VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.5% → 99.6%

Time: 15.3s

Alternatives: 20

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.45 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 7800000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.45e+15)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 7800000.0)
       (+
        (/ -1.0 (/ (tan B) x))
        (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* 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 <= -1.45e+15) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 7800000.0) {
		tmp = (-1.0 / (tan(B) / x)) + ((F / sin(B)) * pow((((F * F) + 2.0) + (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 <= (-1.45d+15)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 7800000.0d0) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((f / sin(b)) * ((((f * f) + 2.0d0) + (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 <= -1.45e+15) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 7800000.0) {
		tmp = (-1.0 / (Math.tan(B) / x)) + ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (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 <= -1.45e+15:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 7800000.0:
		tmp = (-1.0 / (math.tan(B) / x)) + ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (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 <= -1.45e+15)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 7800000.0)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + 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 <= -1.45e+15)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 7800000.0)
		tmp = (-1.0 / (tan(B) / x)) + ((F / sin(B)) * ((((F * F) + 2.0) + (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, -1.45e+15], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 7800000.0], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $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 < -1.45e15

   1. Initial program 62.6%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in F around -inf 99.8%

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

   if -1.45e15 < F < 7.8e6

   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 99.6%

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

      4. un-div-inv 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. metadata-eval 99.6%

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

      2. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

   if 7.8e6 < F

   1. Initial program 70.8%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in F around inf 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%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.45 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 7800000:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.00037:\\ \;\;\;\;\frac{{\left(2 + x \cdot 2\right)}^{-0.5}}{\frac{\sin B}{F}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.42)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.00037)
       (- (/ (pow (+ 2.0 (* x 2.0)) -0.5) (/ (sin B) F)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.00037) {
		tmp = (pow((2.0 + (x * 2.0)), -0.5) / (sin(B) / F)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1.42d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.00037d0) then
        tmp = (((2.0d0 + (x * 2.0d0)) ** (-0.5d0)) / (sin(b) / f)) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1.42) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.00037) {
		tmp = (Math.pow((2.0 + (x * 2.0)), -0.5) / (Math.sin(B) / F)) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1.42:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.00037:
		tmp = (math.pow((2.0 + (x * 2.0)), -0.5) / (math.sin(B) / F)) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.42)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.00037)
		tmp = Float64(Float64((Float64(2.0 + Float64(x * 2.0)) ^ -0.5) / Float64(sin(B) / F)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.42)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.00037)
		tmp = (((2.0 + (x * 2.0)) ^ -0.5) / (sin(B) / F)) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.42], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.00037], N[(N[(N[Power[N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.4199999999999999

   1. Initial program 63.6%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in F around -inf 99.1%

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

   if -1.4199999999999999 < F < 3.6999999999999999e-4

   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. associate-*l/ 99.6%

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

      3. *-commutative 99.6%

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

      4. clear-num 99.5%

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

      5. un-div-inv 99.6%

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

      6. fma-define 99.6%

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

      7. fma-undefine 99.6%

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

      8. *-commutative 99.6%

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

      9. fma-define 99.6%

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

      10. fma-define 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in F around 0 99.2%

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

   if 3.6999999999999999e-4 < F

   1. Initial program 71.5%

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

   3. Simplified 77.9%

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

   5. Taylor expanded in F around inf 99.3%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.00037:\\ \;\;\;\;\frac{{\left(2 + x \cdot 2\right)}^{-0.5}}{\frac{\sin B}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.42:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.00037:\\ \;\;\;\;\frac{F}{\sin B \cdot \sqrt{2 + x \cdot 2}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.42)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.00037)
       (- (/ 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 <= -1.42) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.00037) {
		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 <= (-1.42d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.00037d0) 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 <= -1.42) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.00037) {
		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 <= -1.42:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.00037:
		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 <= -1.42)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.00037)
		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 <= -1.42)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.00037)
		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, -1.42], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.00037], 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 < -1.4199999999999999

   1. Initial program 63.6%

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

   3. Simplified 79.3%

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

   5. Taylor expanded in F around -inf 99.1%

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

   if -1.4199999999999999 < F < 3.6999999999999999e-4

   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. 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 99.2%

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

      1. *-commutative 99.2%

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

   9. Simplified 99.2%

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

   if 3.6999999999999999e-4 < F

   1. Initial program 71.5%

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

   3. Simplified 77.9%

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

   5. Taylor expanded in F around inf 99.3%

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

Alternative 4: 91.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.45 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{-178}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + t\_0 \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 23500:\\ \;\;\;\;\frac{F}{\sin B} \cdot t\_0 - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5)) (t_1 (/ x (tan B))))
   (if (<= F -1.45e+15)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F 2.05e-178)
       (+ (/ -1.0 (/ (tan B) x)) (* t_0 (/ F B)))
       (if (<= F 23500.0)
         (- (* (/ F (sin B)) t_0) (/ x B))
         (- (/ 1.0 (sin B)) t_1))))))

C

double code(double F, double B, double x) {
	double t_0 = pow((((F * F) + 2.0) + (x * 2.0)), -0.5);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -1.45e+15) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= 2.05e-178) {
		tmp = (-1.0 / (tan(B) / x)) + (t_0 * (F / B));
	} else if (F <= 23500.0) {
		tmp = ((F / sin(B)) * t_0) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - t_1;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0)
    t_1 = x / tan(b)
    if (f <= (-1.45d+15)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= 2.05d-178) then
        tmp = ((-1.0d0) / (tan(b) / x)) + (t_0 * (f / b))
    else if (f <= 23500.0d0) then
        tmp = ((f / sin(b)) * t_0) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -1.45e+15) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= 2.05e-178) {
		tmp = (-1.0 / (Math.tan(B) / x)) + (t_0 * (F / B));
	} else if (F <= 23500.0) {
		tmp = ((F / Math.sin(B)) * t_0) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -1.45e+15:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= 2.05e-178:
		tmp = (-1.0 / (math.tan(B) / x)) + (t_0 * (F / B))
	elif F <= 23500.0:
		tmp = ((F / math.sin(B)) * t_0) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.45e+15)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= 2.05e-178)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(t_0 * Float64(F / B)));
	elseif (F <= 23500.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * t_0) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (((F * F) + 2.0) + (x * 2.0)) ^ -0.5;
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -1.45e+15)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= 2.05e-178)
		tmp = (-1.0 / (tan(B) / x)) + (t_0 * (F / B));
	elseif (F <= 23500.0)
		tmp = ((F / sin(B)) * t_0) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.45e+15], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 2.05e-178], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(t$95$0 * N[(F / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 23500.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.45e15

