision])), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.5e-14], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] / (-F)), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e+181], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if F < -7.99999999999999967e28

   1. Initial program 63.5%

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

   2. Taylor expanded in B around 0 43.3%

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

   3. Taylor expanded in F around -inf 79.1%

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

   if -7.99999999999999967e28 < F < -3.6999999999999999e-241

   1. Initial program 99.4%

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

   2. Taylor expanded in F around -inf 34.3%

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

   3. Taylor expanded in B around 0 40.4%

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

      1. div-inv 40.4%

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

      2. clear-num 40.4%

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

   5. Applied egg-rr 40.4%

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

   if -3.6999999999999999e-241 < F < 6.5e-176

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. fma-def 99.7%

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

      3. +-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. fma-def 99.7%

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

      6. fma-def 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

      9. distribute-lft-neg-in 99.7%

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

      10. associate-*r/ 99.9%

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

      11. *-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in F around 0 90.5%

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

      1. mul-1-neg 90.5%

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

      2. *-commutative 90.5%

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

      3. associate-*l/ 90.6%

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

      4. *-commutative 90.6%

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

   6. Simplified 90.6%

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

   7. Taylor expanded in B around 0 59.5%

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

   if 6.5e-176 < F < 3.5000000000000002e-14

   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. Taylor expanded in F around -inf 32.6%

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

      1. expm1-log1p-u 18.9%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{F}{\sin B} \cdot \frac{-1}{F}\right)\right)} \]

      2. expm1-udef 18.9%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{F}{\sin B} \cdot \frac{-1}{F}\right)} - 1\right)} \]

      3. *-commutative 18.9%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{-1}{F} \cdot \frac{F}{\sin B}}\right)} - 1\right) \]

      4. frac-2neg 18.9%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{--1}{-F}} \cdot \frac{F}{\sin B}\right)} - 1\right) \]

      5. metadata-eval 18.9%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{1}}{-F} \cdot \frac{F}{\sin B}\right)} - 1\right) \]

      6. frac-times 18.9%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \left(e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot F}{\left(-F\right) \cdot \sin B}}\right)} - 1\right) \]

      7. *-un-lft-identity 18.9%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{F}}{\left(-F\right) \cdot \sin B}\right)} - 1\right) \]

   4. Applied egg-rr 18.9%

      \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{F}{\left(-F\right) \cdot \sin B}\right)} - 1\right)} \]
   5. Step-by-step derivation

      1. expm1-def 18.9%

         \[\leadsto \left(-x \cdot \frac{1}{\tan B}\right) + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{F}{\left(-F\right) \cdot \sin B}\right)\right)} \]

      2. expm1-log1p 32.6%

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

      3. *-commutative 32.6%

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

      4. associate-/r* 32.6%

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

   6. Simplified 32.6%

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

   7. Taylor expanded in B around 0 44.7%

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

   if 3.5000000000000002e-14 < F < 9.9999999999999992e180

   1. Initial program 80.7%

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

   2. Taylor expanded in B around 0 65.6%

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

   3. Taylor expanded in F around inf 77.6%

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

   if 9.9999999999999992e180 < F

   1. Initial program 32.4%

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

      1. +-commutative 32.4%

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

      2. unsub-neg 32.4%

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

      3. associate-*l/ 50.1%

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

      4. associate-*r/ 50.1%

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

      5. *-commutative 50.1%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in F around inf 99.7%

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

   5. Taylor expanded in B around 0 93.0%

      \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]
3. Recombined 6 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8 \cdot 10^{+28}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.7 \cdot 10^{-241}:\\ \;\;\;\;\frac{F}{B} \cdot \frac{-1}{F} - \frac{1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-176}:\\ \;\;\;\;\cos B \cdot \left(0.16666666666666666 \cdot \left(-x \cdot B\right) - \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{\frac{F}{B}}{-F}\\ \mathbf{elif}\;F \leq 10^{+181}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 19: 61.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4 \cdot 10^{+31}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.2 \cdot 10^{-241}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-174}:\\ \;\;\;\;\cos B \cdot \left(0.16666666666666666 \cdot \left(-x \cdot B\right) - \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{+180}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4e+31)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -2.2e-241)
       (- (/ -1.0 B) t_0)
       (if (<= F 1.7e-174)
         (* (cos B) (- (* 0.16666666666666666 (- (* x B))) (/ x B)))
         (if (<= F 4.3e-14)
           (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))
           (if (<= F 1.45e+180)
             (- (/ 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 <= -1.4e+31) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -2.2e-241) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 1.7e-174) {
		tmp = cos(B) * ((0.16666666666666666 * -(x * B)) - (x / B));
	} else if (F <= 4.3e-14) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	} else if (F <= 1.45e+180) {
		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 <= (-1.4d+31)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-2.2d-241)) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= 1.7d-174) then
        tmp = cos(b) * ((0.16666666666666666d0 * -(x * b)) - (x / b))
    else if (f <= 4.3d-14) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    else if (f <= 1.45d+180) 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 <= -1.4e+31) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -2.2e-241) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 1.7e-174) {
		tmp = Math.cos(B) * ((0.16666666666666666 * -(x * B)) - (x / B));
	} else if (F <= 4.3e-14) {
		tmp = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	} else if (F <= 1.45e+180) {
		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 <= -1.4e+31:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -2.2e-241:
		tmp = (-1.0 / B) - t_0
	elif F <= 1.7e-174:
		tmp = math.cos(B) * ((0.16666666666666666 * -(x * B)) - (x / B))
	elif F <= 4.3e-14:
		tmp = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	elif F <= 1.45e+180:
		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 <= -1.4e+31)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -2.2e-241)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= 1.7e-174)
		tmp = Float64(cos(B) * Float64(Float64(0.16666666666666666 * Float64(-Float64(x * B))) - Float64(x / B)));
	elseif (F <= 4.3e-14)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	elseif (F <= 1.45e+180)
		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 <= -1.4e+31)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -2.2e-241)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= 1.7e-174)
		tmp = cos(B) * ((0.16666666666666666 * -(x * B)) - (x / B));
	elseif (F <= 4.3e-14)
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	elseif (F <= 1.45e+180)
		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, -1.4e+31], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.2e-241], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.7e-174], N[(N[Cos[B], $MachinePrecision] * N[(N[(0.16666666666666666 * (-N[(x * B), $MachinePrecision])), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.3e-14], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.45e+180], 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 6 regimes

2. if F < -1.40000000000000008e31

   1. Initial program 63.5%

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

   2. Taylor expanded in B around 0 43.3%

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

   3. Taylor expanded in F around -inf 79.1%

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

   if -1.40000000000000008e31 < F < -2.1999999999999999e-241

   1. Initial program 99.4%

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

   2. Taylor expanded in F around -inf 34.3%

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

   3. Taylor expanded in B around 0 40.4%

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

      1. expm1-log1p-u 21.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \frac{-1}{F}\right)\right)} \]

      2. expm1-udef 14.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \frac{-1}{F}\right)} - 1} \]

      3. +-commutative 14.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{F}{B} \cdot \frac{-1}{F} + \left(-x \cdot \frac{1}{\tan B}\right)}\right)} - 1 \]

      4. fma-def 14.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\frac{F}{B}, \frac{-1}{F}, -x \cdot \frac{1}{\tan B}\right)}\right)} - 1 \]

      5. un-div-inv 14.4%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{F}{B}, \frac{-1}{F}, -\color{blue}{\frac{x}{\tan B}}\right)\right)} - 1 \]

   5. Applied egg-rr 14.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{F}{B}, \frac{-1}{F}, -\frac{x}{\tan B}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 21.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{F}{B}, \frac{-1}{F}, -\frac{x}{\tan B}\right)\right)\right)} \]

