VandenBroeck and Keller, Equation (23)

Percentage Accurate: 77.3% → 99.6%

Time: 18.6s

Alternatives: 22

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.3% 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: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{+66}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 14.5:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F \cdot {\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1e+66)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 14.5)
       (- (/ 1.0 (/ (sin B) (* F (pow (fma 2.0 x (fma F F 2.0)) -0.5)))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1e+66) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 14.5) {
		tmp = (1.0 / (sin(B) / (F * pow(fma(2.0, x, fma(F, F, 2.0)), -0.5)))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1e+66)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 14.5)
		tmp = Float64(Float64(1.0 / Float64(sin(B) / Float64(F * (fma(2.0, x, fma(F, F, 2.0)) ^ -0.5)))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e+66], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 14.5], N[(N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[(F * N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.99999999999999945e65

   1. Initial program 54.3%

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

   3. Simplified 72.9%

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

   5. Taylor expanded in F around -inf 99.9%

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

   if -9.99999999999999945e65 < F < 14.5

   1. Initial program 98.7%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 99.5%

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

      2. clear-num 99.6%

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

      3. fma-define 99.6%

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

      4. fma-undefine 99.6%

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

      5. *-commutative 99.6%

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

      6. fma-define 99.6%

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

      7. fma-define 99.6%

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

   6. Applied egg-rr 99.6%

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

   if 14.5 < F

   1. Initial program 67.0%

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

   3. Simplified 76.9%

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

   5. Taylor expanded in F around inf 99.8%

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

Alternative 2: 99.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2 \cdot 10^{+66}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 14.5:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2e+66)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 14.5)
       (- (* F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2e+66) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 14.5) {
		tmp = (F * (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2e+66)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 14.5)
		tmp = Float64(Float64(F * Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2e+66], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 14.5], 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] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.99999999999999989e66

   1. Initial program 53.6%

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

   3. Simplified 72.5%

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

   5. Taylor expanded in F around -inf 99.9%

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

   if -1.99999999999999989e66 < F < 14.5

   1. Initial program 98.7%

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

   3. Simplified 99.6%

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

   if 14.5 < F

   1. Initial program 67.0%

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

   3. Simplified 76.9%

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

   5. Taylor expanded in F around inf 99.8%

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

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.65 \cdot 10^{+21}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 14.5:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.65e+21)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 14.5)
       (+
        (/ -1.0 (/ (tan B) x))
        (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.65e+21) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 14.5) {
		tmp = (-1.0 / (tan(B) / x)) + ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.65d+21)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 14.5d0) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.65e+21) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 14.5) {
		tmp = (-1.0 / (Math.tan(B) / x)) + ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.65e+21:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 14.5:
		tmp = (-1.0 / (math.tan(B) / x)) + ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.65e+21)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 14.5)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.65e+21)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 14.5)
		tmp = (-1.0 / (tan(B) / x)) + ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.65e+21], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 14.5], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + 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[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.65e21

   1. Initial program 58.7%

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

   3. Simplified 76.1%

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

   5. Taylor expanded in F around -inf 99.9%

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

   if -1.65e21 < F < 14.5

   1. Initial program 99.4%

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

      1. div-inv 99.6%

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

      2. neg-mul-1 99.6%

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

      3. clear-num 99.4%

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

      4. un-div-inv 99.4%

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

   4. Applied egg-rr 99.4%

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

   if 14.5 < F

   1. Initial program 67.0%

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

   3. Simplified 76.9%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.65 \cdot 10^{+21}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 14.5:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.7 \cdot 10^{+34}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 14.5:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.7e+34)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 14.5)
       (+
        (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
        (* x (/ -1.0 (tan B))))
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.7e+34) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 14.5) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / tan(B)));
	} else {
		tmp = (1.0 / sin(B)) - 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 <= (-1.7d+34)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 14.5d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) + (x * ((-1.0d0) / tan(b)))
    else
        tmp = (1.0d0 / sin(b)) - 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 <= -1.7e+34) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 14.5) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / Math.tan(B)));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.7e+34:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 14.5:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / math.tan(B)))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.7e+34)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 14.5)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) + Float64(x * Float64(-1.0 / tan(B))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.7e+34)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 14.5)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) + (x * (-1.0 / tan(B)));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.7e+34], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 14.5], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] + N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.7e34

   1. Initial program 56.9%

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

   3. Simplified 75.2%

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

   5. Taylor expanded in F around -inf 99.9%

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

   if -1.7e34 < F < 14.5

   1. Initial program 99.4%

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

      1. metadata-eval 99.4%

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

      2. metadata-eval 99.4%

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

   4. Applied egg-rr 99.4%

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

   if 14.5 < F

   1. Initial program 67.0%

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

   3. Simplified 76.9%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{+34}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 14.5:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.45:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F} \cdot \sqrt{2 + x \cdot 2}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.45)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.45)
       (- (/ 1.0 (* (/ (sin B) F) (sqrt (+ 2.0 (* x 2.0))))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.45) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.45) {
		tmp = (1.0 / ((sin(B) / F) * sqrt((2.0 + (x * 2.0))))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.45d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.45d0) then
        tmp = (1.0d0 / ((sin(b) / f) * sqrt((2.0d0 + (x * 2.0d0))))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.45) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.45) {
		tmp = (1.0 / ((Math.sin(B) / F) * Math.sqrt((2.0 + (x * 2.0))))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.45:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.45:
		tmp = (1.0 / ((math.sin(B) / F) * math.sqrt((2.0 + (x * 2.0))))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.45)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.45)
		tmp = Float64(Float64(1.0 / Float64(Float64(sin(B) / F) * sqrt(Float64(2.0 + Float64(x * 2.0))))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.45)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.45)
		tmp = (1.0 / ((sin(B) / F) * sqrt((2.0 + (x * 2.0))))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.45], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.45], N[(N[(1.0 / N[(N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision] * N[Sqrt[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.44999999999999996

   1. Initial program 61.3%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in F around -inf 98.3%

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

   if -1.44999999999999996 < F < 1.44999999999999996

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

      1. associate-*r/ 99.5%

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

      2. clear-num 99.6%

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

      3. fma-define 99.6%

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

      4. fma-undefine 99.6%

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

      5. *-commutative 99.6%

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

      6. fma-define 99.6%

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

      7. fma-define 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in F around 0 99.0%