   1. Initial program 62.6%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in F around -inf 99.8%

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

   if -1.45e15 < F < 2.05e-178

   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 99.6%

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

      4. un-div-inv 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in B around 0 87.9%

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

   if 2.05e-178 < F < 23500

   1. Initial program 99.5%

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

      1. div-inv 99.7%

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

      2. neg-mul-1 99.7%

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

      3. clear-num 99.6%

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

      4. un-div-inv 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. metadata-eval 99.6%

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

      2. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in B around 0 83.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{-0.5} \]
   8. Step-by-step derivation

      1. associate-*r/ 83.8%

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

      2. neg-mul-1 83.8%

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

   9. Simplified 83.8%

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

   if 23500 < F

   1. Initial program 70.8%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.45 \cdot 10^{+15}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.05 \cdot 10^{-178}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B}\\ \mathbf{elif}\;F \leq 23500:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.02 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-178}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{elif}\;F \leq 215000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.02e-16)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 2.8e-178)
       (+ (/ -1.0 (/ (tan B) x)) (* (/ F B) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))))
       (if (<= F 215000.0)
         (- (* (/ F (sin B)) (pow (+ (+ (* F F) 2.0) (* x 2.0)) -0.5)) (/ 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 <= -1.02e-16) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 2.8e-178) {
		tmp = (-1.0 / (tan(B) / x)) + ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0)))));
	} else if (F <= 215000.0) {
		tmp = ((F / sin(B)) * pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (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 <= (-1.02d-16)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 2.8d-178) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((f / b) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))))
    else if (f <= 215000.0d0) then
        tmp = ((f / sin(b)) * ((((f * f) + 2.0d0) + (x * 2.0d0)) ** (-0.5d0))) - (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 <= -1.02e-16) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 2.8e-178) {
		tmp = (-1.0 / (Math.tan(B) / x)) + ((F / B) * Math.sqrt((1.0 / (2.0 + (x * 2.0)))));
	} else if (F <= 215000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (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 <= -1.02e-16:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 2.8e-178:
		tmp = (-1.0 / (math.tan(B) / x)) + ((F / B) * math.sqrt((1.0 / (2.0 + (x * 2.0)))))
	elif F <= 215000.0:
		tmp = ((F / math.sin(B)) * math.pow((((F * F) + 2.0) + (x * 2.0)), -0.5)) - (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 <= -1.02e-16)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 2.8e-178)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(Float64(F / B) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))));
	elseif (F <= 215000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(Float64(F * F) + 2.0) + Float64(x * 2.0)) ^ -0.5)) - 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 <= -1.02e-16)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 2.8e-178)
		tmp = (-1.0 / (tan(B) / x)) + ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0)))));
	elseif (F <= 215000.0)
		tmp = ((F / sin(B)) * ((((F * F) + 2.0) + (x * 2.0)) ^ -0.5)) - (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, -1.02e-16], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2.8e-178], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 215000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(N[(F * F), $MachinePrecision] + 2.0), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $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 < -1.0200000000000001e-16

   1. Initial program 64.6%

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

   3. Simplified 79.8%

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

   5. Taylor expanded in F around -inf 96.7%

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

   if -1.0200000000000001e-16 < F < 2.80000000000000019e-178

   1. Initial program 99.5%

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

      1. 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 99.7%

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

      4. un-div-inv 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in B around 0 89.7%

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

   6. Taylor expanded in F around 0 89.7%

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

   if 2.80000000000000019e-178 < F < 215000

   1. Initial program 99.5%

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

      1. div-inv 99.7%

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

      2. neg-mul-1 99.7%

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

      3. clear-num 99.6%

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

      4. un-div-inv 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. metadata-eval 99.6%

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

      2. metadata-eval 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in B around 0 83.8%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{B}} + \frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + 2 \cdot x\right)}^{-0.5} \]
   8. Step-by-step derivation

      1. associate-*r/ 83.8%

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

      2. neg-mul-1 83.8%

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

   9. Simplified 83.8%

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

   if 215000 < F

   1. Initial program 70.8%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.02 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-178}:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{elif}\;F \leq 215000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(F \cdot F + 2\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.02 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 0.00037:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.02e-16)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 0.00037)
       (+ (/ -1.0 (/ (tan B) x)) (* (/ F B) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))))
       (- (/ 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 <= -1.02e-16) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 0.00037) {
		tmp = (-1.0 / (tan(B) / x)) + ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0)))));
	} 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 <= (-1.02d-16)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 0.00037d0) then
        tmp = ((-1.0d0) / (tan(b) / x)) + ((f / b) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))))
    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 <= -1.02e-16) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 0.00037) {
		tmp = (-1.0 / (Math.tan(B) / x)) + ((F / B) * Math.sqrt((1.0 / (2.0 + (x * 2.0)))));
	} 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 <= -1.02e-16:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 0.00037:
		tmp = (-1.0 / (math.tan(B) / x)) + ((F / B) * math.sqrt((1.0 / (2.0 + (x * 2.0)))))
	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 <= -1.02e-16)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 0.00037)
		tmp = Float64(Float64(-1.0 / Float64(tan(B) / x)) + Float64(Float64(F / B) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))));
	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 <= -1.02e-16)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 0.00037)
		tmp = (-1.0 / (tan(B) / x)) + ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0)))));
	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, -1.02e-16], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.00037], N[(N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] + N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $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 < -1.0200000000000001e-16

   1. Initial program 64.6%

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

   3. Simplified 79.8%

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

   5. Taylor expanded in F around -inf 96.7%

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

   if -1.0200000000000001e-16 < F < 3.6999999999999999e-4

   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 99.6%

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

      4. un-div-inv 99.6%

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

   4. Applied egg-rr 99.6%

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

   5. Taylor expanded in B around 0 83.8%

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

   6. Taylor expanded in F around 0 83.4%

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

   if 3.6999999999999999e-4 < F

   1. Initial program 71.5%

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

   3. Simplified 77.9%

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

   5. Taylor expanded in F around inf 99.3%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.02 \cdot 10^{-16}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.00037:\\ \;\;\;\;\frac{-1}{\frac{\tan B}{x}} + \frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6 \cdot 10^{-83}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 1.4 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -6e-83)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1.4e-55) (/ x (- (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 <= -6e-83) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1.4e-55) {
		tmp = x / -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 <= (-6d-83)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 1.4d-55) then
        tmp = x / -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 <= -6e-83) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 1.4e-55) {
		tmp = x / -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 <= -6e-83:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 1.4e-55:
		tmp = x / -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 <= -6e-83)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1.4e-55)
		tmp = Float64(x / Float64(-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 <= -6e-83)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 1.4e-55)
		tmp = x / -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, -6e-83], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1.4e-55], N[(x / (-N[Tan[B], $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 < -6.00000000000000021e-83