      2. expm1-log1p 40.4%

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

      3. fma-neg 40.4%

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

      4. times-frac 38.3%

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

      5. *-commutative 38.3%

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

      6. times-frac 38.4%

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

      7. *-inverses 38.4%

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

      8. associate-*r/ 38.4%

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

      9. metadata-eval 38.4%

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

   7. Simplified 38.4%

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

   if -2.1999999999999999e-241 < F < 1.7000000000000001e-174

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. fma-def 99.7%

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

      3. +-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. fma-def 99.7%

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

      6. fma-def 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

      9. distribute-lft-neg-in 99.7%

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

      10. associate-*r/ 99.9%

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

      11. *-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in F around 0 90.5%

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

      1. mul-1-neg 90.5%

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

      2. *-commutative 90.5%

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

      3. associate-*l/ 90.6%

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

      4. *-commutative 90.6%

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

   6. Simplified 90.6%

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

   7. Taylor expanded in B around 0 59.5%

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

   if 1.7000000000000001e-174 < F < 4.29999999999999998e-14

   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. Taylor expanded in F around -inf 32.6%

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

   3. Taylor expanded in B around 0 44.7%

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

   if 4.29999999999999998e-14 < F < 1.45000000000000004e180

   1. Initial program 80.7%

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

   2. Taylor expanded in B around 0 65.6%

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

   3. Taylor expanded in F around inf 77.6%

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

   if 1.45000000000000004e180 < F

   1. Initial program 32.4%

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

      1. +-commutative 32.4%

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

      2. unsub-neg 32.4%

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

      3. associate-*l/ 50.1%

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

      4. associate-*r/ 50.1%

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

      5. *-commutative 50.1%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in F around inf 99.7%

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

   5. Taylor expanded in B around 0 93.0%

      \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]
3. Recombined 6 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4 \cdot 10^{+31}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.2 \cdot 10^{-241}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{-174}:\\ \;\;\;\;\cos B \cdot \left(0.16666666666666666 \cdot \left(-x \cdot B\right) - \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 1.45 \cdot 10^{+180}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 20: 61.7% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -8.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-241}:\\ \;\;\;\;\frac{F}{B} \cdot \frac{-1}{F} - t_0\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-174}:\\ \;\;\;\;\cos B \cdot \left(0.16666666666666666 \cdot \left(-x \cdot B\right) - \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{+180}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -8.5e+29)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -2.1e-241)
       (- (* (/ F B) (/ -1.0 F)) t_0)
       (if (<= F 1.75e-174)
         (* (cos B) (- (* 0.16666666666666666 (- (* x B))) (/ x B)))
         (if (<= F 3.5e-14)
           (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))
           (if (<= F 7.2e+180)
             (- (/ 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 <= -8.5e+29) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -2.1e-241) {
		tmp = ((F / B) * (-1.0 / F)) - t_0;
	} else if (F <= 1.75e-174) {
		tmp = cos(B) * ((0.16666666666666666 * -(x * B)) - (x / B));
	} else if (F <= 3.5e-14) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	} else if (F <= 7.2e+180) {
		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 <= (-8.5d+29)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-2.1d-241)) then
        tmp = ((f / b) * ((-1.0d0) / f)) - t_0
    else if (f <= 1.75d-174) then
        tmp = cos(b) * ((0.16666666666666666d0 * -(x * b)) - (x / b))
    else if (f <= 3.5d-14) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    else if (f <= 7.2d+180) 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 <= -8.5e+29) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -2.1e-241) {
		tmp = ((F / B) * (-1.0 / F)) - t_0;
	} else if (F <= 1.75e-174) {
		tmp = Math.cos(B) * ((0.16666666666666666 * -(x * B)) - (x / B));
	} else if (F <= 3.5e-14) {
		tmp = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	} else if (F <= 7.2e+180) {
		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 <= -8.5e+29:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -2.1e-241:
		tmp = ((F / B) * (-1.0 / F)) - t_0
	elif F <= 1.75e-174:
		tmp = math.cos(B) * ((0.16666666666666666 * -(x * B)) - (x / B))
	elif F <= 3.5e-14:
		tmp = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	elif F <= 7.2e+180:
		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 <= -8.5e+29)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -2.1e-241)
		tmp = Float64(Float64(Float64(F / B) * Float64(-1.0 / F)) - t_0);
	elseif (F <= 1.75e-174)
		tmp = Float64(cos(B) * Float64(Float64(0.16666666666666666 * Float64(-Float64(x * B))) - Float64(x / B)));
	elseif (F <= 3.5e-14)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	elseif (F <= 7.2e+180)
		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 <= -8.5e+29)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -2.1e-241)
		tmp = ((F / B) * (-1.0 / F)) - t_0;
	elseif (F <= 1.75e-174)
		tmp = cos(B) * ((0.16666666666666666 * -(x * B)) - (x / B));
	elseif (F <= 3.5e-14)
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	elseif (F <= 7.2e+180)
		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, -8.5e+29], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.1e-241], N[(N[(N[(F / B), $MachinePrecision] * N[(-1.0 / F), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.75e-174], N[(N[Cos[B], $MachinePrecision] * N[(N[(0.16666666666666666 * (-N[(x * B), $MachinePrecision])), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.5e-14], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.2e+180], 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 6 regimes

2. if F < -8.5000000000000006e29

   1. Initial program 63.5%

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

   2. Taylor expanded in B around 0 43.3%

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

   3. Taylor expanded in F around -inf 79.1%

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

   if -8.5000000000000006e29 < F < -2.0999999999999999e-241

   1. Initial program 99.4%

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

   2. Taylor expanded in F around -inf 34.3%

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

   3. Taylor expanded in B around 0 40.4%

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

      1. div-inv 40.4%

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

      2. expm1-log1p-u 26.4%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{F}{B} \cdot \frac{-1}{F} \]

      3. expm1-udef 20.1%

         \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{F}{B} \cdot \frac{-1}{F} \]

   5. Applied egg-rr 20.1%

      \[\leadsto \left(-\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\tan B}\right)} - 1\right)}\right) + \frac{F}{B} \cdot \frac{-1}{F} \]
   6. Step-by-step derivation

      1. expm1-def 26.4%

         \[\leadsto \left(-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\tan B}\right)\right)}\right) + \frac{F}{B} \cdot \frac{-1}{F} \]

      2. expm1-log1p 40.4%

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

   7. Simplified 40.4%

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

   if -2.0999999999999999e-241 < F < 1.74999999999999994e-174

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. fma-def 99.7%

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

      3. +-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. fma-def 99.7%

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

      6. fma-def 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

      9. distribute-lft-neg-in 99.7%

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

      10. associate-*r/ 99.9%

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

      11. *-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in F around 0 90.5%

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

      1. mul-1-neg 90.5%

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

      2. *-commutative 90.5%

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

      3. associate-*l/ 90.6%

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

      4. *-commutative 90.6%

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

   6. Simplified 90.6%

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

   7. Taylor expanded in B around 0 59.5%

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

   if 1.74999999999999994e-174 < F < 3.5000000000000002e-14

   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. Taylor expanded in F around -inf 32.6%

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

   3. Taylor expanded in B around 0 44.7%

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

   if 3.5000000000000002e-14 < F < 7.2000000000000004e180

   1. Initial program 80.7%

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

   2. Taylor expanded in B around 0 65.6%

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

   3. Taylor expanded in F around inf 77.6%

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

   if 7.2000000000000004e180 < F

   1. Initial program 32.4%

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

      1. +-commutative 32.4%

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

      2. unsub-neg 32.4%

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

      3. associate-*l/ 50.1%

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

      4. associate-*r/ 50.1%

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

      5. *-commutative 50.1%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in F around inf 99.7%

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

   5. Taylor expanded in B around 0 93.0%

      \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]
3. Recombined 6 regimes into one program.