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

   if 1.44999999999999996 < F

   1. Initial program 67.0%

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

   3. Simplified 76.9%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.45:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.45:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F} \cdot \sqrt{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.55:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.55:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.55)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.55)
       (- (/ F (* (sin B) (sqrt (+ 2.0 (* x 2.0))))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.55) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.55) {
		tmp = (F / (sin(B) * sqrt((2.0 + (x * 2.0))))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.55d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.55d0) then
        tmp = (f / (sin(b) * sqrt((2.0d0 + (x * 2.0d0))))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.55) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.55) {
		tmp = (F / (Math.sin(B) * Math.sqrt((2.0 + (x * 2.0))))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.55:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.55:
		tmp = (F / (math.sin(B) * math.sqrt((2.0 + (x * 2.0))))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.55)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.55)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(Float64(2.0 + Float64(x * 2.0))))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.55)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.55)
		tmp = (F / (sin(B) * sqrt((2.0 + (x * 2.0))))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.55], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.55], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.55000000000000004

   1. Initial program 61.3%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in F around -inf 98.3%

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

   if -1.55000000000000004 < F < 1.55000000000000004

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. un-div-inv 99.5%

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

      3. fma-define 99.5%

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

      4. fma-undefine 99.5%

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

      5. *-commutative 99.5%

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

      6. fma-define 99.5%

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

      7. fma-define 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in F around 0 98.9%

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

   if 1.55000000000000004 < F

   1. Initial program 67.0%

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

   3. Simplified 76.9%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.55:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.55:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.42:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.42)
       (- (/ F (* (sin B) (sqrt 2.0))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.42) {
		tmp = (F / (sin(B) * sqrt(2.0))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.4d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.42d0) then
        tmp = (f / (sin(b) * sqrt(2.0d0))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.42) {
		tmp = (F / (Math.sin(B) * Math.sqrt(2.0))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.4:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.42:
		tmp = (F / (math.sin(B) * math.sqrt(2.0))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.42)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(2.0))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.4)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.42)
		tmp = (F / (sin(B) * sqrt(2.0))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.42], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 61.3%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in F around -inf 98.3%

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

   if -1.3999999999999999 < F < 1.4199999999999999

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

      1. clear-num 99.5%

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

      2. un-div-inv 99.5%

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

      3. fma-define 99.5%

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

      4. fma-undefine 99.5%

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

      5. *-commutative 99.5%

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

      6. fma-define 99.5%

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

      7. fma-define 99.5%

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

   6. Applied egg-rr 99.5%

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

   7. Taylor expanded in F around 0 98.9%

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

   8. Taylor expanded in x around 0 98.5%

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

   if 1.4199999999999999 < F

   1. Initial program 67.0%

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

   3. Simplified 76.9%

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

   5. Taylor expanded in F around inf 99.8%

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

Alternative 8: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.4:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.42:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.4)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.42)
       (- (* F (/ (sqrt 0.5) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.42) {
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.4d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.42d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.4) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.42) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.4:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.42:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.4)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.42)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.4)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.42)
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.4], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.42], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.3999999999999999

   1. Initial program 61.3%

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

   3. Simplified 77.7%

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

   5. Taylor expanded in F around -inf 98.3%

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

   if -1.3999999999999999 < F < 1.4199999999999999

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in F around 0 98.9%

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

      1. *-commutative 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in x around 0 98.4%

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

   if 1.4199999999999999 < F

   1. Initial program 67.0%

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

   3. Simplified 76.9%

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

   5. Taylor expanded in F around inf 99.8%

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

Alternative 9: 85.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -7.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{-51}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 0.116:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -7.5e-52)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.5e-51)
       (/ x (- (tan B)))
       (if (<= F 0.116)
         (- (/ F (* (sin B) (sqrt (+ 2.0 (* x 2.0))))) (/ x B))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -7.5e-52) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.5e-51) {
		tmp = x / -tan(B);
	} else if (F <= 0.116) {
		tmp = (F / (sin(B) * sqrt((2.0 + (x * 2.0))))) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-7.5d-52)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.5d-51) then
        tmp = x / -tan(b)
    else if (f <= 0.116d0) then
        tmp = (f / (sin(b) * sqrt((2.0d0 + (x * 2.0d0))))) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -7.5e-52) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.5e-51) {
		tmp = x / -Math.tan(B);
	} else if (F <= 0.116) {
		tmp = (F / (Math.sin(B) * Math.sqrt((2.0 + (x * 2.0))))) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -7.5e-52:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.5e-51:
		tmp = x / -math.tan(B)
	elif F <= 0.116:
		tmp = (F / (math.sin(B) * math.sqrt((2.0 + (x * 2.0))))) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -7.5e-52)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.5e-51)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 0.116)
		tmp = Float64(Float64(F / Float64(sin(B) * sqrt(Float64(2.0 + Float64(x * 2.0))))) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -7.5e-52)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.5e-51)
		tmp = x / -tan(B);
	elseif (F <= 0.116)
		tmp = (F / (sin(B) * sqrt((2.0 + (x * 2.0))))) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -7.5e-52], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.5e-51], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 0.116], N[(N[(F / N[(N[Sin[B], $MachinePrecision] * N[Sqrt[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7.50000000000000006e-52

   1. Initial program 63.9%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in F around -inf 95.1%

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

   if -7.50000000000000006e-52 < F < 1.50000000000000001e-51

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 99.6%

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

      2. clear-num 99.7%

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

      3. fma-define 99.7%

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

      4. fma-undefine 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. fma-define 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in F around 0 77.0%