   1. Initial program 69.2%

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

   3. Simplified 82.4%

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

   5. Taylor expanded in F around -inf 89.4%

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

   if -6.00000000000000021e-83 < F < 1.39999999999999992e-55

   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. clear-num 99.6%

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

      2. un-div-inv 99.6%

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

      3. fma-define 99.6%

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

      4. fma-undefine 99.6%

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

      5. *-commutative 99.6%

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

      6. fma-define 99.6%

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

      7. fma-define 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in F around 0 70.3%

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

      1. mul-1-neg 70.3%

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

      2. associate-/l* 70.1%

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

      3. distribute-rgt-neg-in 70.1%

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

   9. Simplified 70.1%

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

       1. distribute-rgt-neg-out 70.1%

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

       2. clear-num 70.1%

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

       3. tan-quot 70.1%

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

       4. div-inv 70.3%

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

   11. Applied egg-rr 70.3%

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

       1. distribute-neg-frac2 70.3%

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

   13. Simplified 70.3%

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

   if 1.39999999999999992e-55 < F

   1. Initial program 73.8%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in F around inf 93.0%

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

Alternative 8: 77.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -6 \cdot 10^{-83}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 8.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -6e-83)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 8.2e-58) (/ x (- (tan B))) (- (* F (/ 1.0 (* F B))) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -6e-83) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 8.2e-58) {
		tmp = x / -tan(B);
	} else {
		tmp = (F * (1.0 / (F * B))) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-6d-83)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 8.2d-58) then
        tmp = x / -tan(b)
    else
        tmp = (f * (1.0d0 / (f * b))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -6e-83) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 8.2e-58) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = (F * (1.0 / (F * B))) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -6e-83:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 8.2e-58:
		tmp = x / -math.tan(B)
	else:
		tmp = (F * (1.0 / (F * B))) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -6e-83)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 8.2e-58)
		tmp = Float64(x / Float64(-tan(B)));
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * B))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -6e-83)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 8.2e-58)
		tmp = x / -tan(B);
	else
		tmp = (F * (1.0 / (F * B))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -6e-83], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 8.2e-58], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(N[(F * N[(1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -6.00000000000000021e-83

   1. Initial program 69.2%

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

   3. Simplified 82.4%

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

   5. Taylor expanded in F around -inf 89.4%

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

   if -6.00000000000000021e-83 < F < 8.20000000000000056e-58

   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. clear-num 99.6%

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

      2. un-div-inv 99.6%

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

      3. fma-define 99.6%

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

      4. fma-undefine 99.6%

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

      5. *-commutative 99.6%

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

      6. fma-define 99.6%

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

      7. fma-define 99.6%

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

   6. Applied egg-rr 99.6%

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

   7. Taylor expanded in F around 0 70.3%

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

      1. mul-1-neg 70.3%

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

      2. associate-/l* 70.1%

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

      3. distribute-rgt-neg-in 70.1%

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

   9. Simplified 70.1%

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

       1. distribute-rgt-neg-out 70.1%

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

       2. clear-num 70.1%

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

       3. tan-quot 70.1%

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

       4. div-inv 70.3%

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

   11. Applied egg-rr 70.3%

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

       1. distribute-neg-frac2 70.3%

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

   13. Simplified 70.3%

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

   if 8.20000000000000056e-58 < F

   1. Initial program 73.8%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in F around inf 92.8%

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

   6. Taylor expanded in B around 0 68.3%

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

4. Final simplification 75.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -6 \cdot 10^{-83}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 8.2 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -2.55 \cdot 10^{+165}:\\ \;\;\;\;F \cdot \left(\frac{1}{F} \cdot \frac{-1}{B}\right) - t\_0\\ \mathbf{elif}\;F \leq -1.1 \cdot 10^{+51}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-57}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -2.55e+165)
     (- (* F (* (/ 1.0 F) (/ -1.0 B))) t_0)
     (if (<= F -1.1e+51)
       (- (/ -1.0 (sin B)) (/ x B))
       (if (<= F 7e-57) (/ x (- (tan B))) (- (* F (/ 1.0 (* F B))) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -2.55e+165) {
		tmp = (F * ((1.0 / F) * (-1.0 / B))) - t_0;
	} else if (F <= -1.1e+51) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 7e-57) {
		tmp = x / -tan(B);
	} else {
		tmp = (F * (1.0 / (F * B))) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-2.55d+165)) then
        tmp = (f * ((1.0d0 / f) * ((-1.0d0) / b))) - t_0
    else if (f <= (-1.1d+51)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 7d-57) then
        tmp = x / -tan(b)
    else
        tmp = (f * (1.0d0 / (f * b))) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -2.55e+165) {
		tmp = (F * ((1.0 / F) * (-1.0 / B))) - t_0;
	} else if (F <= -1.1e+51) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 7e-57) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = (F * (1.0 / (F * B))) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -2.55e+165:
		tmp = (F * ((1.0 / F) * (-1.0 / B))) - t_0
	elif F <= -1.1e+51:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 7e-57:
		tmp = x / -math.tan(B)
	else:
		tmp = (F * (1.0 / (F * B))) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -2.55e+165)
		tmp = Float64(Float64(F * Float64(Float64(1.0 / F) * Float64(-1.0 / B))) - t_0);
	elseif (F <= -1.1e+51)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 7e-57)
		tmp = Float64(x / Float64(-tan(B)));
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * B))) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -2.55e+165)
		tmp = (F * ((1.0 / F) * (-1.0 / B))) - t_0;
	elseif (F <= -1.1e+51)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 7e-57)
		tmp = x / -tan(B);
	else
		tmp = (F * (1.0 / (F * B))) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.55e+165], N[(N[(F * N[(N[(1.0 / F), $MachinePrecision] * N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -1.1e+51], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7e-57], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(N[(F * N[(1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.5500000000000002e165

   1. Initial program 50.2%

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

   3. Simplified 64.8%

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

   5. Taylor expanded in F around inf 64.0%

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

      1. div-inv 64.0%

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

      2. add-sqr-sqrt 27.9%

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

      3. sqrt-unprod 67.2%

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

      4. frac-times 67.2%

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

      5. metadata-eval 67.2%

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

      6. metadata-eval 67.2%

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

      7. frac-times 67.2%

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

      8. sqrt-unprod 52.6%

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

      9. add-sqr-sqrt 99.6%

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in B around 0 83.4%

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

   if -2.5500000000000002e165 < F < -1.09999999999999996e51

   1. Initial program 72.9%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in F around -inf 99.9%

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

   6. Taylor expanded in B around 0 84.1%

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

   if -1.09999999999999996e51 < F < 6.99999999999999983e-57

   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. 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 65.6%

