VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.3% → 99.7%

Time: 19.5s

Alternatives: 22

Speedup: 1.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 76.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1.55 \cdot 10^{+21}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 10^{+46}:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\sin B} \cdot \cos B\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1.55e+21)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 1e+46)
       (- (* F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B))) t_0)
       (- (/ 1.0 (sin B)) (* (/ x (sin B)) (cos B)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1.55e+21) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 1e+46) {
		tmp = (F * (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - ((x / sin(B)) * cos(B));
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1.55e+21)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 1e+46)
		tmp = Float64(Float64(F * Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(Float64(x / sin(B)) * cos(B)));
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1.55e+21], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 1e+46], N[(N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.55e21

   1. Initial program 50.3%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -1.55e21 < F < 9.9999999999999999e45

   1. Initial program 98.2%

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

   3. Simplified 99.6%

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

   if 9.9999999999999999e45 < F

   1. Initial program 55.1%

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

   3. Simplified 70.8%

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

      1. tan-quot 70.8%

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

      2. associate-/r/ 70.9%

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

   6. Applied egg-rr 70.9%

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

   7. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.55 \cdot 10^{+21}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 10^{+46}:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\sin B} \cdot \cos B\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 23000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\sin B} \cdot \cos B\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5e+32)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 23000000.0)
     (-
      (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
      (/ (* x (cos B)) (sin B)))
     (- (/ 1.0 (sin B)) (* (/ x (sin B)) (cos B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5e+32) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 23000000.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - ((x * cos(B)) / sin(B));
	} else {
		tmp = (1.0 / sin(B)) - ((x / sin(B)) * cos(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 <= (-5d+32)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 23000000.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - ((x * cos(b)) / sin(b))
    else
        tmp = (1.0d0 / sin(b)) - ((x / sin(b)) * cos(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -5e+32) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 23000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - ((x * Math.cos(B)) / Math.sin(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - ((x / Math.sin(B)) * Math.cos(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5e+32:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 23000000.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - ((x * math.cos(B)) / math.sin(B))
	else:
		tmp = (1.0 / math.sin(B)) - ((x / math.sin(B)) * math.cos(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5e+32)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 23000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(Float64(x * cos(B)) / sin(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(Float64(x / sin(B)) * cos(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5e+32)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 23000000.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - ((x * cos(B)) / sin(B));
	else
		tmp = (1.0 / sin(B)) - ((x / sin(B)) * cos(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5e+32], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 23000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.9999999999999997e32

   1. Initial program 50.3%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -4.9999999999999997e32 < F < 2.3e7

   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. Taylor expanded in x around 0 99.5%

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

   if 2.3e7 < F

   1. Initial program 57.5%

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

   3. Simplified 74.1%

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

      1. tan-quot 74.1%

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

      2. associate-/r/ 74.1%

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

   6. Applied egg-rr 74.1%

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

   7. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{+32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 23000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\sin B} \cdot \cos B\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.6 \cdot 10^{+21}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 38000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\sin B} \cdot \cos B\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.6e+21)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 38000000.0)
     (+
      (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
      (/ -1.0 (/ (tan B) x)))
     (- (/ 1.0 (sin B)) (* (/ x (sin B)) (cos B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.6e+21) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 38000000.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (-1.0 / (tan(B) / x));
	} else {
		tmp = (1.0 / sin(B)) - ((x / sin(B)) * cos(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 <= (-1.6d+21)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 38000000.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) + ((-1.0d0) / (tan(b) / x))
    else
        tmp = (1.0d0 / sin(b)) - ((x / sin(b)) * cos(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.6e+21) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 38000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (-1.0 / (Math.tan(B) / x));
	} else {
		tmp = (1.0 / Math.sin(B)) - ((x / Math.sin(B)) * Math.cos(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.6e+21:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 38000000.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (-1.0 / (math.tan(B) / x))
	else:
		tmp = (1.0 / math.sin(B)) - ((x / math.sin(B)) * math.cos(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.6e+21)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 38000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) + Float64(-1.0 / Float64(tan(B) / x)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(Float64(x / sin(B)) * cos(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.6e+21)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 38000000.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) + (-1.0 / (tan(B) / x));
	else
		tmp = (1.0 / sin(B)) - ((x / sin(B)) * cos(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.6e+21], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 38000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.6e21