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

      1. mul-1-neg 77.0%

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

      2. associate-/l* 76.9%

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

      3. distribute-rgt-neg-in 76.9%

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

      4. distribute-neg-frac2 76.9%

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

   9. Simplified 76.9%

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

       1. clear-num 76.9%

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

       2. add-sqr-sqrt 42.5%

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

       3. sqrt-unprod 35.9%

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

       4. sqr-neg 35.9%

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

       5. sqrt-unprod 1.3%

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

       6. add-sqr-sqrt 2.3%

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

       7. tan-quot 2.3%

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

       8. div-inv 2.3%

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

       9. add-sqr-sqrt 1.3%

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

       10. sqrt-unprod 30.3%

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

       11. clear-num 30.2%

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

       12. clear-num 30.2%

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

       13. frac-times 30.3%

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

       14. metadata-eval 30.3%

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

       15. metadata-eval 30.3%

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

       16. frac-times 30.2%

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

       17. sqrt-unprod 36.0%

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

       18. add-sqr-sqrt 76.9%

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

   11. Applied egg-rr 76.9%

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

       1. div-inv 76.9%

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

       2. clear-num 77.1%

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

   13. Applied egg-rr 77.1%

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

       1. neg-mul-1 77.1%

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

       2. distribute-neg-frac 77.1%

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

   15. Simplified 77.1%

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

   if 1.50000000000000001e-51 < F < 0.116000000000000006

   1. Initial program 98.8%

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

   3. Simplified 98.7%

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

      1. clear-num 98.3%

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

      2. un-div-inv 98.5%

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

      3. fma-define 98.5%

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

      4. fma-undefine 98.5%

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

      5. *-commutative 98.5%

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

      6. fma-define 98.5%

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

      7. fma-define 98.5%

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

   6. Applied egg-rr 98.5%

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

   7. Taylor expanded in F around 0 92.3%

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

   8. Taylor expanded in B around 0 82.6%

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

   if 0.116000000000000006 < F

   1. Initial program 67.0%

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

   3. Simplified 76.9%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.5 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.5 \cdot 10^{-51}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 0.116:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 91.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.1 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.275:\\ \;\;\;\;F \cdot \left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2.1e-15)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.275)
       (- (* F (* (/ 1.0 B) (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2.1e-15) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.275) {
		tmp = (F * ((1.0 / B) * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-2.1d-15)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.275d0) then
        tmp = (f * ((1.0d0 / b) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -2.1e-15) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.275) {
		tmp = (F * ((1.0 / B) * Math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -2.1e-15:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.275:
		tmp = (F * ((1.0 / B) * math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.1e-15)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.275)
		tmp = Float64(Float64(F * Float64(Float64(1.0 / B) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -2.1e-15)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.275)
		tmp = (F * ((1.0 / B) * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.1e-15], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.275], N[(N[(F * N[(N[(1.0 / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.09999999999999981e-15

   1. Initial program 62.7%

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

   3. Simplified 78.5%

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

   5. Taylor expanded in F around -inf 96.1%

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

   if -2.09999999999999981e-15 < F < 0.27500000000000002

   1. Initial program 99.4%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in F around 0 98.9%

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

      1. *-commutative 98.9%

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

   7. Simplified 98.9%

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

   8. Taylor expanded in B around 0 83.0%

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

   if 0.27500000000000002 < F

   1. Initial program 67.0%

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

   3. Simplified 76.9%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.1 \cdot 10^{-15}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.275:\\ \;\;\;\;F \cdot \left(\frac{1}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 84.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 0.047:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F \cdot \sqrt{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -6.6e-52)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.4e-46)
       (/ x (- (tan B)))
       (if (<= F 0.047)
         (/ 1.0 (/ (sin B) (* F (sqrt 0.5))))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -6.6e-52) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.4e-46) {
		tmp = x / -tan(B);
	} else if (F <= 0.047) {
		tmp = 1.0 / (sin(B) / (F * sqrt(0.5)));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-6.6d-52)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.4d-46) then
        tmp = x / -tan(b)
    else if (f <= 0.047d0) then
        tmp = 1.0d0 / (sin(b) / (f * sqrt(0.5d0)))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -6.6e-52) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.4e-46) {
		tmp = x / -Math.tan(B);
	} else if (F <= 0.047) {
		tmp = 1.0 / (Math.sin(B) / (F * Math.sqrt(0.5)));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -6.6e-52:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.4e-46:
		tmp = x / -math.tan(B)
	elif F <= 0.047:
		tmp = 1.0 / (math.sin(B) / (F * math.sqrt(0.5)))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -6.6e-52)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4e-46)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 0.047)
		tmp = Float64(1.0 / Float64(sin(B) / Float64(F * sqrt(0.5))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -6.6e-52)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.4e-46)
		tmp = x / -tan(B);
	elseif (F <= 0.047)
		tmp = 1.0 / (sin(B) / (F * sqrt(0.5)));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -6.6e-52], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4e-46], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 0.047], N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -6.5999999999999999e-52

   1. Initial program 63.9%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in F around -inf 95.1%

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

   if -6.5999999999999999e-52 < F < 1.3999999999999999e-46

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 99.6%

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

      2. clear-num 99.7%

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

      3. fma-define 99.7%

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

      4. fma-undefine 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. fma-define 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in F around 0 77.0%

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

      1. mul-1-neg 77.0%

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

      2. associate-/l* 76.9%

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

      3. distribute-rgt-neg-in 76.9%

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

      4. distribute-neg-frac2 76.9%

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

   9. Simplified 76.9%

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

       1. clear-num 76.9%

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

       2. add-sqr-sqrt 42.5%

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

       3. sqrt-unprod 35.9%

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

       4. sqr-neg 35.9%

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

       5. sqrt-unprod 1.3%

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

       6. add-sqr-sqrt 2.3%

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

       7. tan-quot 2.3%

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

       8. div-inv 2.3%

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

       9. add-sqr-sqrt 1.3%

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

       10. sqrt-unprod 30.3%

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

       11. clear-num 30.2%

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

       12. clear-num 30.2%

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

       13. frac-times 30.3%

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

       14. metadata-eval 30.3%

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

       15. metadata-eval 30.3%

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

       16. frac-times 30.2%

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

       17. sqrt-unprod 36.0%

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

       18. add-sqr-sqrt 76.9%

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

   11. Applied egg-rr 76.9%

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

       1. div-inv 76.9%

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

       2. clear-num 77.1%

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

   13. Applied egg-rr 77.1%

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

       1. neg-mul-1 77.1%

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

       2. distribute-neg-frac 77.1%

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

   15. Simplified 77.1%

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

   if 1.3999999999999999e-46 < F < 0.047

   1. Initial program 98.8%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in F around 0 91.8%

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

      1. *-commutative 91.8%

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

   7. Simplified 91.8%

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

   8. Taylor expanded in x around 0 75.2%

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

      1. associate-/l* 75.2%

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

   10. Simplified 75.2%

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

       1. associate-*r/ 75.2%

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

       2. clear-num 75.4%

          \[\leadsto \color{blue}{\frac{1}{\frac{\sin B}{F \cdot \sqrt{0.5}}}} \]