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

      1. mul-1-neg 65.6%

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

      2. associate-/l* 65.5%

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

      3. distribute-rgt-neg-in 65.5%

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

   9. Simplified 65.5%

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

       1. distribute-rgt-neg-out 65.5%

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

       2. clear-num 65.4%

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

       3. tan-quot 65.5%

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

       4. div-inv 65.6%

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

   11. Applied egg-rr 65.6%

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

       1. distribute-neg-frac2 65.6%

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

   13. Simplified 65.6%

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

   if 6.99999999999999983e-57 < F

   1. Initial program 73.8%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in F around inf 92.8%

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

   6. Taylor expanded in B around 0 68.3%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.55 \cdot 10^{+165}:\\ \;\;\;\;F \cdot \left(\frac{1}{F} \cdot \frac{-1}{B}\right) - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.1 \cdot 10^{+51}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-57}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 69.5% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.42 \cdot 10^{+51}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.42e+51)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 5.5e-55)
     (/ x (- (tan B)))
     (- (* F (/ 1.0 (* F B))) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.42e+51) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 5.5e-55) {
		tmp = x / -tan(B);
	} else {
		tmp = (F * (1.0 / (F * B))) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.42d+51)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 5.5d-55) then
        tmp = x / -tan(b)
    else
        tmp = (f * (1.0d0 / (f * b))) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.42e+51) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 5.5e-55) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = (F * (1.0 / (F * B))) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.42e+51:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 5.5e-55:
		tmp = x / -math.tan(B)
	else:
		tmp = (F * (1.0 / (F * B))) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.42e+51)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 5.5e-55)
		tmp = Float64(x / Float64(-tan(B)));
	else
		tmp = Float64(Float64(F * Float64(1.0 / Float64(F * B))) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.42e+51)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 5.5e-55)
		tmp = x / -tan(B);
	else
		tmp = (F * (1.0 / (F * B))) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.42e+51], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5.5e-55], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(N[(F * N[(1.0 / N[(F * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.41999999999999998e51

   1. Initial program 60.3%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in F around -inf 99.8%

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

   6. Taylor expanded in B around 0 69.9%

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

   if -1.41999999999999998e51 < F < 5.4999999999999999e-55

   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. 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 65.6%

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

      1. mul-1-neg 65.6%

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

      2. associate-/l* 65.5%

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

      3. distribute-rgt-neg-in 65.5%

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

   9. Simplified 65.5%

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

       1. distribute-rgt-neg-out 65.5%

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

       2. clear-num 65.4%

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

       3. tan-quot 65.5%

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

       4. div-inv 65.6%

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

   11. Applied egg-rr 65.6%

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

       1. distribute-neg-frac2 65.6%

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

   13. Simplified 65.6%

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

   if 5.4999999999999999e-55 < F

   1. Initial program 73.8%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in F around inf 92.8%

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

   6. Taylor expanded in B around 0 68.3%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.42 \cdot 10^{+51}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5.5 \cdot 10^{-55}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;F \cdot \frac{1}{F \cdot B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 58.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-62}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \mathbf{elif}\;x \leq -1.02 \cdot 10^{-252}:\\ \;\;\;\;x \cdot \frac{1}{\sin B \cdot x}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-181}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -1.85e-62)
   (* x (/ -1.0 (tan B)))
   (if (<= x -1.02e-252)
     (* x (/ 1.0 (* (sin B) x)))
     (if (<= x 1.55e-181) (/ -1.0 (sin B)) (/ x (- (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -1.85e-62) {
		tmp = x * (-1.0 / tan(B));
	} else if (x <= -1.02e-252) {
		tmp = x * (1.0 / (sin(B) * x));
	} else if (x <= 1.55e-181) {
		tmp = -1.0 / sin(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 (x <= (-1.85d-62)) then
        tmp = x * ((-1.0d0) / tan(b))
    else if (x <= (-1.02d-252)) then
        tmp = x * (1.0d0 / (sin(b) * x))
    else if (x <= 1.55d-181) then
        tmp = (-1.0d0) / sin(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 (x <= -1.85e-62) {
		tmp = x * (-1.0 / Math.tan(B));
	} else if (x <= -1.02e-252) {
		tmp = x * (1.0 / (Math.sin(B) * x));
	} else if (x <= 1.55e-181) {
		tmp = -1.0 / Math.sin(B);
	} else {
		tmp = x / -Math.tan(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if x <= -1.85e-62:
		tmp = x * (-1.0 / math.tan(B))
	elif x <= -1.02e-252:
		tmp = x * (1.0 / (math.sin(B) * x))
	elif x <= 1.55e-181:
		tmp = -1.0 / math.sin(B)
	else:
		tmp = x / -math.tan(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -1.85e-62)
		tmp = Float64(x * Float64(-1.0 / tan(B)));
	elseif (x <= -1.02e-252)
		tmp = Float64(x * Float64(1.0 / Float64(sin(B) * x)));
	elseif (x <= 1.55e-181)
		tmp = Float64(-1.0 / sin(B));
	else
		tmp = Float64(x / Float64(-tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (x <= -1.85e-62)
		tmp = x * (-1.0 / tan(B));
	elseif (x <= -1.02e-252)
		tmp = x * (1.0 / (sin(B) * x));
	elseif (x <= 1.55e-181)
		tmp = -1.0 / sin(B);
	else
		tmp = x / -tan(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -1.85e-62], N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.02e-252], N[(x * N[(1.0 / N[(N[Sin[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e-181], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.8499999999999999e-62

   1. Initial program 79.6%

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

   3. Simplified 94.0%

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

      1. clear-num 94.0%

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

      2. un-div-inv 94.0%

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

      3. fma-define 94.0%

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

      4. fma-undefine 94.0%

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

      5. *-commutative 94.0%

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

      6. fma-define 94.0%

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

      7. fma-define 94.0%

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

   6. Applied egg-rr 94.0%

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

   7. Taylor expanded in F around 0 87.0%

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

      1. mul-1-neg 87.0%

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

      2. associate-/l* 87.0%

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

      3. distribute-rgt-neg-in 87.0%

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

   9. Simplified 87.0%

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

       1. neg-sub0 87.0%

          \[\leadsto x \cdot \color{blue}{\left(0 - \frac{\cos B}{\sin B}\right)} \]