   1. Initial program 50.3%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -1.6e21 < F < 3.8e7

   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. clear-num 99.5%

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

   4. Applied egg-rr 99.5%

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

   if 3.8e7 < F

   1. Initial program 57.5%

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

   3. Simplified 74.1%

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

      1. tan-quot 74.1%

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

      2. associate-/r/ 74.1%

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

   6. Applied egg-rr 74.1%

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

   7. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.6 \cdot 10^{+21}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 38000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\sin B} \cdot \cos B\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.102:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\sin B} \cdot \cos B\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.102)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 100000000.0)
     (+
      (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
      (* x (/ -1.0 (tan B))))
     (- (/ 1.0 (sin B)) (* (/ x (sin B)) (cos B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.102) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 100000000.0) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / tan(B)));
	} else {
		tmp = (1.0 / sin(B)) - ((x / sin(B)) * cos(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-0.102d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 100000000.0d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) + (x * ((-1.0d0) / tan(b)))
    else
        tmp = (1.0d0 / sin(b)) - ((x / sin(b)) * cos(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.102) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 100000000.0) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / Math.tan(B)));
	} else {
		tmp = (1.0 / Math.sin(B)) - ((x / Math.sin(B)) * Math.cos(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.102:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 100000000.0:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) + (x * (-1.0 / math.tan(B)))
	else:
		tmp = (1.0 / math.sin(B)) - ((x / math.sin(B)) * math.cos(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.102)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 100000000.0)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) + Float64(x * Float64(-1.0 / tan(B))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(Float64(x / sin(B)) * cos(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.102)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 100000000.0)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) + (x * (-1.0 / tan(B)));
	else
		tmp = (1.0 / sin(B)) - ((x / sin(B)) * cos(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.102], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 100000000.0], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] + N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.101999999999999993