   12. Applied egg-rr 75.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{\sin B}{F \cdot \sqrt{0.5}}}} \]

   if 0.047 < F

   1. Initial program 67.0%

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

   3. Simplified 76.9%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 0.047:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F \cdot \sqrt{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 84.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 0.096:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -6.6e-52)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.4e-46)
       (/ x (- (tan B)))
       (if (<= F 0.096)
         (* F (/ (sqrt 0.5) (sin B)))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -6.6e-52) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.4e-46) {
		tmp = x / -tan(B);
	} else if (F <= 0.096) {
		tmp = F * (sqrt(0.5) / sin(B));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-6.6d-52)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.4d-46) then
        tmp = x / -tan(b)
    else if (f <= 0.096d0) then
        tmp = f * (sqrt(0.5d0) / sin(b))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -6.6e-52) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.4e-46) {
		tmp = x / -Math.tan(B);
	} else if (F <= 0.096) {
		tmp = F * (Math.sqrt(0.5) / Math.sin(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -6.6e-52:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.4e-46:
		tmp = x / -math.tan(B)
	elif F <= 0.096:
		tmp = F * (math.sqrt(0.5) / math.sin(B))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -6.6e-52)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4e-46)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 0.096)
		tmp = Float64(F * Float64(sqrt(0.5) / sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -6.6e-52)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.4e-46)
		tmp = x / -tan(B);
	elseif (F <= 0.096)
		tmp = F * (sqrt(0.5) / sin(B));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -6.6e-52], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4e-46], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 0.096], N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -6.5999999999999999e-52

   1. Initial program 63.9%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in F around -inf 95.1%

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

   if -6.5999999999999999e-52 < F < 1.3999999999999999e-46

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 99.6%

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

      2. clear-num 99.7%

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

      3. fma-define 99.7%

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

      4. fma-undefine 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. fma-define 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in F around 0 77.0%

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

      1. mul-1-neg 77.0%

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

      2. associate-/l* 76.9%

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

      3. distribute-rgt-neg-in 76.9%

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

      4. distribute-neg-frac2 76.9%

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

   9. Simplified 76.9%

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

       1. clear-num 76.9%

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

       2. add-sqr-sqrt 42.5%

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

       3. sqrt-unprod 35.9%

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

       4. sqr-neg 35.9%

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

       5. sqrt-unprod 1.3%

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

       6. add-sqr-sqrt 2.3%

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

       7. tan-quot 2.3%

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

       8. div-inv 2.3%

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

       9. add-sqr-sqrt 1.3%

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

       10. sqrt-unprod 30.3%

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

       11. clear-num 30.2%

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

       12. clear-num 30.2%

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

       13. frac-times 30.3%

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

       14. metadata-eval 30.3%

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

       15. metadata-eval 30.3%

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

       16. frac-times 30.2%

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

       17. sqrt-unprod 36.0%

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

       18. add-sqr-sqrt 76.9%

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

   11. Applied egg-rr 76.9%

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

       1. div-inv 76.9%

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

       2. clear-num 77.1%

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

   13. Applied egg-rr 77.1%

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

       1. neg-mul-1 77.1%

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

       2. distribute-neg-frac 77.1%

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

   15. Simplified 77.1%

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

   if 1.3999999999999999e-46 < F < 0.096000000000000002

   1. Initial program 98.8%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in F around 0 91.8%

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

      1. *-commutative 91.8%

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

   7. Simplified 91.8%

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

   8. Taylor expanded in x around 0 75.2%

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

      1. associate-/l* 75.2%

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

   10. Simplified 75.2%

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

   if 0.096000000000000002 < F

   1. Initial program 67.0%

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

   3. Simplified 76.9%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-46}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 0.096:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 78.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-47}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 0.047:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.8e-52)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 1.4e-47)
     (/ x (- (tan B)))
     (if (<= F 0.047)
       (* F (/ (sqrt 0.5) (sin B)))
       (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.8e-52) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 1.4e-47) {
		tmp = x / -tan(B);
	} else if (F <= 0.047) {
		tmp = F * (sqrt(0.5) / sin(B));
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-7.8d-52)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 1.4d-47) then
        tmp = x / -tan(b)
    else if (f <= 0.047d0) then
        tmp = f * (sqrt(0.5d0) / sin(b))
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.8e-52) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 1.4e-47) {
		tmp = x / -Math.tan(B);
	} else if (F <= 0.047) {
		tmp = F * (Math.sqrt(0.5) / Math.sin(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -7.8e-52:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 1.4e-47:
		tmp = x / -math.tan(B)
	elif F <= 0.047:
		tmp = F * (math.sqrt(0.5) / math.sin(B))
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.8e-52)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 1.4e-47)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 0.047)
		tmp = Float64(F * Float64(sqrt(0.5) / sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -7.8e-52)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 1.4e-47)
		tmp = x / -tan(B);
	elseif (F <= 0.047)
		tmp = F * (sqrt(0.5) / sin(B));
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.8e-52], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.4e-47], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 0.047], N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7.80000000000000036e-52

   1. Initial program 63.9%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in F around -inf 95.1%

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

   if -7.80000000000000036e-52 < F < 1.39999999999999996e-47

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 99.6%

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

      2. clear-num 99.7%

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

      3. fma-define 99.7%

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

      4. fma-undefine 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. fma-define 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in F around 0 77.0%