       2. clear-num 87.0%

          \[\leadsto x \cdot \left(0 - \color{blue}{\frac{1}{\frac{\sin B}{\cos B}}}\right) \]

       3. tan-quot 87.1%

          \[\leadsto x \cdot \left(0 - \frac{1}{\color{blue}{\tan B}}\right) \]

   11. Applied egg-rr 87.1%

       \[\leadsto x \cdot \color{blue}{\left(0 - \frac{1}{\tan B}\right)} \]
   12. Step-by-step derivation

       1. sub0-neg 87.1%

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

       2. mul-1-neg 87.1%

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

       3. associate-*r/ 87.1%

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

       4. metadata-eval 87.1%

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

   13. Simplified 87.1%

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

   if -1.8499999999999999e-62 < x < -1.02000000000000002e-252

   1. Initial program 72.5%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in F around inf 40.0%

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

   6. Taylor expanded in x around inf 35.4%

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

   7. Taylor expanded in x around 0 35.4%

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

   if -1.02000000000000002e-252 < x < 1.55000000000000011e-181

   1. Initial program 68.4%

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

   3. Simplified 72.8%

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

   5. Taylor expanded in F around inf 25.4%

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

      1. div-inv 25.4%

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

      2. add-sqr-sqrt 13.6%

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

      3. sqrt-unprod 17.8%

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

      4. frac-times 17.8%

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

      5. metadata-eval 17.8%

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

      6. metadata-eval 17.8%

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

      7. frac-times 17.8%

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

      8. sqrt-unprod 16.1%

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

      9. add-sqr-sqrt 28.5%

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

   7. Applied egg-rr 28.5%

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

   8. Taylor expanded in B around 0 28.5%

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

   9. Taylor expanded in B around inf 28.7%

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

   if 1.55000000000000011e-181 < x

   1. Initial program 89.0%

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

   3. Simplified 92.6%

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

      1. clear-num 92.6%

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

      2. un-div-inv 92.6%

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

      3. fma-define 92.6%

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

      4. fma-undefine 92.6%

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

      5. *-commutative 92.6%

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

      6. fma-define 92.6%

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

      7. fma-define 92.6%

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

   6. Applied egg-rr 92.6%

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

   7. Taylor expanded in F around 0 78.0%

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

      1. mul-1-neg 78.0%

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

      2. associate-/l* 78.0%

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

      3. distribute-rgt-neg-in 78.0%

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

   9. Simplified 78.0%

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

       1. distribute-rgt-neg-out 78.0%

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

       2. clear-num 77.8%

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

       3. tan-quot 78.0%

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

       4. div-inv 78.2%

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

   11. Applied egg-rr 78.2%

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

       1. distribute-neg-frac2 78.2%

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

   13. Simplified 78.2%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.85 \cdot 10^{-62}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \mathbf{elif}\;x \leq -1.02 \cdot 10^{-252}:\\ \;\;\;\;x \cdot \frac{1}{\sin B \cdot x}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-181}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.16 \cdot 10^{+51}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.16e+51)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 5e-57) (/ x (- (tan B))) (- (/ 1.0 B) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.16e+51) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 5e-57) {
		tmp = x / -tan(B);
	} else {
		tmp = (1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.16d+51)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 5d-57) then
        tmp = x / -tan(b)
    else
        tmp = (1.0d0 / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.16e+51) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 5e-57) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.16e+51:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 5e-57:
		tmp = x / -math.tan(B)
	else:
		tmp = (1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.16e+51)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 5e-57)
		tmp = Float64(x / Float64(-tan(B)));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.16e+51)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 5e-57)
		tmp = x / -tan(B);
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.16e+51], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 5e-57], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.16e51

   1. Initial program 60.3%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in F around -inf 99.8%

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

   6. Taylor expanded in B around 0 69.9%

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

   if -1.16e51 < F < 5.0000000000000002e-57

   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. 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 65.6%

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

      1. mul-1-neg 65.6%

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

      2. associate-/l* 65.5%

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

      3. distribute-rgt-neg-in 65.5%

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

   9. Simplified 65.5%

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

       1. distribute-rgt-neg-out 65.5%

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

       2. clear-num 65.4%

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

       3. tan-quot 65.5%

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

       4. div-inv 65.6%

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

   11. Applied egg-rr 65.6%

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

       1. distribute-neg-frac2 65.6%

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

   13. Simplified 65.6%

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

   if 5.0000000000000002e-57 < F

   1. Initial program 73.8%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in F around inf 92.8%

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

   6. Taylor expanded in B around 0 68.3%

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

Alternative 13: 60.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.1 \cdot 10^{+51}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 8.6 \cdot 10^{+201}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\sin B \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.1e+51)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 8.6e+201) (/ x (- (tan B))) (* x (/ 1.0 (* (sin B) x))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.1e+51) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 8.6e+201) {
		tmp = x / -tan(B);
	} else {
		tmp = x * (1.0 / (sin(B) * x));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.1d+51)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 8.6d+201) then
        tmp = x / -tan(b)
    else
        tmp = x * (1.0d0 / (sin(b) * x))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.1e+51) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 8.6e+201) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = x * (1.0 / (Math.sin(B) * x));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.1e+51:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 8.6e+201:
		tmp = x / -math.tan(B)
	else:
		tmp = x * (1.0 / (math.sin(B) * x))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.1e+51)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 8.6e+201)
		tmp = Float64(x / Float64(-tan(B)));
	else
		tmp = Float64(x * Float64(1.0 / Float64(sin(B) * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.1e+51)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 8.6e+201)
		tmp = x / -tan(B);
	else
		tmp = x * (1.0 / (sin(B) * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.1e+51], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 8.6e+201], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(x * N[(1.0 / N[(N[Sin[B], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.09999999999999996e51