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.5 \cdot 10^{+29}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{-241}:\\ \;\;\;\;\frac{F}{B} \cdot \frac{-1}{F} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.75 \cdot 10^{-174}:\\ \;\;\;\;\cos B \cdot \left(0.16666666666666666 \cdot \left(-x \cdot B\right) - \frac{x}{B}\right)\\ \mathbf{elif}\;F \leq 3.5 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 7.2 \cdot 10^{+180}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 21: 61.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{-1}{B} - t_0\\ \mathbf{if}\;F \leq -9.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.95 \cdot 10^{-241}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-176}:\\ \;\;\;\;\frac{-x}{B} - \left(x \cdot B\right) \cdot -0.3333333333333333\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{+180}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))) (t_1 (- (/ -1.0 B) t_0)))
   (if (<= F -9.6e+30)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -1.95e-241)
       t_1
       (if (<= F 6.5e-176)
         (- (/ (- x) B) (* (* x B) -0.3333333333333333))
         (if (<= F 4.3e-14)
           t_1
           (if (<= F 6.8e+180)
             (- (/ 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 t_1 = (-1.0 / B) - t_0;
	double tmp;
	if (F <= -9.6e+30) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -1.95e-241) {
		tmp = t_1;
	} else if (F <= 6.5e-176) {
		tmp = (-x / B) - ((x * B) * -0.3333333333333333);
	} else if (F <= 4.3e-14) {
		tmp = t_1;
	} else if (F <= 6.8e+180) {
		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) :: t_1
    real(8) :: tmp
    t_0 = x / tan(b)
    t_1 = ((-1.0d0) / b) - t_0
    if (f <= (-9.6d+30)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-1.95d-241)) then
        tmp = t_1
    else if (f <= 6.5d-176) then
        tmp = (-x / b) - ((x * b) * (-0.3333333333333333d0))
    else if (f <= 4.3d-14) then
        tmp = t_1
    else if (f <= 6.8d+180) 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 t_1 = (-1.0 / B) - t_0;
	double tmp;
	if (F <= -9.6e+30) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -1.95e-241) {
		tmp = t_1;
	} else if (F <= 6.5e-176) {
		tmp = (-x / B) - ((x * B) * -0.3333333333333333);
	} else if (F <= 4.3e-14) {
		tmp = t_1;
	} else if (F <= 6.8e+180) {
		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)
	t_1 = (-1.0 / B) - t_0
	tmp = 0
	if F <= -9.6e+30:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -1.95e-241:
		tmp = t_1
	elif F <= 6.5e-176:
		tmp = (-x / B) - ((x * B) * -0.3333333333333333)
	elif F <= 4.3e-14:
		tmp = t_1
	elif F <= 6.8e+180:
		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))
	t_1 = Float64(Float64(-1.0 / B) - t_0)
	tmp = 0.0
	if (F <= -9.6e+30)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -1.95e-241)
		tmp = t_1;
	elseif (F <= 6.5e-176)
		tmp = Float64(Float64(Float64(-x) / B) - Float64(Float64(x * B) * -0.3333333333333333));
	elseif (F <= 4.3e-14)
		tmp = t_1;
	elseif (F <= 6.8e+180)
		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);
	t_1 = (-1.0 / B) - t_0;
	tmp = 0.0;
	if (F <= -9.6e+30)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -1.95e-241)
		tmp = t_1;
	elseif (F <= 6.5e-176)
		tmp = (-x / B) - ((x * B) * -0.3333333333333333);
	elseif (F <= 4.3e-14)
		tmp = t_1;
	elseif (F <= 6.8e+180)
		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]}, Block[{t$95$1 = N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[F, -9.6e+30], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.95e-241], t$95$1, If[LessEqual[F, 6.5e-176], N[(N[((-x) / B), $MachinePrecision] - N[(N[(x * B), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.3e-14], t$95$1, If[LessEqual[F, 6.8e+180], 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 5 regimes

2. if F < -9.5999999999999997e30

   1. Initial program 63.5%

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

   2. Taylor expanded in B around 0 43.3%

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

   3. Taylor expanded in F around -inf 79.1%

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

   if -9.5999999999999997e30 < F < -1.9499999999999999e-241 or 6.5e-176 < F < 4.29999999999999998e-14

   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. Taylor expanded in F around -inf 33.6%

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

   3. Taylor expanded in B around 0 42.1%

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

      1. expm1-log1p-u 23.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \frac{-1}{F}\right)\right)} \]

      2. expm1-udef 15.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \frac{-1}{F}\right)} - 1} \]

      3. +-commutative 15.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{F}{B} \cdot \frac{-1}{F} + \left(-x \cdot \frac{1}{\tan B}\right)}\right)} - 1 \]

      4. fma-def 15.3%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\frac{F}{B}, \frac{-1}{F}, -x \cdot \frac{1}{\tan B}\right)}\right)} - 1 \]

      5. un-div-inv 15.3%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{F}{B}, \frac{-1}{F}, -\color{blue}{\frac{x}{\tan B}}\right)\right)} - 1 \]

   5. Applied egg-rr 15.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{F}{B}, \frac{-1}{F}, -\frac{x}{\tan B}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 23.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{F}{B}, \frac{-1}{F}, -\frac{x}{\tan B}\right)\right)\right)} \]

      2. expm1-log1p 42.1%

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

      3. fma-neg 42.1%

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

      4. times-frac 40.8%

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

      5. *-commutative 40.8%

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

      6. times-frac 40.9%

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

      7. *-inverses 40.9%

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

      8. associate-*r/ 40.9%

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

      9. metadata-eval 40.9%

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

   7. Simplified 40.9%

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

   if -1.9499999999999999e-241 < F < 6.5e-176

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. fma-def 99.7%

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

      3. +-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. fma-def 99.7%

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

      6. fma-def 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

      9. distribute-lft-neg-in 99.7%

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

      10. associate-*r/ 99.9%

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

      11. *-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in F around 0 90.5%

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

      1. mul-1-neg 90.5%

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

      2. *-commutative 90.5%

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

      3. associate-*l/ 90.6%

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

      4. *-commutative 90.6%

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

   6. Simplified 90.6%

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

   7. Taylor expanded in B around 0 59.0%

      \[\leadsto -\color{blue}{\left(\frac{x}{B} + \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right)} \]
   8. Step-by-step derivation

      1. *-commutative 59.0%

         \[\leadsto -\left(\frac{x}{B} + \color{blue}{B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}\right) \]

      2. distribute-rgt-out-- 59.0%

         \[\leadsto -\left(\frac{x}{B} + B \cdot \color{blue}{\left(x \cdot \left(-0.5 - -0.16666666666666666\right)\right)}\right) \]

      3. metadata-eval 59.0%

         \[\leadsto -\left(\frac{x}{B} + B \cdot \left(x \cdot \color{blue}{-0.3333333333333333}\right)\right) \]

   9. Simplified 59.0%

      \[\leadsto -\color{blue}{\left(\frac{x}{B} + B \cdot \left(x \cdot -0.3333333333333333\right)\right)} \]

   10. Taylor expanded in x around 0 59.0%

       \[\leadsto -\left(\frac{x}{B} + \color{blue}{-0.3333333333333333 \cdot \left(x \cdot B\right)}\right) \]

   if 4.29999999999999998e-14 < F < 6.79999999999999969e180

   1. Initial program 80.7%

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

   2. Taylor expanded in B around 0 65.6%

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

   3. Taylor expanded in F around inf 77.6%

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

   if 6.79999999999999969e180 < F

   1. Initial program 32.4%

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

      1. +-commutative 32.4%

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

      2. unsub-neg 32.4%

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

      3. associate-*l/ 50.1%

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

      4. associate-*r/ 50.1%

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

      5. *-commutative 50.1%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in F around inf 99.7%