   1. Initial program 53.9%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -0.101999999999999993 < F < 1e8

   1. Initial program 99.4%

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

   if 1e8 < F

   1. Initial program 57.5%

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

   3. Simplified 74.1%

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

      1. tan-quot 74.1%

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

      2. associate-/r/ 74.1%

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

   6. Applied egg-rr 74.1%

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

   7. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.102:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 100000000:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} + x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\sin B} \cdot \cos B\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -8.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.25 \cdot 10^{-186}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-228}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.47:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot t\_0\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \frac{x}{\sin B} \cdot \cos B\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))))
   (if (<= F -8.8e-32)
     (- (/ -1.0 (sin B)) (/ x (tan B)))
     (if (<= F -1.25e-186)
       (- (* F (/ (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (sin B))) (/ x B))
       (if (<= F 3.1e-228)
         (/ (* (cos B) (- x)) (sin B))
         (if (<= F 0.47)
           (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (* F t_0)) (/ x B))
           (- t_0 (* (/ x (sin B)) (cos B)))))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double tmp;
	if (F <= -8.8e-32) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -1.25e-186) {
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) / sin(B))) - (x / B);
	} else if (F <= 3.1e-228) {
		tmp = (cos(B) * -x) / sin(B);
	} else if (F <= 0.47) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F * t_0)) - (x / B);
	} else {
		tmp = t_0 - ((x / sin(B)) * cos(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / sin(b)
    if (f <= (-8.8d-32)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-1.25d-186)) then
        tmp = (f * (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) / sin(b))) - (x / b)
    else if (f <= 3.1d-228) then
        tmp = (cos(b) * -x) / sin(b)
    else if (f <= 0.47d0) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f * t_0)) - (x / b)
    else
        tmp = t_0 - ((x / sin(b)) * cos(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 1.0 / Math.sin(B);
	double tmp;
	if (F <= -8.8e-32) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -1.25e-186) {
		tmp = (F * (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) / Math.sin(B))) - (x / B);
	} else if (F <= 3.1e-228) {
		tmp = (Math.cos(B) * -x) / Math.sin(B);
	} else if (F <= 0.47) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F * t_0)) - (x / B);
	} else {
		tmp = t_0 - ((x / Math.sin(B)) * Math.cos(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 1.0 / math.sin(B)
	tmp = 0
	if F <= -8.8e-32:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -1.25e-186:
		tmp = (F * (math.sqrt((1.0 / (2.0 + (x * 2.0)))) / math.sin(B))) - (x / B)
	elif F <= 3.1e-228:
		tmp = (math.cos(B) * -x) / math.sin(B)
	elif F <= 0.47:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F * t_0)) - (x / B)
	else:
		tmp = t_0 - ((x / math.sin(B)) * math.cos(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -8.8e-32)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -1.25e-186)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) / sin(B))) - Float64(x / B));
	elseif (F <= 3.1e-228)
		tmp = Float64(Float64(cos(B) * Float64(-x)) / sin(B));
	elseif (F <= 0.47)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F * t_0)) - Float64(x / B));
	else
		tmp = Float64(t_0 - Float64(Float64(x / sin(B)) * cos(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 1.0 / sin(B);
	tmp = 0.0;
	if (F <= -8.8e-32)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -1.25e-186)
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) / sin(B))) - (x / B);
	elseif (F <= 3.1e-228)
		tmp = (cos(B) * -x) / sin(B);
	elseif (F <= 0.47)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F * t_0)) - (x / B);
	else
		tmp = t_0 - ((x / sin(B)) * cos(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -8.8e-32], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.25e-186], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.1e-228], N[(N[(N[Cos[B], $MachinePrecision] * (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.47], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(t$95$0 - N[(N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -8.7999999999999999e-32

   1. Initial program 55.5%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -8.7999999999999999e-32 < F < -1.25e-186

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 90.7%

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

   4. Taylor expanded in F around 0 90.7%

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

      1. associate-*l/ 90.8%

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

      2. associate-*r/ 90.8%

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

   6. Simplified 90.8%

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

   if -1.25e-186 < F < 3.0999999999999998e-228

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 47.9%

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

   4. Taylor expanded in x around inf 92.5%

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

      1. associate-*r/ 92.5%

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

      2. neg-mul-1 92.5%

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

      3. distribute-rgt-neg-in 92.5%

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

   6. Simplified 92.5%

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

   if 3.0999999999999998e-228 < F < 0.46999999999999997

   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. Taylor expanded in B around 0 86.5%

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

      1. div-inv 86.5%

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

   5. Applied egg-rr 86.5%

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

   if 0.46999999999999997 < F

   1. Initial program 58.7%

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

   3. Simplified 74.8%

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

      1. tan-quot 74.8%

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

      2. associate-/r/ 74.9%

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

   6. Applied egg-rr 74.9%

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

   7. Taylor expanded in F around inf 98.9%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.25 \cdot 10^{-186}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-228}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.47:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{\sin B}\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\sin B} \cdot \cos B\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sin B}\\ t_1 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -33000000000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_1\\ \mathbf{elif}\;F \leq 3.7:\\ \;\;\;\;F \cdot \left(t\_0 \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0 - \frac{x}{\sin B} \cdot \cos B\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sin B))) (t_1 (/ x (tan B))))
   (if (<= F -33000000000.0)
     (- (/ -1.0 (sin B)) t_1)
     (if (<= F 3.7)
       (- (* F (* t_0 (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))))) t_1)
       (- t_0 (* (/ x (sin B)) (cos B)))))))