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

      1. mul-1-neg 77.0%

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

      2. associate-/l* 76.9%

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

      3. distribute-rgt-neg-in 76.9%

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

      4. distribute-neg-frac2 76.9%

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

   9. Simplified 76.9%

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

       1. clear-num 76.9%

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

       2. add-sqr-sqrt 42.5%

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

       3. sqrt-unprod 35.9%

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

       4. sqr-neg 35.9%

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

       5. sqrt-unprod 1.3%

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

       6. add-sqr-sqrt 2.3%

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

       7. tan-quot 2.3%

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

       8. div-inv 2.3%

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

       9. add-sqr-sqrt 1.3%

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

       10. sqrt-unprod 30.3%

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

       11. clear-num 30.2%

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

       12. clear-num 30.2%

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

       13. frac-times 30.3%

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

       14. metadata-eval 30.3%

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

       15. metadata-eval 30.3%

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

       16. frac-times 30.2%

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

       17. sqrt-unprod 36.0%

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

       18. add-sqr-sqrt 76.9%

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

   11. Applied egg-rr 76.9%

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

       1. div-inv 76.9%

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

       2. clear-num 77.1%

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

   13. Applied egg-rr 77.1%

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

       1. neg-mul-1 77.1%

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

       2. distribute-neg-frac 77.1%

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

   15. Simplified 77.1%

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

   if 1.39999999999999996e-47 < F < 0.047

   1. Initial program 98.8%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in F around 0 91.8%

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

      1. *-commutative 91.8%

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

   7. Simplified 91.8%

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

   8. Taylor expanded in x around 0 75.2%

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

      1. associate-/l* 75.2%

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

   10. Simplified 75.2%

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

   if 0.047 < F

   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. Add Preprocessing

   3. Taylor expanded in B around 0 42.4%

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

      1. associate-*r/ 42.4%

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

      2. neg-mul-1 42.4%

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

   5. Simplified 42.4%

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

   6. Taylor expanded in F around inf 75.1%

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

      1. neg-mul-1 75.1%

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

      2. distribute-frac-neg 75.1%

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

      3. +-commutative 75.1%

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

      4. distribute-frac-neg 75.1%

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

      5. unsub-neg 75.1%

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

   8. Simplified 75.1%

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

4. Final simplification 82.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-47}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 0.047:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 71.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.85 \cdot 10^{-53}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-51}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 0.08:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.85e-53)
   (- (/ -1.0 B) (/ x (tan B)))
   (if (<= F 3.6e-51)
     (/ x (- (tan B)))
     (if (<= F 0.08)
       (* F (/ (sqrt 0.5) (sin B)))
       (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.85e-53) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= 3.6e-51) {
		tmp = x / -tan(B);
	} else if (F <= 0.08) {
		tmp = F * (sqrt(0.5) / sin(B));
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.85d-53)) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (f <= 3.6d-51) then
        tmp = x / -tan(b)
    else if (f <= 0.08d0) then
        tmp = f * (sqrt(0.5d0) / sin(b))
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.85e-53) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (F <= 3.6e-51) {
		tmp = x / -Math.tan(B);
	} else if (F <= 0.08) {
		tmp = F * (Math.sqrt(0.5) / Math.sin(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.85e-53:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif F <= 3.6e-51:
		tmp = x / -math.tan(B)
	elif F <= 0.08:
		tmp = F * (math.sqrt(0.5) / math.sin(B))
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.85e-53)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= 3.6e-51)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 0.08)
		tmp = Float64(F * Float64(sqrt(0.5) / sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.85e-53)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (F <= 3.6e-51)
		tmp = x / -tan(B);
	elseif (F <= 0.08)
		tmp = F * (sqrt(0.5) / sin(B));
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.85e-53], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.6e-51], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 0.08], N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.84999999999999991e-53

   1. Initial program 63.9%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in F around -inf 95.1%

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

   6. Taylor expanded in B around 0 71.8%

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

   if -1.84999999999999991e-53 < F < 3.6e-51

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 99.6%

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

      2. clear-num 99.7%

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

      3. fma-define 99.7%

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

      4. fma-undefine 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. fma-define 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in F around 0 77.0%