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

   5. Taylor expanded in B around 0 93.0%

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

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -9.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -1.95 \cdot 10^{-241}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{-176}:\\ \;\;\;\;\frac{-x}{B} - \left(x \cdot B\right) \cdot -0.3333333333333333\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{-14}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 6.8 \cdot 10^{+180}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 22: 61.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.52 \cdot 10^{+29}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -9.5 \cdot 10^{-241}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-174}:\\ \;\;\;\;\frac{-x}{B} - \left(x \cdot B\right) \cdot -0.3333333333333333\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 10^{+181}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.52e+29)
     (- (/ -1.0 (sin B)) (/ x B))
     (if (<= F -9.5e-241)
       (- (/ -1.0 B) t_0)
       (if (<= F 1.4e-174)
         (- (/ (- x) B) (* (* x B) -0.3333333333333333))
         (if (<= F 4.3e-14)
           (+ (* x (/ -1.0 (tan B))) (/ -1.0 B))
           (if (<= F 1e+181)
             (- (/ 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 <= -1.52e+29) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -9.5e-241) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 1.4e-174) {
		tmp = (-x / B) - ((x * B) * -0.3333333333333333);
	} else if (F <= 4.3e-14) {
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	} else if (F <= 1e+181) {
		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 <= (-1.52d+29)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-9.5d-241)) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= 1.4d-174) then
        tmp = (-x / b) - ((x * b) * (-0.3333333333333333d0))
    else if (f <= 4.3d-14) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((-1.0d0) / b)
    else if (f <= 1d+181) 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 <= -1.52e+29) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -9.5e-241) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 1.4e-174) {
		tmp = (-x / B) - ((x * B) * -0.3333333333333333);
	} else if (F <= 4.3e-14) {
		tmp = (x * (-1.0 / Math.tan(B))) + (-1.0 / B);
	} else if (F <= 1e+181) {
		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 <= -1.52e+29:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -9.5e-241:
		tmp = (-1.0 / B) - t_0
	elif F <= 1.4e-174:
		tmp = (-x / B) - ((x * B) * -0.3333333333333333)
	elif F <= 4.3e-14:
		tmp = (x * (-1.0 / math.tan(B))) + (-1.0 / B)
	elif F <= 1e+181:
		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 <= -1.52e+29)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -9.5e-241)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= 1.4e-174)
		tmp = Float64(Float64(Float64(-x) / B) - Float64(Float64(x * B) * -0.3333333333333333));
	elseif (F <= 4.3e-14)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(-1.0 / B));
	elseif (F <= 1e+181)
		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 <= -1.52e+29)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -9.5e-241)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= 1.4e-174)
		tmp = (-x / B) - ((x * B) * -0.3333333333333333);
	elseif (F <= 4.3e-14)
		tmp = (x * (-1.0 / tan(B))) + (-1.0 / B);
	elseif (F <= 1e+181)
		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, -1.52e+29], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -9.5e-241], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4e-174], N[(N[((-x) / B), $MachinePrecision] - N[(N[(x * B), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.3e-14], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e+181], 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 6 regimes

2. if F < -1.52e29

   1. Initial program 63.5%

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

   2. Taylor expanded in B around 0 43.3%

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

   3. Taylor expanded in F around -inf 79.1%

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

   if -1.52e29 < F < -9.49999999999999971e-241

   1. Initial program 99.4%

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

   2. Taylor expanded in F around -inf 34.3%

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

   3. Taylor expanded in B around 0 40.4%

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

      1. expm1-log1p-u 21.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \frac{-1}{F}\right)\right)} \]

      2. expm1-udef 14.4%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \frac{-1}{F}\right)} - 1} \]

      3. +-commutative 14.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{F}{B} \cdot \frac{-1}{F} + \left(-x \cdot \frac{1}{\tan B}\right)}\right)} - 1 \]

      4. fma-def 14.4%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\frac{F}{B}, \frac{-1}{F}, -x \cdot \frac{1}{\tan B}\right)}\right)} - 1 \]

      5. un-div-inv 14.4%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{F}{B}, \frac{-1}{F}, -\color{blue}{\frac{x}{\tan B}}\right)\right)} - 1 \]

   5. Applied egg-rr 14.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{F}{B}, \frac{-1}{F}, -\frac{x}{\tan B}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 21.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{F}{B}, \frac{-1}{F}, -\frac{x}{\tan B}\right)\right)\right)} \]

      2. expm1-log1p 40.4%

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

      3. fma-neg 40.4%

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

      4. times-frac 38.3%

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

      5. *-commutative 38.3%

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

      6. times-frac 38.4%

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

      7. *-inverses 38.4%

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

      8. associate-*r/ 38.4%

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

      9. metadata-eval 38.4%

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

   7. Simplified 38.4%

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

   if -9.49999999999999971e-241 < F < 1.39999999999999999e-174

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. fma-def 99.7%

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

      3. +-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. fma-def 99.7%

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

      6. fma-def 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

      9. distribute-lft-neg-in 99.7%

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

      10. associate-*r/ 99.9%

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

      11. *-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in F around 0 90.5%

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

      1. mul-1-neg 90.5%

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

      2. *-commutative 90.5%

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

      3. associate-*l/ 90.6%

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

      4. *-commutative 90.6%

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

   6. Simplified 90.6%

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

   7. Taylor expanded in B around 0 59.0%

      \[\leadsto -\color{blue}{\left(\frac{x}{B} + \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right)} \]
   8. Step-by-step derivation

      1. *-commutative 59.0%

         \[\leadsto -\left(\frac{x}{B} + \color{blue}{B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}\right) \]

      2. distribute-rgt-out-- 59.0%

         \[\leadsto -\left(\frac{x}{B} + B \cdot \color{blue}{\left(x \cdot \left(-0.5 - -0.16666666666666666\right)\right)}\right) \]

      3. metadata-eval 59.0%

         \[\leadsto -\left(\frac{x}{B} + B \cdot \left(x \cdot \color{blue}{-0.3333333333333333}\right)\right) \]

   9. Simplified 59.0%

      \[\leadsto -\color{blue}{\left(\frac{x}{B} + B \cdot \left(x \cdot -0.3333333333333333\right)\right)} \]

   10. Taylor expanded in x around 0 59.0%

       \[\leadsto -\left(\frac{x}{B} + \color{blue}{-0.3333333333333333 \cdot \left(x \cdot B\right)}\right) \]

   if 1.39999999999999999e-174 < F < 4.29999999999999998e-14

   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. Taylor expanded in F around -inf 32.6%

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

   3. Taylor expanded in B around 0 44.7%

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

   if 4.29999999999999998e-14 < F < 9.9999999999999992e180

   1. Initial program 80.7%

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

   2. Taylor expanded in B around 0 65.6%

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

   3. Taylor expanded in F around inf 77.6%

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

   if 9.9999999999999992e180 < F

   1. Initial program 32.4%

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

      1. +-commutative 32.4%

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

      2. unsub-neg 32.4%

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

      3. associate-*l/ 50.1%

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

      4. associate-*r/ 50.1%

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

      5. *-commutative 50.1%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in F around inf 99.7%

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

   5. Taylor expanded in B around 0 93.0%

      \[\leadsto \color{blue}{\frac{1}{B}} - \frac{x}{\tan B} \]
3. Recombined 6 regimes into one program.