C

double code(double F, double B, double x) {
	double t_0 = 1.0 / sin(B);
	double t_1 = x / tan(B);
	double tmp;
	if (F <= -33000000000.0) {
		tmp = (-1.0 / sin(B)) - t_1;
	} else if (F <= 3.7) {
		tmp = (F * (t_0 * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	} else {
		tmp = t_0 - ((x / sin(B)) * cos(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 / sin(b)
    t_1 = x / tan(b)
    if (f <= (-33000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_1
    else if (f <= 3.7d0) then
        tmp = (f * (t_0 * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))))) - t_1
    else
        tmp = t_0 - ((x / sin(b)) * cos(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = 1.0 / Math.sin(B);
	double t_1 = x / Math.tan(B);
	double tmp;
	if (F <= -33000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_1;
	} else if (F <= 3.7) {
		tmp = (F * (t_0 * Math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	} else {
		tmp = t_0 - ((x / Math.sin(B)) * Math.cos(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = 1.0 / math.sin(B)
	t_1 = x / math.tan(B)
	tmp = 0
	if F <= -33000000000.0:
		tmp = (-1.0 / math.sin(B)) - t_1
	elif F <= 3.7:
		tmp = (F * (t_0 * math.sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1
	else:
		tmp = t_0 - ((x / math.sin(B)) * math.cos(B))
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(1.0 / sin(B))
	t_1 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -33000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_1);
	elseif (F <= 3.7)
		tmp = Float64(Float64(F * Float64(t_0 * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))))) - t_1);
	else
		tmp = Float64(t_0 - Float64(Float64(x / sin(B)) * cos(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = 1.0 / sin(B);
	t_1 = x / tan(B);
	tmp = 0.0;
	if (F <= -33000000000.0)
		tmp = (-1.0 / sin(B)) - t_1;
	elseif (F <= 3.7)
		tmp = (F * (t_0 * sqrt((1.0 / (2.0 + (x * 2.0)))))) - t_1;
	else
		tmp = t_0 - ((x / sin(B)) * cos(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -33000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], If[LessEqual[F, 3.7], N[(N[(F * N[(t$95$0 * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision], N[(t$95$0 - N[(N[(x / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Cos[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.3e10