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

      1. mul-1-neg 77.0%

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

      2. associate-/l* 76.9%

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

      3. distribute-rgt-neg-in 76.9%

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

      4. distribute-neg-frac2 76.9%

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

   9. Simplified 76.9%

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

       1. clear-num 76.9%

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

       2. add-sqr-sqrt 42.5%

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

       3. sqrt-unprod 35.9%

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

       4. sqr-neg 35.9%

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

       5. sqrt-unprod 1.3%

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

       6. add-sqr-sqrt 2.3%

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

       7. tan-quot 2.3%

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

       8. div-inv 2.3%

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

       9. add-sqr-sqrt 1.3%

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

       10. sqrt-unprod 30.3%

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

       11. clear-num 30.2%

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

       12. clear-num 30.2%

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

       13. frac-times 30.3%

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

       14. metadata-eval 30.3%

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

       15. metadata-eval 30.3%

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

       16. frac-times 30.2%

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

       17. sqrt-unprod 36.0%

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

       18. add-sqr-sqrt 76.9%

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

   11. Applied egg-rr 76.9%

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

       1. div-inv 76.9%

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

       2. clear-num 77.1%

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

   13. Applied egg-rr 77.1%

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

       1. neg-mul-1 77.1%

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

       2. distribute-neg-frac 77.1%

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

   15. Simplified 77.1%

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

   if 3.6e-51 < F < 0.0800000000000000017

   1. Initial program 98.8%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in F around 0 91.8%

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

      1. *-commutative 91.8%

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

   7. Simplified 91.8%

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

   8. Taylor expanded in x around 0 75.2%

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

      1. associate-/l* 75.2%

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

   10. Simplified 75.2%

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

   if 0.0800000000000000017 < F

   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. Add Preprocessing

   3. Taylor expanded in B around 0 42.4%

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

      1. associate-*r/ 42.4%

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

      2. neg-mul-1 42.4%

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

   5. Simplified 42.4%

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

   6. Taylor expanded in F around inf 75.1%

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

      1. neg-mul-1 75.1%

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

      2. distribute-frac-neg 75.1%

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

      3. +-commutative 75.1%

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

      4. distribute-frac-neg 75.1%

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

      5. unsub-neg 75.1%

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

   8. Simplified 75.1%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.85 \cdot 10^{-53}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.6 \cdot 10^{-51}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 0.08:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 71.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -8.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 13.5:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -8.5e-53)
   (- (/ -1.0 B) (/ x (tan B)))
   (if (<= F 1.1e-44)
     (/ x (- (tan B)))
     (if (<= F 13.5)
       (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)) (/ x B))
       (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.5e-53) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= 1.1e-44) {
		tmp = x / -tan(B);
	} else if (F <= 13.5) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-8.5d-53)) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (f <= 1.1d-44) then
        tmp = x / -tan(b)
    else if (f <= 13.5d0) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -8.5e-53) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (F <= 1.1e-44) {
		tmp = x / -Math.tan(B);
	} else if (F <= 13.5) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -8.5e-53:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif F <= 1.1e-44:
		tmp = x / -math.tan(B)
	elif F <= 13.5:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -8.5e-53)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= 1.1e-44)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 13.5)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -8.5e-53)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (F <= 1.1e-44)
		tmp = x / -tan(B);
	elseif (F <= 13.5)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -8.5e-53], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.1e-44], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 13.5], 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], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -8.50000000000000044e-53

   1. Initial program 63.9%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in F around -inf 95.1%

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

   6. Taylor expanded in B around 0 71.8%

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

   if -8.50000000000000044e-53 < F < 1.10000000000000006e-44

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 99.6%

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

      2. clear-num 99.7%

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

      3. fma-define 99.7%

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

      4. fma-undefine 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. fma-define 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in F around 0 77.0%

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

      1. mul-1-neg 77.0%

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

      2. associate-/l* 76.9%

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

      3. distribute-rgt-neg-in 76.9%

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

      4. distribute-neg-frac2 76.9%

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

   9. Simplified 76.9%

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

       1. clear-num 76.9%

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

       2. add-sqr-sqrt 42.5%

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

       3. sqrt-unprod 35.9%

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

       4. sqr-neg 35.9%

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

       5. sqrt-unprod 1.3%

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

       6. add-sqr-sqrt 2.3%

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

       7. tan-quot 2.3%

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

       8. div-inv 2.3%

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

       9. add-sqr-sqrt 1.3%

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

       10. sqrt-unprod 30.3%

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

       11. clear-num 30.2%

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

       12. clear-num 30.2%

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

       13. frac-times 30.3%

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

       14. metadata-eval 30.3%

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

       15. metadata-eval 30.3%

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

       16. frac-times 30.2%

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

       17. sqrt-unprod 36.0%

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

       18. add-sqr-sqrt 76.9%

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

   11. Applied egg-rr 76.9%

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

       1. div-inv 76.9%

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

       2. clear-num 77.1%

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

   13. Applied egg-rr 77.1%

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

       1. neg-mul-1 77.1%

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

       2. distribute-neg-frac 77.1%

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

   15. Simplified 77.1%

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

   if 1.10000000000000006e-44 < F < 13.5

   1. Initial program 98.8%

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

   3. Taylor expanded in B around 0 89.1%

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

      1. associate-*r/ 89.1%

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

      2. neg-mul-1 89.1%

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

   5. Simplified 89.1%

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

      1. metadata-eval 98.8%

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

      2. metadata-eval 98.8%

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

   7. Applied egg-rr 89.1%

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

   8. Taylor expanded in B around 0 60.1%

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

   if 13.5 < F

   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. Add Preprocessing

   3. Taylor expanded in B around 0 42.4%

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

      1. associate-*r/ 42.4%

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

      2. neg-mul-1 42.4%

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

   5. Simplified 42.4%

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

   6. Taylor expanded in F around inf 75.1%

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

      1. neg-mul-1 75.1%

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

      2. distribute-frac-neg 75.1%

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

      3. +-commutative 75.1%

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

      4. distribute-frac-neg 75.1%

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

      5. unsub-neg 75.1%

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

   8. Simplified 75.1%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.5 \cdot 10^{-53}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 1.1 \cdot 10^{-44}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 13.5:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 71.5% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.4 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 8.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 0.17:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.4e-52)
   (- (/ -1.0 B) (/ x (tan B)))
   (if (<= F 8.2e-51)
     (/ x (- (tan B)))
     (if (<= F 0.17)
       (/ (- (* F (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) x) B)
       (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.4e-52) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= 8.2e-51) {
		tmp = x / -tan(B);
	} else if (F <= 0.17) {
		tmp = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-6.4d-52)) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (f <= 8.2d-51) then
        tmp = x / -tan(b)
    else if (f <= 0.17d0) then
        tmp = ((f * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - x) / b
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.4e-52) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (F <= 8.2e-51) {
		tmp = x / -Math.tan(B);
	} else if (F <= 0.17) {
		tmp = ((F * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6.4e-52:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif F <= 8.2e-51:
		tmp = x / -math.tan(B)
	elif F <= 0.17:
		tmp = ((F * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.4e-52)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= 8.2e-51)
		tmp = Float64(x / Float64(-tan(B)));
	elseif (F <= 0.17)
		tmp = Float64(Float64(Float64(F * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - x) / B);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -6.4e-52)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (F <= 8.2e-51)
		tmp = x / -tan(B);
	elseif (F <= 0.17)
		tmp = ((F * sqrt((1.0 / (2.0 + (x * 2.0))))) - x) / B;
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.4e-52], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8.2e-51], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 0.17], N[(N[(N[(F * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -6.4000000000000002e-52