4. Final simplification 66.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.52 \cdot 10^{+29}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -9.5 \cdot 10^{-241}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-174}:\\ \;\;\;\;\frac{-x}{B} - \left(x \cdot B\right) \cdot -0.3333333333333333\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{-14}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{-1}{B}\\ \mathbf{elif}\;F \leq 10^{+181}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 23: 61.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{-1}{B} - t_0\\ \mathbf{if}\;F \leq -1.8 \cdot 10^{-241}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-173}:\\ \;\;\;\;\frac{-x}{B} - \left(x \cdot B\right) \cdot -0.3333333333333333\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{-14}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;F \leq 10^{+181}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))) (t_1 (- (/ -1.0 B) t_0)))
   (if (<= F -1.8e-241)
     t_1
     (if (<= F 1.3e-173)
       (- (/ (- x) B) (* (* x B) -0.3333333333333333))
       (if (<= F 4.3e-14)
         t_1
         (if (<= F 1e+181) (- (/ 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 t_1 = (-1.0 / B) - t_0;
	double tmp;
	if (F <= -1.8e-241) {
		tmp = t_1;
	} else if (F <= 1.3e-173) {
		tmp = (-x / B) - ((x * B) * -0.3333333333333333);
	} else if (F <= 4.3e-14) {
		tmp = t_1;
	} else if (F <= 1e+181) {
		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) :: t_1
    real(8) :: tmp
    t_0 = x / tan(b)
    t_1 = ((-1.0d0) / b) - t_0
    if (f <= (-1.8d-241)) then
        tmp = t_1
    else if (f <= 1.3d-173) then
        tmp = (-x / b) - ((x * b) * (-0.3333333333333333d0))
    else if (f <= 4.3d-14) then
        tmp = t_1
    else if (f <= 1d+181) 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 t_1 = (-1.0 / B) - t_0;
	double tmp;
	if (F <= -1.8e-241) {
		tmp = t_1;
	} else if (F <= 1.3e-173) {
		tmp = (-x / B) - ((x * B) * -0.3333333333333333);
	} else if (F <= 4.3e-14) {
		tmp = t_1;
	} else if (F <= 1e+181) {
		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)
	t_1 = (-1.0 / B) - t_0
	tmp = 0
	if F <= -1.8e-241:
		tmp = t_1
	elif F <= 1.3e-173:
		tmp = (-x / B) - ((x * B) * -0.3333333333333333)
	elif F <= 4.3e-14:
		tmp = t_1
	elif F <= 1e+181:
		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))
	t_1 = Float64(Float64(-1.0 / B) - t_0)
	tmp = 0.0
	if (F <= -1.8e-241)
		tmp = t_1;
	elseif (F <= 1.3e-173)
		tmp = Float64(Float64(Float64(-x) / B) - Float64(Float64(x * B) * -0.3333333333333333));
	elseif (F <= 4.3e-14)
		tmp = t_1;
	elseif (F <= 1e+181)
		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);
	t_1 = (-1.0 / B) - t_0;
	tmp = 0.0;
	if (F <= -1.8e-241)
		tmp = t_1;
	elseif (F <= 1.3e-173)
		tmp = (-x / B) - ((x * B) * -0.3333333333333333);
	elseif (F <= 4.3e-14)
		tmp = t_1;
	elseif (F <= 1e+181)
		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]}, Block[{t$95$1 = N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]}, If[LessEqual[F, -1.8e-241], t$95$1, If[LessEqual[F, 1.3e-173], N[(N[((-x) / B), $MachinePrecision] - N[(N[(x * B), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.3e-14], t$95$1, If[LessEqual[F, 1e+181], 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 4 regimes

2. if F < -1.7999999999999999e-241 or 1.30000000000000002e-173 < F < 4.29999999999999998e-14

   1. Initial program 83.1%

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

   2. Taylor expanded in F around -inf 54.2%

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

   3. Taylor expanded in B around 0 43.3%

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

      1. expm1-log1p-u 24.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \frac{-1}{F}\right)\right)} \]

      2. expm1-udef 19.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \frac{-1}{F}\right)} - 1} \]

      3. +-commutative 19.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{F}{B} \cdot \frac{-1}{F} + \left(-x \cdot \frac{1}{\tan B}\right)}\right)} - 1 \]

      4. fma-def 19.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\frac{F}{B}, \frac{-1}{F}, -x \cdot \frac{1}{\tan B}\right)}\right)} - 1 \]

      5. un-div-inv 19.9%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{F}{B}, \frac{-1}{F}, -\color{blue}{\frac{x}{\tan B}}\right)\right)} - 1 \]

   5. Applied egg-rr 19.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{F}{B}, \frac{-1}{F}, -\frac{x}{\tan B}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 24.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{F}{B}, \frac{-1}{F}, -\frac{x}{\tan B}\right)\right)\right)} \]

      2. expm1-log1p 43.3%

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

      3. fma-neg 43.3%

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

      4. times-frac 52.0%

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

      5. *-commutative 52.0%

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

      6. times-frac 52.1%

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

      7. *-inverses 52.1%

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

      8. associate-*r/ 52.1%

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

      9. metadata-eval 52.1%

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

   7. Simplified 52.1%

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

   if -1.7999999999999999e-241 < F < 1.30000000000000002e-173

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. fma-def 99.7%

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

      3. +-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. fma-def 99.7%

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

      6. fma-def 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

      9. distribute-lft-neg-in 99.7%

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

      10. associate-*r/ 99.9%

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

      11. *-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in F around 0 90.5%

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

      1. mul-1-neg 90.5%

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

      2. *-commutative 90.5%

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

      3. associate-*l/ 90.6%

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

      4. *-commutative 90.6%

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

   6. Simplified 90.6%

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

   7. Taylor expanded in B around 0 59.0%

      \[\leadsto -\color{blue}{\left(\frac{x}{B} + \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right)} \]
   8. Step-by-step derivation

      1. *-commutative 59.0%

         \[\leadsto -\left(\frac{x}{B} + \color{blue}{B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}\right) \]

      2. distribute-rgt-out-- 59.0%

         \[\leadsto -\left(\frac{x}{B} + B \cdot \color{blue}{\left(x \cdot \left(-0.5 - -0.16666666666666666\right)\right)}\right) \]

      3. metadata-eval 59.0%

         \[\leadsto -\left(\frac{x}{B} + B \cdot \left(x \cdot \color{blue}{-0.3333333333333333}\right)\right) \]

   9. Simplified 59.0%

      \[\leadsto -\color{blue}{\left(\frac{x}{B} + B \cdot \left(x \cdot -0.3333333333333333\right)\right)} \]

   10. Taylor expanded in x around 0 59.0%

       \[\leadsto -\left(\frac{x}{B} + \color{blue}{-0.3333333333333333 \cdot \left(x \cdot B\right)}\right) \]

   if 4.29999999999999998e-14 < F < 9.9999999999999992e180

   1. Initial program 80.7%

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

   2. Taylor expanded in B around 0 65.6%

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

   3. Taylor expanded in F around inf 77.6%

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

   if 9.9999999999999992e180 < F

   1. Initial program 32.4%

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

      1. +-commutative 32.4%

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

      2. unsub-neg 32.4%

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

      3. associate-*l/ 50.1%

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

      4. associate-*r/ 50.1%

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

      5. *-commutative 50.1%

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

   3. Simplified 50.3%

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

   4. Taylor expanded in F around inf 99.7%

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

   5. Taylor expanded in B around 0 93.0%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.8 \cdot 10^{-241}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.3 \cdot 10^{-173}:\\ \;\;\;\;\frac{-x}{B} - \left(x \cdot B\right) \cdot -0.3333333333333333\\ \mathbf{elif}\;F \leq 4.3 \cdot 10^{-14}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 10^{+181}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 24: 53.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-241} \lor \neg \left(F \leq 2.5 \cdot 10^{-176}\right):\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B} - \left(x \cdot B\right) \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= F -2e-241) (not (<= F 2.5e-176)))
   (- (/ -1.0 B) (/ x (tan B)))
   (- (/ (- x) B) (* (* x B) -0.3333333333333333))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((F <= -2e-241) || !(F <= 2.5e-176)) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else {
		tmp = (-x / B) - ((x * B) * -0.3333333333333333);
	}
	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 <= (-2d-241)) .or. (.not. (f <= 2.5d-176))) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else
        tmp = (-x / b) - ((x * b) * (-0.3333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((F <= -2e-241) || !(F <= 2.5e-176)) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = (-x / B) - ((x * B) * -0.3333333333333333);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (F <= -2e-241) or not (F <= 2.5e-176):
		tmp = (-1.0 / B) - (x / math.tan(B))
	else:
		tmp = (-x / B) - ((x * B) * -0.3333333333333333)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((F <= -2e-241) || !(F <= 2.5e-176))
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(Float64(Float64(-x) / B) - Float64(Float64(x * B) * -0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((F <= -2e-241) || ~((F <= 2.5e-176)))
		tmp = (-1.0 / B) - (x / tan(B));
	else
		tmp = (-x / B) - ((x * B) * -0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[F, -2e-241], N[Not[LessEqual[F, 2.5e-176]], $MachinePrecision]], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / B), $MachinePrecision] - N[(N[(x * B), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.9999999999999999e-241 or 2.5e-176 < F