   1. Initial program 52.2%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -3.3e10 < F < 3.7000000000000002

   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 0 99.1%

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

   if 3.7000000000000002 < F

   1. Initial program 57.5%

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

   3. Simplified 74.1%

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

      1. tan-quot 74.1%

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

      2. associate-/r/ 74.1%

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

   6. Applied egg-rr 74.1%

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

   7. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -33000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.7:\\ \;\;\;\;F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}}\right) - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\sin B} \cdot \cos B\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sin B}\\ \mathbf{if}\;F \leq -8.8 \cdot 10^{-32}:\\ \;\;\;\;t\_0 - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.05 \cdot 10^{-187}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-229}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.47:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{\sin B}\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (sin B))))
   (if (<= F -8.8e-32)
     (- t_0 (/ x (tan B)))
     (if (<= F -2.05e-187)
       (- (* F (/ (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (sin B))) (/ x B))
       (if (<= F 4.6e-229)
         (/ (* (cos B) (- x)) (sin B))
         (if (<= F 0.47)
           (-
            (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (* F (/ 1.0 (sin B))))
            (/ x B))
           (- (/ (- x) (tan B)) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 / sin(B);
	double tmp;
	if (F <= -8.8e-32) {
		tmp = t_0 - (x / tan(B));
	} else if (F <= -2.05e-187) {
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) / sin(B))) - (x / B);
	} else if (F <= 4.6e-229) {
		tmp = (cos(B) * -x) / sin(B);
	} else if (F <= 0.47) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F * (1.0 / sin(B)))) - (x / B);
	} else {
		tmp = (-x / tan(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 = (-1.0d0) / sin(b)
    if (f <= (-8.8d-32)) then
        tmp = t_0 - (x / tan(b))
    else if (f <= (-2.05d-187)) then
        tmp = (f * (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) / sin(b))) - (x / b)
    else if (f <= 4.6d-229) then
        tmp = (cos(b) * -x) / sin(b)
    else if (f <= 0.47d0) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f * (1.0d0 / sin(b)))) - (x / b)
    else
        tmp = (-x / tan(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -1.0 / Math.sin(B);
	double tmp;
	if (F <= -8.8e-32) {
		tmp = t_0 - (x / Math.tan(B));
	} else if (F <= -2.05e-187) {
		tmp = (F * (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) / Math.sin(B))) - (x / B);
	} else if (F <= 4.6e-229) {
		tmp = (Math.cos(B) * -x) / Math.sin(B);
	} else if (F <= 0.47) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F * (1.0 / Math.sin(B)))) - (x / B);
	} else {
		tmp = (-x / Math.tan(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -1.0 / math.sin(B)
	tmp = 0
	if F <= -8.8e-32:
		tmp = t_0 - (x / math.tan(B))
	elif F <= -2.05e-187:
		tmp = (F * (math.sqrt((1.0 / (2.0 + (x * 2.0)))) / math.sin(B))) - (x / B)
	elif F <= 4.6e-229:
		tmp = (math.cos(B) * -x) / math.sin(B)
	elif F <= 0.47:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F * (1.0 / math.sin(B)))) - (x / B)
	else:
		tmp = (-x / math.tan(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 / sin(B))
	tmp = 0.0
	if (F <= -8.8e-32)
		tmp = Float64(t_0 - Float64(x / tan(B)));
	elseif (F <= -2.05e-187)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) / sin(B))) - Float64(x / B));
	elseif (F <= 4.6e-229)
		tmp = Float64(Float64(cos(B) * Float64(-x)) / sin(B));
	elseif (F <= 0.47)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F * Float64(1.0 / sin(B)))) - Float64(x / B));
	else
		tmp = Float64(Float64(Float64(-x) / tan(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -1.0 / sin(B);
	tmp = 0.0;
	if (F <= -8.8e-32)
		tmp = t_0 - (x / tan(B));
	elseif (F <= -2.05e-187)
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) / sin(B))) - (x / B);
	elseif (F <= 4.6e-229)
		tmp = (cos(B) * -x) / sin(B);
	elseif (F <= 0.47)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F * (1.0 / sin(B)))) - (x / B);
	else
		tmp = (-x / tan(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -8.8e-32], N[(t$95$0 - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.05e-187], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 4.6e-229], N[(N[(N[Cos[B], $MachinePrecision] * (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.47], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F * N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -8.7999999999999999e-32

   1. Initial program 55.5%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -8.7999999999999999e-32 < F < -2.0500000000000001e-187

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 90.7%

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

   4. Taylor expanded in F around 0 90.7%

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

      1. associate-*l/ 90.8%

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

      2. associate-*r/ 90.8%

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

   6. Simplified 90.8%

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

   if -2.0500000000000001e-187 < F < 4.59999999999999992e-229

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 47.9%

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

   4. Taylor expanded in x around inf 92.5%

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

      1. associate-*r/ 92.5%

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

      2. neg-mul-1 92.5%

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

      3. distribute-rgt-neg-in 92.5%

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

   6. Simplified 92.5%

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

   if 4.59999999999999992e-229 < F < 0.46999999999999997

   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. Taylor expanded in B around 0 86.5%