   1. Initial program 60.3%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in F around -inf 99.8%

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

   6. Taylor expanded in B around 0 69.9%

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

   if -1.09999999999999996e51 < F < 8.59999999999999981e201

   1. Initial program 97.2%

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

   3. Simplified 99.0%

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

      1. clear-num 99.0%

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

      2. un-div-inv 99.0%

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

      3. fma-define 99.0%

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

      4. fma-undefine 99.0%

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

      5. *-commutative 99.0%

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

      6. fma-define 99.0%

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

      7. fma-define 99.0%

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

   6. Applied egg-rr 99.0%

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

   7. Taylor expanded in F around 0 62.2%

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

      1. mul-1-neg 62.2%

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

      2. associate-/l* 62.2%

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

      3. distribute-rgt-neg-in 62.2%

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

   9. Simplified 62.2%

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

       1. distribute-rgt-neg-out 62.2%

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

       2. clear-num 62.1%

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

       3. tan-quot 62.2%

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

       4. div-inv 62.3%

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

   11. Applied egg-rr 62.3%

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

       1. distribute-neg-frac2 62.3%

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

   13. Simplified 62.3%

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

   if 8.59999999999999981e201 < F

   1. Initial program 23.0%

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

   3. Simplified 32.3%

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

   5. Taylor expanded in F around inf 99.5%

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

   6. Taylor expanded in x around inf 84.3%

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

   7. Taylor expanded in x around 0 54.1%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.1 \cdot 10^{+51}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 8.6 \cdot 10^{+201}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{\sin B \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 59.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-224} \lor \neg \left(x \leq 2.7 \cdot 10^{-181}\right):\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -6.8e-224) (not (<= x 2.7e-181)))
   (/ x (- (tan B)))
   (/ -1.0 (sin B))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -6.8e-224) || !(x <= 2.7e-181)) {
		tmp = x / -tan(B);
	} else {
		tmp = -1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-6.8d-224)) .or. (.not. (x <= 2.7d-181))) then
        tmp = x / -tan(b)
    else
        tmp = (-1.0d0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -6.8e-224) || !(x <= 2.7e-181)) {
		tmp = x / -Math.tan(B);
	} else {
		tmp = -1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -6.8e-224) or not (x <= 2.7e-181):
		tmp = x / -math.tan(B)
	else:
		tmp = -1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -6.8e-224) || !(x <= 2.7e-181))
		tmp = Float64(x / Float64(-tan(B)));
	else
		tmp = Float64(-1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -6.8e-224) || ~((x <= 2.7e-181)))
		tmp = x / -tan(B);
	else
		tmp = -1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -6.8e-224], N[Not[LessEqual[x, 2.7e-181]], $MachinePrecision]], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.79999999999999984e-224 or 2.6999999999999999e-181 < x

   1. Initial program 83.4%

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

   3. Simplified 90.6%

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

      1. clear-num 90.6%

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

      2. un-div-inv 90.6%

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

      3. fma-define 90.6%

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

      4. fma-undefine 90.6%

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

      5. *-commutative 90.6%

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

      6. fma-define 90.6%

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

      7. fma-define 90.6%

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

   6. Applied egg-rr 90.6%

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

   7. Taylor expanded in F around 0 72.2%

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

      1. mul-1-neg 72.2%

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

      2. associate-/l* 72.2%

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

      3. distribute-rgt-neg-in 72.2%

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

   9. Simplified 72.2%

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

       1. distribute-rgt-neg-out 72.2%

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

       2. clear-num 72.1%

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

       3. tan-quot 72.2%

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

       4. div-inv 72.3%

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

   11. Applied egg-rr 72.3%

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

       1. distribute-neg-frac2 72.3%

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

   13. Simplified 72.3%

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

   if -6.79999999999999984e-224 < x < 2.6999999999999999e-181

   1. Initial program 72.1%

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

   3. Simplified 75.7%

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

   5. Taylor expanded in F around inf 26.4%

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

      1. div-inv 26.5%

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

      2. add-sqr-sqrt 13.0%

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

      3. sqrt-unprod 16.8%

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

      4. frac-times 16.8%

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

      5. metadata-eval 16.8%

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

      6. metadata-eval 16.8%

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

      7. frac-times 16.8%

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

      8. sqrt-unprod 13.7%

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

      9. add-sqr-sqrt 25.7%

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

   7. Applied egg-rr 25.7%

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

   8. Taylor expanded in B around 0 25.7%

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

   9. Taylor expanded in B around inf 25.9%

      \[\leadsto \color{blue}{\frac{-1}{\sin B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.8 \cdot 10^{-224} \lor \neg \left(x \leq 2.7 \cdot 10^{-181}\right):\\ \;\;\;\;\frac{x}{-\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 59.0% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{-218}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-182}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= x -3e-218)
   (* x (/ -1.0 (tan B)))
   (if (<= x 6.5e-182) (/ -1.0 (sin B)) (/ x (- (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (x <= -3e-218) {
		tmp = x * (-1.0 / tan(B));
	} else if (x <= 6.5e-182) {
		tmp = -1.0 / sin(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 (x <= (-3d-218)) then
        tmp = x * ((-1.0d0) / tan(b))
    else if (x <= 6.5d-182) then
        tmp = (-1.0d0) / sin(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 (x <= -3e-218) {
		tmp = x * (-1.0 / Math.tan(B));
	} else if (x <= 6.5e-182) {
		tmp = -1.0 / Math.sin(B);
	} else {
		tmp = x / -Math.tan(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if x <= -3e-218:
		tmp = x * (-1.0 / math.tan(B))
	elif x <= 6.5e-182:
		tmp = -1.0 / math.sin(B)
	else:
		tmp = x / -math.tan(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (x <= -3e-218)
		tmp = Float64(x * Float64(-1.0 / tan(B)));
	elseif (x <= 6.5e-182)
		tmp = Float64(-1.0 / sin(B));
	else
		tmp = Float64(x / Float64(-tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (x <= -3e-218)
		tmp = x * (-1.0 / tan(B));
	elseif (x <= 6.5e-182)
		tmp = -1.0 / sin(B);
	else
		tmp = x / -tan(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[x, -3e-218], N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.5e-182], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[(x / (-N[Tan[B], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.9999999999999998e-218

   1. Initial program 74.7%

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

   3. Simplified 87.4%

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

      1. clear-num 87.4%

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

      2. un-div-inv 87.4%

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

      3. fma-define 87.4%

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

      4. fma-undefine 87.4%

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

      5. *-commutative 87.4%

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

      6. fma-define 87.4%

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

      7. fma-define 87.4%

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

   6. Applied egg-rr 87.4%

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

   7. Taylor expanded in F around 0 63.3%

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

      1. mul-1-neg 63.3%

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

      2. associate-/l* 63.3%

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

      3. distribute-rgt-neg-in 63.3%

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

   9. Simplified 63.3%

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

       1. neg-sub0 63.3%

          \[\leadsto x \cdot \color{blue}{\left(0 - \frac{\cos B}{\sin B}\right)} \]

       2. clear-num 63.2%

          \[\leadsto x \cdot \left(0 - \color{blue}{\frac{1}{\frac{\sin B}{\cos B}}}\right) \]

       3. tan-quot 63.3%

          \[\leadsto x \cdot \left(0 - \frac{1}{\color{blue}{\tan B}}\right) \]

   11. Applied egg-rr 63.3%

       \[\leadsto x \cdot \color{blue}{\left(0 - \frac{1}{\tan B}\right)} \]
   12. Step-by-step derivation