   1. Initial program 76.2%

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

   2. Taylor expanded in F around -inf 46.9%

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

   3. Taylor expanded in B around 0 40.2%

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

      1. expm1-log1p-u 22.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \frac{-1}{F}\right)\right)} \]

      2. expm1-udef 19.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \frac{-1}{F}\right)} - 1} \]

      3. +-commutative 19.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{F}{B} \cdot \frac{-1}{F} + \left(-x \cdot \frac{1}{\tan B}\right)}\right)} - 1 \]

      4. fma-def 19.6%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\frac{F}{B}, \frac{-1}{F}, -x \cdot \frac{1}{\tan B}\right)}\right)} - 1 \]

      5. un-div-inv 19.6%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{F}{B}, \frac{-1}{F}, -\color{blue}{\frac{x}{\tan B}}\right)\right)} - 1 \]

   5. Applied egg-rr 19.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{F}{B}, \frac{-1}{F}, -\frac{x}{\tan B}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 22.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{F}{B}, \frac{-1}{F}, -\frac{x}{\tan B}\right)\right)\right)} \]

      2. expm1-log1p 40.3%

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

      3. fma-neg 40.3%

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

      4. times-frac 48.2%

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

      5. *-commutative 48.2%

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

      6. times-frac 48.2%

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

      7. *-inverses 48.2%

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

      8. associate-*r/ 48.2%

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

      9. metadata-eval 48.2%

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

   7. Simplified 48.2%

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

   if -1.9999999999999999e-241 < F < 2.5e-176

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. fma-def 99.7%

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

      3. +-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. fma-def 99.7%

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

      6. fma-def 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

      9. distribute-lft-neg-in 99.7%

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

      10. associate-*r/ 99.9%

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

      11. *-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in F around 0 90.5%

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

      1. mul-1-neg 90.5%

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

      2. *-commutative 90.5%

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

      3. associate-*l/ 90.6%

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

      4. *-commutative 90.6%

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

   6. Simplified 90.6%

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

   7. Taylor expanded in B around 0 59.0%

      \[\leadsto -\color{blue}{\left(\frac{x}{B} + \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right)} \]
   8. Step-by-step derivation

      1. *-commutative 59.0%

         \[\leadsto -\left(\frac{x}{B} + \color{blue}{B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}\right) \]

      2. distribute-rgt-out-- 59.0%

         \[\leadsto -\left(\frac{x}{B} + B \cdot \color{blue}{\left(x \cdot \left(-0.5 - -0.16666666666666666\right)\right)}\right) \]

      3. metadata-eval 59.0%

         \[\leadsto -\left(\frac{x}{B} + B \cdot \left(x \cdot \color{blue}{-0.3333333333333333}\right)\right) \]

   9. Simplified 59.0%

      \[\leadsto -\color{blue}{\left(\frac{x}{B} + B \cdot \left(x \cdot -0.3333333333333333\right)\right)} \]

   10. Taylor expanded in x around 0 59.0%

       \[\leadsto -\left(\frac{x}{B} + \color{blue}{-0.3333333333333333 \cdot \left(x \cdot B\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-241} \lor \neg \left(F \leq 2.5 \cdot 10^{-176}\right):\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B} - \left(x \cdot B\right) \cdot -0.3333333333333333\\ \end{array} \]

Alternative 25: 61.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -8.2 \cdot 10^{-241}:\\ \;\;\;\;\frac{-1}{B} - t_0\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-174}:\\ \;\;\;\;\frac{-x}{B} - \left(x \cdot B\right) \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -8.2e-241)
     (- (/ -1.0 B) t_0)
     (if (<= F 4.2e-174)
       (- (/ (- x) B) (* (* x B) -0.3333333333333333))
       (- (/ 1.0 B) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -8.2e-241) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 4.2e-174) {
		tmp = (-x / B) - ((x * B) * -0.3333333333333333);
	} 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 <= (-8.2d-241)) then
        tmp = ((-1.0d0) / b) - t_0
    else if (f <= 4.2d-174) then
        tmp = (-x / b) - ((x * b) * (-0.3333333333333333d0))
    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 <= -8.2e-241) {
		tmp = (-1.0 / B) - t_0;
	} else if (F <= 4.2e-174) {
		tmp = (-x / B) - ((x * B) * -0.3333333333333333);
	} 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 <= -8.2e-241:
		tmp = (-1.0 / B) - t_0
	elif F <= 4.2e-174:
		tmp = (-x / B) - ((x * B) * -0.3333333333333333)
	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 <= -8.2e-241)
		tmp = Float64(Float64(-1.0 / B) - t_0);
	elseif (F <= 4.2e-174)
		tmp = Float64(Float64(Float64(-x) / B) - Float64(Float64(x * B) * -0.3333333333333333));
	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 <= -8.2e-241)
		tmp = (-1.0 / B) - t_0;
	elseif (F <= 4.2e-174)
		tmp = (-x / B) - ((x * B) * -0.3333333333333333);
	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, -8.2e-241], N[(N[(-1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 4.2e-174], N[(N[((-x) / B), $MachinePrecision] - N[(N[(x * B), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.1999999999999997e-241

   1. Initial program 78.6%

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

   2. Taylor expanded in F around -inf 60.2%

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

   3. Taylor expanded in B around 0 42.9%

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

      1. expm1-log1p-u 23.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \frac{-1}{F}\right)\right)} \]

      2. expm1-udef 20.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(-x \cdot \frac{1}{\tan B}\right) + \frac{F}{B} \cdot \frac{-1}{F}\right)} - 1} \]

      3. +-commutative 20.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{F}{B} \cdot \frac{-1}{F} + \left(-x \cdot \frac{1}{\tan B}\right)}\right)} - 1 \]

      4. fma-def 20.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\mathsf{fma}\left(\frac{F}{B}, \frac{-1}{F}, -x \cdot \frac{1}{\tan B}\right)}\right)} - 1 \]

      5. un-div-inv 20.7%

         \[\leadsto e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{F}{B}, \frac{-1}{F}, -\color{blue}{\frac{x}{\tan B}}\right)\right)} - 1 \]

   5. Applied egg-rr 20.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\mathsf{fma}\left(\frac{F}{B}, \frac{-1}{F}, -\frac{x}{\tan B}\right)\right)} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 23.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{fma}\left(\frac{F}{B}, \frac{-1}{F}, -\frac{x}{\tan B}\right)\right)\right)} \]

      2. expm1-log1p 42.9%

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

      3. fma-neg 42.9%

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

      4. times-frac 54.0%

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

      5. *-commutative 54.0%

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

      6. times-frac 54.1%

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

      7. *-inverses 54.1%

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

      8. associate-*r/ 54.1%

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

      9. metadata-eval 54.1%

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

   7. Simplified 54.1%

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

   if -8.1999999999999997e-241 < F < 4.20000000000000021e-174

   1. Initial program 99.7%

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

      1. +-commutative 99.7%

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

      2. fma-def 99.7%

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

      3. +-commutative 99.7%

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

      4. *-commutative 99.7%

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

      5. fma-def 99.7%

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

      6. fma-def 99.7%

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

      7. metadata-eval 99.7%

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

      8. metadata-eval 99.7%

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

      9. distribute-lft-neg-in 99.7%

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

      10. associate-*r/ 99.9%

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

      11. *-rgt-identity 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in F around 0 90.5%