   1. Initial program 63.9%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in F around -inf 95.1%

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

   6. Taylor expanded in B around 0 71.8%

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

   if -6.4000000000000002e-52 < F < 8.19999999999999947e-51

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

      1. associate-*r/ 99.6%

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

      2. clear-num 99.7%

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

      3. fma-define 99.7%

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

      4. fma-undefine 99.7%

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

      5. *-commutative 99.7%

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

      6. fma-define 99.7%

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

      7. fma-define 99.7%

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

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in F around 0 77.0%

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

      1. mul-1-neg 77.0%

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

      2. associate-/l* 76.9%

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

      3. distribute-rgt-neg-in 76.9%

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

      4. distribute-neg-frac2 76.9%

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

   9. Simplified 76.9%

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

       1. clear-num 76.9%

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

       2. add-sqr-sqrt 42.5%

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

       3. sqrt-unprod 35.9%

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

       4. sqr-neg 35.9%

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

       5. sqrt-unprod 1.3%

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

       6. add-sqr-sqrt 2.3%

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

       7. tan-quot 2.3%

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

       8. div-inv 2.3%

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

       9. add-sqr-sqrt 1.3%

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

       10. sqrt-unprod 30.3%

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

       11. clear-num 30.2%

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

       12. clear-num 30.2%

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

       13. frac-times 30.3%

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

       14. metadata-eval 30.3%

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

       15. metadata-eval 30.3%

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

       16. frac-times 30.2%

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

       17. sqrt-unprod 36.0%

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

       18. add-sqr-sqrt 76.9%

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

   11. Applied egg-rr 76.9%

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

       1. div-inv 76.9%

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

       2. clear-num 77.1%

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

   13. Applied egg-rr 77.1%

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

       1. neg-mul-1 77.1%

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

       2. distribute-neg-frac 77.1%

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

   15. Simplified 77.1%

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

   if 8.19999999999999947e-51 < F < 0.170000000000000012

   1. Initial program 98.8%

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

   3. Simplified 98.7%

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

   5. Taylor expanded in F around 0 91.8%

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

      1. *-commutative 91.8%

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

   7. Simplified 91.8%

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

   8. Taylor expanded in B around 0 54.0%

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

   if 0.170000000000000012 < F

   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. Add Preprocessing

   3. Taylor expanded in B around 0 42.4%

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

      1. associate-*r/ 42.4%

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

      2. neg-mul-1 42.4%

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

   5. Simplified 42.4%

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

   6. Taylor expanded in F around inf 75.1%

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

      1. neg-mul-1 75.1%

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

      2. distribute-frac-neg 75.1%

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

      3. +-commutative 75.1%

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

      4. distribute-frac-neg 75.1%

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

      5. unsub-neg 75.1%

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

   8. Simplified 75.1%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.4 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 8.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{elif}\;F \leq 0.17:\\ \;\;\;\;\frac{F \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 71.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -6.4 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.023:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -6.4e-52)
   (- (/ -1.0 B) (/ x (tan B)))
   (if (<= F 0.023) (/ x (- (tan B))) (- (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.4e-52) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= 0.023) {
		tmp = x / -tan(B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-6.4d-52)) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (f <= 0.023d0) then
        tmp = x / -tan(b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -6.4e-52) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (F <= 0.023) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -6.4e-52:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif F <= 0.023:
		tmp = x / -math.tan(B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -6.4e-52)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= 0.023)
		tmp = Float64(x / Float64(-tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -6.4e-52)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (F <= 0.023)
		tmp = x / -tan(B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -6.4e-52], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.023], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.4000000000000002e-52

   1. Initial program 63.9%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in F around -inf 95.1%

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

   6. Taylor expanded in B around 0 71.8%

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

   if -6.4000000000000002e-52 < F < 0.023

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 99.6%

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

      2. clear-num 99.6%

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

      3. fma-define 99.6%

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

      4. fma-undefine 99.6%

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

      5. *-commutative 99.6%

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

      6. fma-define 99.6%

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

      7. fma-define 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in F around 0 70.8%

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

      1. mul-1-neg 70.8%

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

      2. associate-/l* 70.7%

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

      3. distribute-rgt-neg-in 70.7%

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

      4. distribute-neg-frac2 70.7%

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

   9. Simplified 70.7%

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

       1. clear-num 70.7%

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

       2. add-sqr-sqrt 39.2%

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

       3. sqrt-unprod 33.3%

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

       4. sqr-neg 33.3%

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

       5. sqrt-unprod 1.3%

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

       6. add-sqr-sqrt 2.5%

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

       7. tan-quot 2.5%

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

       8. div-inv 2.5%

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

       9. add-sqr-sqrt 1.3%

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

       10. sqrt-unprod 28.3%

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

       11. clear-num 28.2%

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

       12. clear-num 28.2%

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

       13. frac-times 28.3%

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

       14. metadata-eval 28.3%

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

       15. metadata-eval 28.3%

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

       16. frac-times 28.2%

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

       17. sqrt-unprod 33.4%

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

       18. add-sqr-sqrt 70.7%

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

   11. Applied egg-rr 70.7%

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

       1. div-inv 70.7%

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

       2. clear-num 70.8%

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

   13. Applied egg-rr 70.8%

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

       1. neg-mul-1 70.8%

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

       2. distribute-neg-frac 70.8%

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

   15. Simplified 70.8%

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

   if 0.023 < F

   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. Add Preprocessing

   3. Taylor expanded in B around 0 43.2%

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

      1. associate-*r/ 43.2%

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

      2. neg-mul-1 43.2%

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

   5. Simplified 43.2%

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

   6. Taylor expanded in F around inf 74.3%

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

      1. neg-mul-1 74.3%

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

      2. distribute-frac-neg 74.3%

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

      3. +-commutative 74.3%

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

      4. distribute-frac-neg 74.3%

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

      5. unsub-neg 74.3%

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

   8. Simplified 74.3%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6.4 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.023:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 62.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.4e-52) (- (/ -1.0 B) (/ x (tan B))) (/ x (- (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.4e-52) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else {
		tmp = 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 <= (-3.4d-52)) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else
        tmp = x / -tan(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.4e-52) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = x / -Math.tan(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.4e-52:
		tmp = (-1.0 / B) - (x / math.tan(B))
	else:
		tmp = x / -math.tan(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.4e-52)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(x / Float64(-tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -3.4e-52)
		tmp = (-1.0 / B) - (x / tan(B));
	else
		tmp = x / -tan(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.4e-52], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -3.40000000000000017e-52