       1. sub0-neg 63.3%

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

       2. mul-1-neg 63.3%

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

       3. associate-*r/ 63.3%

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

       4. metadata-eval 63.3%

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

   13. Simplified 63.3%

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

   if -2.9999999999999998e-218 < x < 6.49999999999999997e-182

   1. Initial program 72.1%

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

   3. Simplified 75.7%

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

   5. Taylor expanded in F around inf 26.4%

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

      1. div-inv 26.5%

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

      2. add-sqr-sqrt 13.0%

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

      3. sqrt-unprod 16.8%

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

      4. frac-times 16.8%

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

      5. metadata-eval 16.8%

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

      6. metadata-eval 16.8%

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

      7. frac-times 16.8%

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

      8. sqrt-unprod 13.7%

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

      9. add-sqr-sqrt 25.7%

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

   7. Applied egg-rr 25.7%

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

   8. Taylor expanded in B around 0 25.7%

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

   9. Taylor expanded in B around inf 25.9%

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

   if 6.49999999999999997e-182 < x

   1. Initial program 89.0%

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

   3. Simplified 92.6%

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

      1. clear-num 92.6%

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

      2. un-div-inv 92.6%

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

      3. fma-define 92.6%

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

      4. fma-undefine 92.6%

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

      5. *-commutative 92.6%

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

      6. fma-define 92.6%

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

      7. fma-define 92.6%

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

   6. Applied egg-rr 92.6%

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

   7. Taylor expanded in F around 0 78.0%

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

      1. mul-1-neg 78.0%

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

      2. associate-/l* 78.0%

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

      3. distribute-rgt-neg-in 78.0%

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

   9. Simplified 78.0%

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

       1. distribute-rgt-neg-out 78.0%

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

       2. clear-num 77.8%

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

       3. tan-quot 78.0%

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

       4. div-inv 78.2%

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

   11. Applied egg-rr 78.2%

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

       1. distribute-neg-frac2 78.2%

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

   13. Simplified 78.2%

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

Alternative 16: 43.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{+106}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{+36}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-56}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.3e+106)
   (/ (- -1.0 x) B)
   (if (<= F -2.1e+36)
     (/ -1.0 (sin B))
     (if (<= F 3.4e-56) (/ x (- B)) (/ (- 1.0 x) B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.3e+106) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -2.1e+36) {
		tmp = -1.0 / sin(B);
	} else if (F <= 3.4e-56) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - 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 <= (-1.3d+106)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= (-2.1d+36)) then
        tmp = (-1.0d0) / sin(b)
    else if (f <= 3.4d-56) then
        tmp = x / -b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.3e+106) {
		tmp = (-1.0 - x) / B;
	} else if (F <= -2.1e+36) {
		tmp = -1.0 / Math.sin(B);
	} else if (F <= 3.4e-56) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.3e+106:
		tmp = (-1.0 - x) / B
	elif F <= -2.1e+36:
		tmp = -1.0 / math.sin(B)
	elif F <= 3.4e-56:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.3e+106)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= -2.1e+36)
		tmp = Float64(-1.0 / sin(B));
	elseif (F <= 3.4e-56)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.3e+106)
		tmp = (-1.0 - x) / B;
	elseif (F <= -2.1e+36)
		tmp = -1.0 / sin(B);
	elseif (F <= 3.4e-56)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.3e+106], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, -2.1e+36], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.4e-56], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.3000000000000001e106

   1. Initial program 53.2%

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

   3. Simplified 71.3%

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

   5. Taylor expanded in F around inf 63.0%

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

      1. div-inv 63.0%

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

      2. add-sqr-sqrt 27.6%

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

      3. sqrt-unprod 67.1%

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

      4. frac-times 67.1%

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

      5. metadata-eval 67.1%

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

      6. metadata-eval 67.1%

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

      7. frac-times 67.1%

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

      8. sqrt-unprod 52.8%

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

      9. add-sqr-sqrt 99.6%

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

   7. Applied egg-rr 99.6%

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

   8. Taylor expanded in B around 0 49.3%

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

      1. associate-*r/ 49.3%

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

      2. neg-mul-1 49.3%

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

      3. distribute-neg-in 49.3%

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

      4. metadata-eval 49.3%

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

      5. unsub-neg 49.3%

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

   10. Simplified 49.3%

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

   if -1.3000000000000001e106 < F < -2.10000000000000004e36

   1. Initial program 88.7%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in F around inf 30.2%

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

      1. div-inv 30.2%

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

      2. add-sqr-sqrt 0.2%

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

      3. sqrt-unprod 71.5%

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

      4. frac-times 71.4%

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

      5. metadata-eval 71.4%

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

      6. metadata-eval 71.4%

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

      7. frac-times 71.5%

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

      8. sqrt-unprod 81.9%

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

      9. add-sqr-sqrt 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in B around 0 80.4%

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

   9. Taylor expanded in B around inf 68.7%

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

   if -2.10000000000000004e36 < F < 3.39999999999999982e-56

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around inf 26.7%

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

   6. Taylor expanded in B around 0 16.2%

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

   7. Taylor expanded in x around inf 28.7%

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

      1. associate-*r/ 28.7%

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

      2. neg-mul-1 28.7%

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

   9. Simplified 28.7%

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

   if 3.39999999999999982e-56 < F

   1. Initial program 73.8%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in F around inf 92.8%

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

   6. Taylor expanded in B around 0 40.3%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{+106}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq -2.1 \cdot 10^{+36}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{-56}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 44.0% accurate, 21.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{-93}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.4e-93)
   (/ (- -1.0 x) B)
   (if (<= F 4.5e-58) (/ x (- B)) (/ (- 1.0 x) B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.4e-93) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.5e-58) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - 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 <= (-2.4d-93)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 4.5d-58) then
        tmp = x / -b
    else
        tmp = (1.0d0 - x) / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.4e-93) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 4.5e-58) {
		tmp = x / -B;
	} else {
		tmp = (1.0 - x) / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.4e-93:
		tmp = (-1.0 - x) / B
	elif F <= 4.5e-58:
		tmp = x / -B
	else:
		tmp = (1.0 - x) / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.4e-93)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 4.5e-58)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(Float64(1.0 - x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.4e-93)
		tmp = (-1.0 - x) / B;
	elseif (F <= 4.5e-58)
		tmp = x / -B;
	else
		tmp = (1.0 - x) / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.4e-93], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 4.5e-58], N[(x / (-B)), $MachinePrecision], N[(N[(1.0 - x), $MachinePrecision] / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.4000000000000001e-93