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

      1. mul-1-neg 90.5%

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

      2. *-commutative 90.5%

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

      3. associate-*l/ 90.6%

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

      4. *-commutative 90.6%

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

   6. Simplified 90.6%

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

   7. Taylor expanded in B around 0 59.0%

      \[\leadsto -\color{blue}{\left(\frac{x}{B} + \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right)} \]
   8. Step-by-step derivation

      1. *-commutative 59.0%

         \[\leadsto -\left(\frac{x}{B} + \color{blue}{B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}\right) \]

      2. distribute-rgt-out-- 59.0%

         \[\leadsto -\left(\frac{x}{B} + B \cdot \color{blue}{\left(x \cdot \left(-0.5 - -0.16666666666666666\right)\right)}\right) \]

      3. metadata-eval 59.0%

         \[\leadsto -\left(\frac{x}{B} + B \cdot \left(x \cdot \color{blue}{-0.3333333333333333}\right)\right) \]

   9. Simplified 59.0%

      \[\leadsto -\color{blue}{\left(\frac{x}{B} + B \cdot \left(x \cdot -0.3333333333333333\right)\right)} \]

   10. Taylor expanded in x around 0 59.0%

       \[\leadsto -\left(\frac{x}{B} + \color{blue}{-0.3333333333333333 \cdot \left(x \cdot B\right)}\right) \]

   if 4.20000000000000021e-174 < F

   1. Initial program 74.0%

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

      1. +-commutative 74.0%

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

      2. unsub-neg 74.0%

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

      3. associate-*l/ 81.6%

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

      4. associate-*r/ 81.6%

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

      5. *-commutative 81.6%

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

   3. Simplified 81.6%

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

   4. Taylor expanded in F around inf 77.8%

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

   5. Taylor expanded in B around 0 64.1%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.2 \cdot 10^{-241}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 4.2 \cdot 10^{-174}:\\ \;\;\;\;\frac{-x}{B} - \left(x \cdot B\right) \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]

Alternative 26: 37.8% accurate, 24.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.4 \cdot 10^{-34}:\\ \;\;\;\;\left(x \cdot B\right) \cdot 0.3333333333333333 - \frac{x + 1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B} - \left(x \cdot B\right) \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.4e-34)
   (- (* (* x B) 0.3333333333333333) (/ (+ x 1.0) B))
   (- (/ (- x) B) (* (* x B) -0.3333333333333333))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4e-34) {
		tmp = ((x * B) * 0.3333333333333333) - ((x + 1.0) / B);
	} else {
		tmp = (-x / B) - ((x * B) * -0.3333333333333333);
	}
	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 <= (-1.4d-34)) then
        tmp = ((x * b) * 0.3333333333333333d0) - ((x + 1.0d0) / b)
    else
        tmp = (-x / b) - ((x * b) * (-0.3333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.4e-34) {
		tmp = ((x * B) * 0.3333333333333333) - ((x + 1.0) / B);
	} else {
		tmp = (-x / B) - ((x * B) * -0.3333333333333333);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.4e-34:
		tmp = ((x * B) * 0.3333333333333333) - ((x + 1.0) / B)
	else:
		tmp = (-x / B) - ((x * B) * -0.3333333333333333)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.4e-34)
		tmp = Float64(Float64(Float64(x * B) * 0.3333333333333333) - Float64(Float64(x + 1.0) / B));
	else
		tmp = Float64(Float64(Float64(-x) / B) - Float64(Float64(x * B) * -0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.4e-34)
		tmp = ((x * B) * 0.3333333333333333) - ((x + 1.0) / B);
	else
		tmp = (-x / B) - ((x * B) * -0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.4e-34], N[(N[(N[(x * B), $MachinePrecision] * 0.3333333333333333), $MachinePrecision] - N[(N[(x + 1.0), $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / B), $MachinePrecision] - N[(N[(x * B), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.39999999999999998e-34

   1. Initial program 67.0%

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

   2. Taylor expanded in F around -inf 78.6%

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

   3. Taylor expanded in B around 0 47.8%

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

   4. Taylor expanded in B around 0 44.5%

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

   if -1.39999999999999998e-34 < F

   1. Initial program 83.7%

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

      1. +-commutative 83.7%

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

      2. fma-def 83.7%

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

      3. +-commutative 83.7%

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

      4. *-commutative 83.7%

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

      5. fma-def 83.7%

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

      6. fma-def 83.7%

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

      7. metadata-eval 83.7%

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

      8. metadata-eval 83.7%

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

      9. distribute-lft-neg-in 83.7%

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

      10. associate-*r/ 83.8%

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

      11. *-rgt-identity 83.8%

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

   3. Simplified 83.8%

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

   4. Taylor expanded in F around 0 52.1%

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

      1. mul-1-neg 52.1%

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

      2. *-commutative 52.1%

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

      3. associate-*l/ 52.2%

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

      4. *-commutative 52.2%

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

   6. Simplified 52.2%

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

   7. Taylor expanded in B around 0 29.6%

      \[\leadsto -\color{blue}{\left(\frac{x}{B} + \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right)} \]
   8. Step-by-step derivation

      1. *-commutative 29.6%

         \[\leadsto -\left(\frac{x}{B} + \color{blue}{B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}\right) \]

      2. distribute-rgt-out-- 29.6%

         \[\leadsto -\left(\frac{x}{B} + B \cdot \color{blue}{\left(x \cdot \left(-0.5 - -0.16666666666666666\right)\right)}\right) \]

      3. metadata-eval 29.6%

         \[\leadsto -\left(\frac{x}{B} + B \cdot \left(x \cdot \color{blue}{-0.3333333333333333}\right)\right) \]

   9. Simplified 29.6%

      \[\leadsto -\color{blue}{\left(\frac{x}{B} + B \cdot \left(x \cdot -0.3333333333333333\right)\right)} \]

   10. Taylor expanded in x around 0 29.6%

       \[\leadsto -\left(\frac{x}{B} + \color{blue}{-0.3333333333333333 \cdot \left(x \cdot B\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.4 \cdot 10^{-34}:\\ \;\;\;\;\left(x \cdot B\right) \cdot 0.3333333333333333 - \frac{x + 1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B} - \left(x \cdot B\right) \cdot -0.3333333333333333\\ \end{array} \]

Alternative 27: 37.7% accurate, 26.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{-21}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B} - \left(x \cdot B\right) \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.3e-21)
   (/ (- -1.0 x) B)
   (- (/ (- x) B) (* (* x B) -0.3333333333333333))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.3e-21) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = (-x / B) - ((x * B) * -0.3333333333333333);
	}
	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 <= (-1.3d-21)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = (-x / b) - ((x * b) * (-0.3333333333333333d0))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.3e-21) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = (-x / B) - ((x * B) * -0.3333333333333333);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.3e-21:
		tmp = (-1.0 - x) / B
	else:
		tmp = (-x / B) - ((x * B) * -0.3333333333333333)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.3e-21)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(Float64(Float64(-x) / B) - Float64(Float64(x * B) * -0.3333333333333333));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.3e-21)
		tmp = (-1.0 - x) / B;
	else
		tmp = (-x / B) - ((x * B) * -0.3333333333333333);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.3e-21], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(N[((-x) / B), $MachinePrecision] - N[(N[(x * B), $MachinePrecision] * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.30000000000000009e-21