   1. Initial program 63.9%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in F around -inf 95.1%

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

   6. Taylor expanded in B around 0 71.8%

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

   if -3.40000000000000017e-52 < F

   1. Initial program 85.8%

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

   3. Simplified 90.0%

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

      1. associate-*r/ 90.0%

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

      2. clear-num 90.1%

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

      3. fma-define 90.1%

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

      4. fma-undefine 90.1%

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

      5. *-commutative 90.1%

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

      6. fma-define 90.1%

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

      7. fma-define 90.1%

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

   6. Applied egg-rr 90.1%

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

   7. Taylor expanded in F around 0 63.2%

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

      1. mul-1-neg 63.2%

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

      2. associate-/l* 63.1%

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

      3. distribute-rgt-neg-in 63.1%

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

      4. distribute-neg-frac2 63.1%

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

   9. Simplified 63.1%

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

       1. clear-num 63.1%

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

       2. add-sqr-sqrt 32.5%

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

       3. sqrt-unprod 29.5%

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

       4. sqr-neg 29.5%

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

       5. sqrt-unprod 1.0%

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

       6. add-sqr-sqrt 2.3%

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

       7. tan-quot 2.3%

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

       8. div-inv 2.3%

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

       9. add-sqr-sqrt 1.1%

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

       10. sqrt-unprod 25.9%

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

       11. clear-num 25.9%

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

       12. clear-num 25.8%

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

       13. frac-times 25.9%

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

       14. metadata-eval 25.9%

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

       15. metadata-eval 25.9%

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

       16. frac-times 25.8%

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

       17. sqrt-unprod 31.1%

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

       18. add-sqr-sqrt 63.1%

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

   11. Applied egg-rr 63.1%

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

       1. div-inv 63.1%

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

       2. clear-num 63.3%

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

   13. Applied egg-rr 63.3%

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

       1. neg-mul-1 63.3%

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

       2. distribute-neg-frac 63.3%

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

   15. Simplified 63.3%

       \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{-52}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 42.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 7.2 \cdot 10^{-186}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 7.2e-186) (/ (- -1.0 x) B) (/ x (- (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 7.2e-186) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = 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 (b <= 7.2d-186) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = x / -tan(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 7.2e-186) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = x / -Math.tan(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 7.2e-186:
		tmp = (-1.0 - x) / B
	else:
		tmp = x / -math.tan(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 7.2e-186)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(x / Float64(-tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 7.2e-186)
		tmp = (-1.0 - x) / B;
	else
		tmp = x / -tan(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 7.2e-186], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 7.1999999999999997e-186

   1. Initial program 73.5%

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

   3. Simplified 83.5%

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

   5. Taylor expanded in F around -inf 59.1%

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

   6. Taylor expanded in B around 0 34.9%

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

      1. mul-1-neg 34.9%

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

      2. distribute-neg-frac2 34.9%

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

   8. Simplified 34.9%

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

   if 7.1999999999999997e-186 < B

   1. Initial program 87.1%

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

   3. Simplified 91.5%

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

      1. associate-*r/ 91.5%

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

      2. clear-num 91.5%

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

      3. fma-define 91.5%

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

      4. fma-undefine 91.5%

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

      5. *-commutative 91.5%

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

      6. fma-define 91.5%

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

      7. fma-define 91.5%

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

   6. Applied egg-rr 91.5%

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

   7. Taylor expanded in F around 0 59.6%

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

      1. mul-1-neg 59.6%

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

      2. associate-/l* 59.5%

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

      3. distribute-rgt-neg-in 59.5%

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

      4. distribute-neg-frac2 59.5%

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

   9. Simplified 59.5%

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

       1. clear-num 59.5%

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

       2. add-sqr-sqrt 17.8%

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

       3. sqrt-unprod 19.3%

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

       4. sqr-neg 19.3%

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

       5. sqrt-unprod 1.4%

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

       6. add-sqr-sqrt 2.3%

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

       7. tan-quot 2.3%

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

       8. div-inv 2.3%

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

       9. add-sqr-sqrt 1.0%

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

       10. sqrt-unprod 20.7%

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

       11. clear-num 20.7%

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

       12. clear-num 20.6%

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

       13. frac-times 20.6%

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

       14. metadata-eval 20.6%

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

       15. metadata-eval 20.6%

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

       16. frac-times 20.6%

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

       17. sqrt-unprod 23.5%

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

       18. add-sqr-sqrt 59.5%

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

   11. Applied egg-rr 59.5%

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

       1. div-inv 59.5%

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

       2. clear-num 59.6%

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

   13. Applied egg-rr 59.6%

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

       1. neg-mul-1 59.6%

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

       2. distribute-neg-frac 59.6%

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

   15. Simplified 59.6%

       \[\leadsto \color{blue}{\frac{-x}{\tan B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7.2 \cdot 10^{-186}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 37.0% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{-53}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -3.4e-53) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -3.4e-53) {
		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 <= (-3.4d-53)) 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 <= -3.4e-53) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -3.4e-53:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -3.4e-53)
		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 <= -3.4e-53)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -3.4e-53], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -3.4e-53

   1. Initial program 63.9%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in F around -inf 95.1%

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

   6. Taylor expanded in B around 0 45.7%

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

      1. mul-1-neg 45.7%

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

      2. distribute-neg-frac2 45.7%

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

   8. Simplified 45.7%

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

   if -3.4e-53 < F

   1. Initial program 85.8%

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

   3. Simplified 90.0%

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

   5. Taylor expanded in F around -inf 40.1%

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

   6. Taylor expanded in B around 0 22.4%

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

      1. mul-1-neg 22.4%

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

      2. distribute-neg-frac2 22.4%

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

   8. Simplified 22.4%

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

   9. Taylor expanded in x around inf 33.7%

      \[\leadsto \frac{\color{blue}{x}}{-B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 37.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.4 \cdot 10^{-53}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 29.9% 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 78.5%

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

3. Simplified 86.4%

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

5. Taylor expanded in F around -inf 58.4%

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

6. Taylor expanded in B around 0 30.2%

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

   1. mul-1-neg 30.2%

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

   2. distribute-neg-frac2 30.2%

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

8. Simplified 30.2%

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

9. Taylor expanded in x around inf 31.1%

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

10. Final simplification 31.1%

    \[\leadsto \frac{-x}{B} \]
11. Add Preprocessing

Alternative 22: 10.4% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 78.5%

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

3. Simplified 86.4%

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

5. Taylor expanded in F around -inf 58.4%

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

6. Taylor expanded in B around 0 30.2%

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

   1. mul-1-neg 30.2%

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

   2. distribute-neg-frac2 30.2%

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

8. Simplified 30.2%

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

9. Taylor expanded in x around 0 10.0%

   \[\leadsto \color{blue}{\frac{-1}{B}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(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))))))