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

      1. div-inv 86.5%

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

   5. Applied egg-rr 86.5%

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

   if 0.46999999999999997 < F

   1. Initial program 58.7%

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

   3. Simplified 74.8%

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

   5. Taylor expanded in F around inf 98.6%

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

      1. associate-/r* 98.7%

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

      2. associate-*r/ 98.8%

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

      3. rgt-mult-inverse 98.9%

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

      4. frac-2neg 98.9%

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

      5. metadata-eval 98.9%

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

   7. Applied egg-rr 98.9%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.05 \cdot 10^{-187}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 4.6 \cdot 10^{-229}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.47:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \left(F \cdot \frac{1}{\sin B}\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B} - \frac{-1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sin B}\\ \mathbf{if}\;F \leq -8.8 \cdot 10^{-32}:\\ \;\;\;\;t\_0 - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.4 \cdot 10^{-182}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.25 \cdot 10^{-228}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.47:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (sin B))))
   (if (<= F -8.8e-32)
     (- t_0 (/ x (tan B)))
     (if (<= F -2.4e-182)
       (- (* F (/ (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (sin B))) (/ x B))
       (if (<= F 2.25e-228)
         (/ (* (cos B) (- x)) (sin B))
         (if (<= F 0.47)
           (-
            (* (/ F (sin B)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
            (/ x B))
           (- (/ (- x) (tan B)) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 / sin(B);
	double tmp;
	if (F <= -8.8e-32) {
		tmp = t_0 - (x / tan(B));
	} else if (F <= -2.4e-182) {
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) / sin(B))) - (x / B);
	} else if (F <= 2.25e-228) {
		tmp = (cos(B) * -x) / sin(B);
	} else if (F <= 0.47) {
		tmp = ((F / sin(B)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else {
		tmp = (-x / tan(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 = (-1.0d0) / sin(b)
    if (f <= (-8.8d-32)) then
        tmp = t_0 - (x / tan(b))
    else if (f <= (-2.4d-182)) then
        tmp = (f * (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) / sin(b))) - (x / b)
    else if (f <= 2.25d-228) then
        tmp = (cos(b) * -x) / sin(b)
    else if (f <= 0.47d0) then
        tmp = ((f / sin(b)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    else
        tmp = (-x / tan(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -1.0 / Math.sin(B);
	double tmp;
	if (F <= -8.8e-32) {
		tmp = t_0 - (x / Math.tan(B));
	} else if (F <= -2.4e-182) {
		tmp = (F * (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) / Math.sin(B))) - (x / B);
	} else if (F <= 2.25e-228) {
		tmp = (Math.cos(B) * -x) / Math.sin(B);
	} else if (F <= 0.47) {
		tmp = ((F / Math.sin(B)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else {
		tmp = (-x / Math.tan(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -1.0 / math.sin(B)
	tmp = 0
	if F <= -8.8e-32:
		tmp = t_0 - (x / math.tan(B))
	elif F <= -2.4e-182:
		tmp = (F * (math.sqrt((1.0 / (2.0 + (x * 2.0)))) / math.sin(B))) - (x / B)
	elif F <= 2.25e-228:
		tmp = (math.cos(B) * -x) / math.sin(B)
	elif F <= 0.47:
		tmp = ((F / math.sin(B)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B)
	else:
		tmp = (-x / math.tan(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 / sin(B))
	tmp = 0.0
	if (F <= -8.8e-32)
		tmp = Float64(t_0 - Float64(x / tan(B)));
	elseif (F <= -2.4e-182)
		tmp = Float64(Float64(F * Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) / sin(B))) - Float64(x / B));
	elseif (F <= 2.25e-228)
		tmp = Float64(Float64(cos(B) * Float64(-x)) / sin(B));
	elseif (F <= 0.47)
		tmp = Float64(Float64(Float64(F / sin(B)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B));
	else
		tmp = Float64(Float64(Float64(-x) / tan(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -1.0 / sin(B);
	tmp = 0.0;
	if (F <= -8.8e-32)
		tmp = t_0 - (x / tan(B));
	elseif (F <= -2.4e-182)
		tmp = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) / sin(B))) - (x / B);
	elseif (F <= 2.25e-228)
		tmp = (cos(B) * -x) / sin(B);
	elseif (F <= 0.47)
		tmp = ((F / sin(B)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (x / B);
	else
		tmp = (-x / tan(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -8.8e-32], N[(t$95$0 - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.4e-182], N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.25e-228], N[(N[(N[Cos[B], $MachinePrecision] * (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.47], N[(N[(N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -8.7999999999999999e-32

   1. Initial program 55.5%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -8.7999999999999999e-32 < F < -2.3999999999999998e-182