   1. Initial program 66.6%

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

   2. Taylor expanded in F around -inf 78.3%

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

   3. Taylor expanded in B around 0 47.1%

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

   4. Taylor expanded in B around 0 45.0%

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

      1. associate-*r/ 45.0%

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

      2. distribute-lft-in 45.0%

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

      3. metadata-eval 45.0%

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

      4. neg-mul-1 45.0%

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

   6. Simplified 45.0%

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

   if -1.30000000000000009e-21 < F

   1. Initial program 83.8%

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

      1. +-commutative 83.8%

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

      2. fma-def 83.8%

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

      3. +-commutative 83.8%

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

      4. *-commutative 83.8%

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

      5. fma-def 83.8%

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

      6. fma-def 83.8%

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

      7. metadata-eval 83.8%

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

      8. metadata-eval 83.8%

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

      9. distribute-lft-neg-in 83.8%

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

      10. associate-*r/ 83.8%

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

      11. *-rgt-identity 83.8%

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

   3. Simplified 83.8%

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

   4. Taylor expanded in F around 0 52.4%

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

      1. mul-1-neg 52.4%

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

      2. *-commutative 52.4%

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

      3. associate-*l/ 52.4%

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

      4. *-commutative 52.4%

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

   6. Simplified 52.4%

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

   7. Taylor expanded in B around 0 29.4%

      \[\leadsto -\color{blue}{\left(\frac{x}{B} + \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right)} \]
   8. Step-by-step derivation

      1. *-commutative 29.4%

         \[\leadsto -\left(\frac{x}{B} + \color{blue}{B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}\right) \]

      2. distribute-rgt-out-- 29.4%

         \[\leadsto -\left(\frac{x}{B} + B \cdot \color{blue}{\left(x \cdot \left(-0.5 - -0.16666666666666666\right)\right)}\right) \]

      3. metadata-eval 29.4%

         \[\leadsto -\left(\frac{x}{B} + B \cdot \left(x \cdot \color{blue}{-0.3333333333333333}\right)\right) \]

   9. Simplified 29.4%

      \[\leadsto -\color{blue}{\left(\frac{x}{B} + B \cdot \left(x \cdot -0.3333333333333333\right)\right)} \]

   10. Taylor expanded in x around 0 29.4%

       \[\leadsto -\left(\frac{x}{B} + \color{blue}{-0.3333333333333333 \cdot \left(x \cdot B\right)}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{-21}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B} - \left(x \cdot B\right) \cdot -0.3333333333333333\\ \end{array} \]

Alternative 28: 37.7% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.5e-20)
   (/ (- -1.0 x) B)
   (* x (- (/ -1.0 B) (* B -0.3333333333333333)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.5e-20) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333));
	}
	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 <= (-2.5d-20)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = x * (((-1.0d0) / b) - (b * (-0.3333333333333333d0)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.5e-20) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.5e-20:
		tmp = (-1.0 - x) / B
	else:
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.5e-20)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(x * Float64(Float64(-1.0 / B) - Float64(B * -0.3333333333333333)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.5e-20)
		tmp = (-1.0 - x) / B;
	else
		tmp = x * ((-1.0 / B) - (B * -0.3333333333333333));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.5e-20], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(x * N[(N[(-1.0 / B), $MachinePrecision] - N[(B * -0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -2.4999999999999999e-20

   1. Initial program 66.6%

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

   2. Taylor expanded in F around -inf 78.3%

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

   3. Taylor expanded in B around 0 47.1%

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

   4. Taylor expanded in B around 0 45.0%

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

      1. associate-*r/ 45.0%

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

      2. distribute-lft-in 45.0%

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

      3. metadata-eval 45.0%

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

      4. neg-mul-1 45.0%

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

   6. Simplified 45.0%

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

   if -2.4999999999999999e-20 < F

   1. Initial program 83.8%

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

      1. +-commutative 83.8%

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

      2. fma-def 83.8%

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

      3. +-commutative 83.8%

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

      4. *-commutative 83.8%

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

      5. fma-def 83.8%

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

      6. fma-def 83.8%

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

      7. metadata-eval 83.8%

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

      8. metadata-eval 83.8%

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

      9. distribute-lft-neg-in 83.8%

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

      10. associate-*r/ 83.8%

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

      11. *-rgt-identity 83.8%

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

   3. Simplified 83.8%

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

   4. Taylor expanded in F around 0 52.4%

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

      1. mul-1-neg 52.4%

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

      2. *-commutative 52.4%

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

      3. associate-*l/ 52.4%

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

      4. *-commutative 52.4%

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

   6. Simplified 52.4%

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

   7. Taylor expanded in B around 0 29.4%

      \[\leadsto -\color{blue}{\left(\frac{x}{B} + \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right) \cdot B\right)} \]
   8. Step-by-step derivation

      1. *-commutative 29.4%

         \[\leadsto -\left(\frac{x}{B} + \color{blue}{B \cdot \left(-0.5 \cdot x - -0.16666666666666666 \cdot x\right)}\right) \]

      2. distribute-rgt-out-- 29.4%

         \[\leadsto -\left(\frac{x}{B} + B \cdot \color{blue}{\left(x \cdot \left(-0.5 - -0.16666666666666666\right)\right)}\right) \]

      3. metadata-eval 29.4%

         \[\leadsto -\left(\frac{x}{B} + B \cdot \left(x \cdot \color{blue}{-0.3333333333333333}\right)\right) \]

   9. Simplified 29.4%

      \[\leadsto -\color{blue}{\left(\frac{x}{B} + B \cdot \left(x \cdot -0.3333333333333333\right)\right)} \]

   10. Taylor expanded in x around 0 29.4%

       \[\leadsto -\color{blue}{\left(-0.3333333333333333 \cdot B + \frac{1}{B}\right) \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{-20}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{-1}{B} - B \cdot -0.3333333333333333\right)\\ \end{array} \]

Alternative 29: 37.6% accurate, 45.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.35 \cdot 10^{-40}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.35e-40) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.35e-40) {
		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 <= (-1.35d-40)) 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 <= -1.35e-40) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.35e-40:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.35e-40)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.35e-40)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.35e-40], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.35e-40

   1. Initial program 67.4%

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

   2. Taylor expanded in F around -inf 77.7%

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

   3. Taylor expanded in B around 0 47.2%

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

   4. Taylor expanded in B around 0 43.8%

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

      1. associate-*r/ 43.8%

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

      2. distribute-lft-in 43.8%

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

      3. metadata-eval 43.8%

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

      4. neg-mul-1 43.8%

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

   6. Simplified 43.8%

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

   if -1.35e-40 < F

   1. Initial program 83.6%

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

      1. +-commutative 83.6%

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

      2. fma-def 83.6%

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

      3. +-commutative 83.6%

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

      4. *-commutative 83.6%

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

      5. fma-def 83.6%

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

      6. fma-def 83.6%

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

      7. metadata-eval 83.6%

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

      8. metadata-eval 83.6%

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

      9. distribute-lft-neg-in 83.6%

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

      10. associate-*r/ 83.7%

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

      11. *-rgt-identity 83.7%

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

   3. Simplified 83.7%

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

   4. Taylor expanded in F around 0 52.4%

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

      1. mul-1-neg 52.4%

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

      2. *-commutative 52.4%

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

      3. associate-*l/ 52.4%

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

      4. *-commutative 52.4%

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

   6. Simplified 52.4%

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

   7. Taylor expanded in B around 0 29.5%

      \[\leadsto -\color{blue}{\frac{x}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.35 \cdot 10^{-40}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]

Alternative 30: 30.8% 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(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 79.0%

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

   1. +-commutative 79.0%

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

   2. fma-def 79.0%

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

   3. +-commutative 79.0%

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

   4. *-commutative 79.0%

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

   5. fma-def 79.0%

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

   6. fma-def 79.0%

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

   7. metadata-eval 79.0%

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

   8. metadata-eval 79.0%

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

   9. distribute-lft-neg-in 79.0%

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

   10. associate-*r/ 79.1%

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

   11. *-rgt-identity 79.1%

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

3. Simplified 79.1%

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

4. Taylor expanded in F around 0 51.1%

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

   1. mul-1-neg 51.1%

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

   2. *-commutative 51.1%

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

   3. associate-*l/ 51.2%

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

   4. *-commutative 51.2%

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

6. Simplified 51.2%

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

7. Taylor expanded in B around 0 28.6%

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

8. Final simplification 28.6%

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

Reproduce

herbie shell --seed 2023238 
(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))))))