   1. Initial program 70.3%

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

   3. Simplified 83.0%

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

   5. Taylor expanded in F around inf 48.8%

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

      1. div-inv 48.8%

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

      2. add-sqr-sqrt 18.6%

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

      3. sqrt-unprod 60.0%

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

      4. frac-times 59.9%

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

      5. metadata-eval 59.9%

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

      6. metadata-eval 59.9%

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

      7. frac-times 60.0%

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

      8. sqrt-unprod 52.3%

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

      9. add-sqr-sqrt 86.4%

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

   7. Applied egg-rr 86.4%

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

   8. Taylor expanded in B around 0 39.3%

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

      1. associate-*r/ 39.3%

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

      2. neg-mul-1 39.3%

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

      3. distribute-neg-in 39.3%

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

      4. metadata-eval 39.3%

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

      5. unsub-neg 39.3%

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

   10. Simplified 39.3%

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

   if -2.4000000000000001e-93 < F < 4.5000000000000003e-58

   1. Initial program 99.4%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around inf 26.6%

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

   6. Taylor expanded in B around 0 17.1%

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

   7. Taylor expanded in x around inf 32.3%

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

      1. associate-*r/ 32.3%

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

      2. neg-mul-1 32.3%

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

   9. Simplified 32.3%

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

   if 4.5000000000000003e-58 < F

   1. Initial program 73.8%

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

   3. Simplified 79.7%

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

   5. Taylor expanded in F around inf 92.8%

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

   6. Taylor expanded in B around 0 40.3%

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

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.4 \cdot 10^{-93}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 4.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 31.5% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-62} \lor \neg \left(x \leq 1.4 \cdot 10^{-208}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -5.8e-62) (not (<= x 1.4e-208))) (/ x (- B)) (/ 1.0 B)))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -5.8e-62) || !(x <= 1.4e-208)) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-5.8d-62)) .or. (.not. (x <= 1.4d-208))) then
        tmp = x / -b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -5.8e-62) || !(x <= 1.4e-208)) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -5.8e-62) or not (x <= 1.4e-208):
		tmp = x / -B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -5.8e-62) || !(x <= 1.4e-208))
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -5.8e-62) || ~((x <= 1.4e-208)))
		tmp = x / -B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -5.8e-62], N[Not[LessEqual[x, 1.4e-208]], $MachinePrecision]], N[(x / (-B)), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -5.79999999999999971e-62 or 1.40000000000000001e-208 < x

   1. Initial program 86.1%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in F around inf 66.4%

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

   6. Taylor expanded in B around 0 29.6%

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

   7. Taylor expanded in x around inf 34.9%

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

      1. associate-*r/ 34.9%

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

      2. neg-mul-1 34.9%

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

   9. Simplified 34.9%

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

   if -5.79999999999999971e-62 < x < 1.40000000000000001e-208

   1. Initial program 69.4%

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

   3. Simplified 75.3%

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

   5. Taylor expanded in F around inf 34.2%

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

   6. Taylor expanded in B around 0 21.1%

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

   7. Taylor expanded in x around 0 21.1%

      \[\leadsto \color{blue}{\frac{1}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5.8 \cdot 10^{-62} \lor \neg \left(x \leq 1.4 \cdot 10^{-208}\right):\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 36.4% accurate, 23.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.35 \cdot 10^{-93}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{+190}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.35e-93)
   (/ (- -1.0 x) B)
   (if (<= F 6.5e+190) (/ x (- B)) (/ 1.0 B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.35e-93) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 6.5e+190) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / 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 <= (-2.35d-93)) then
        tmp = ((-1.0d0) - x) / b
    else if (f <= 6.5d+190) then
        tmp = x / -b
    else
        tmp = 1.0d0 / b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.35e-93) {
		tmp = (-1.0 - x) / B;
	} else if (F <= 6.5e+190) {
		tmp = x / -B;
	} else {
		tmp = 1.0 / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.35e-93:
		tmp = (-1.0 - x) / B
	elif F <= 6.5e+190:
		tmp = x / -B
	else:
		tmp = 1.0 / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.35e-93)
		tmp = Float64(Float64(-1.0 - x) / B);
	elseif (F <= 6.5e+190)
		tmp = Float64(x / Float64(-B));
	else
		tmp = Float64(1.0 / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.35e-93)
		tmp = (-1.0 - x) / B;
	elseif (F <= 6.5e+190)
		tmp = x / -B;
	else
		tmp = 1.0 / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.35e-93], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], If[LessEqual[F, 6.5e+190], N[(x / (-B)), $MachinePrecision], N[(1.0 / B), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -2.35e-93

   1. Initial program 70.3%

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

   3. Simplified 83.0%

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

   5. Taylor expanded in F around inf 48.8%

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

      1. div-inv 48.8%

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

      2. add-sqr-sqrt 18.6%

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

      3. sqrt-unprod 60.0%

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

      4. frac-times 59.9%

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

      5. metadata-eval 59.9%

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

      6. metadata-eval 59.9%

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

      7. frac-times 60.0%

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

      8. sqrt-unprod 52.3%

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

      9. add-sqr-sqrt 86.4%

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

   7. Applied egg-rr 86.4%

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

   8. Taylor expanded in B around 0 39.3%

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

      1. associate-*r/ 39.3%

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

      2. neg-mul-1 39.3%

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

      3. distribute-neg-in 39.3%

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

      4. metadata-eval 39.3%

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

      5. unsub-neg 39.3%

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

   10. Simplified 39.3%

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

   if -2.35e-93 < F < 6.5000000000000001e190

   1. Initial program 96.8%

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

   3. Simplified 98.9%

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

   5. Taylor expanded in F around inf 53.2%

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

   6. Taylor expanded in B around 0 26.8%

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

   7. Taylor expanded in x around inf 30.0%

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

      1. associate-*r/ 30.0%

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

      2. neg-mul-1 30.0%

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

   9. Simplified 30.0%

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

   if 6.5000000000000001e190 < F

   1. Initial program 29.0%

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

   3. Simplified 37.5%

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

   5. Taylor expanded in F around inf 99.5%

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

   6. Taylor expanded in B around 0 40.4%

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

   7. Taylor expanded in x around 0 33.1%

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

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.35 \cdot 10^{-93}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 6.5 \cdot 10^{+190}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 10.2% 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 80.9%

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

3. Simplified 87.3%

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

5. Taylor expanded in F around inf 56.4%

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

6. Taylor expanded in B around 0 27.0%

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

7. Taylor expanded in x around 0 9.2%

   \[\leadsto \color{blue}{\frac{1}{B}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024154 
(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))))))