   1. Initial program 99.5%

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

   3. Taylor expanded in B around 0 90.7%

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

   4. Taylor expanded in F around 0 90.7%

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

      1. associate-*l/ 90.8%

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

      2. associate-*r/ 90.8%

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

   6. Simplified 90.8%

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

   if -2.3999999999999998e-182 < F < 2.25e-228

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 47.9%

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

   4. Taylor expanded in x around inf 92.5%

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

      1. associate-*r/ 92.5%

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

      2. neg-mul-1 92.5%

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

      3. distribute-rgt-neg-in 92.5%

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

   6. Simplified 92.5%

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

   if 2.25e-228 < F < 0.46999999999999997

   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. Taylor expanded in B around 0 86.5%

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

   if 0.46999999999999997 < F

   1. Initial program 58.7%

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

   3. Simplified 74.8%

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

   5. Taylor expanded in F around inf 98.6%

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

      1. associate-/r* 98.7%

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

      2. associate-*r/ 98.8%

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

      3. rgt-mult-inverse 98.9%

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

      4. frac-2neg 98.9%

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

      5. metadata-eval 98.9%

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

   7. Applied egg-rr 98.9%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.4 \cdot 10^{-182}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.25 \cdot 10^{-228}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.47:\\ \;\;\;\;\frac{F}{\sin B} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B} - \frac{-1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 88.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - \frac{x}{B}\\ t_1 := \frac{-1}{\sin B}\\ \mathbf{if}\;F \leq -8.8 \cdot 10^{-32}:\\ \;\;\;\;t\_1 - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.15 \cdot 10^{-186}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{-228}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.029:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B} - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (- (* F (/ (sqrt (/ 1.0 (+ 2.0 (* x 2.0)))) (sin B))) (/ x B)))
        (t_1 (/ -1.0 (sin B))))
   (if (<= F -8.8e-32)
     (- t_1 (/ x (tan B)))
     (if (<= F -1.15e-186)
       t_0
       (if (<= F 2.4e-228)
         (/ (* (cos B) (- x)) (sin B))
         (if (<= F 0.029) t_0 (- (/ (- x) (tan B)) t_1)))))))

C

double code(double F, double B, double x) {
	double t_0 = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) / sin(B))) - (x / B);
	double t_1 = -1.0 / sin(B);
	double tmp;
	if (F <= -8.8e-32) {
		tmp = t_1 - (x / tan(B));
	} else if (F <= -1.15e-186) {
		tmp = t_0;
	} else if (F <= 2.4e-228) {
		tmp = (cos(B) * -x) / sin(B);
	} else if (F <= 0.029) {
		tmp = t_0;
	} else {
		tmp = (-x / tan(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 * (sqrt((1.0d0 / (2.0d0 + (x * 2.0d0)))) / sin(b))) - (x / b)
    t_1 = (-1.0d0) / sin(b)
    if (f <= (-8.8d-32)) then
        tmp = t_1 - (x / tan(b))
    else if (f <= (-1.15d-186)) then
        tmp = t_0
    else if (f <= 2.4d-228) then
        tmp = (cos(b) * -x) / sin(b)
    else if (f <= 0.029d0) then
        tmp = t_0
    else
        tmp = (-x / tan(b)) - t_1
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (F * (Math.sqrt((1.0 / (2.0 + (x * 2.0)))) / Math.sin(B))) - (x / B);
	double t_1 = -1.0 / Math.sin(B);
	double tmp;
	if (F <= -8.8e-32) {
		tmp = t_1 - (x / Math.tan(B));
	} else if (F <= -1.15e-186) {
		tmp = t_0;
	} else if (F <= 2.4e-228) {
		tmp = (Math.cos(B) * -x) / Math.sin(B);
	} else if (F <= 0.029) {
		tmp = t_0;
	} else {
		tmp = (-x / Math.tan(B)) - t_1;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (F * (math.sqrt((1.0 / (2.0 + (x * 2.0)))) / math.sin(B))) - (x / B)
	t_1 = -1.0 / math.sin(B)
	tmp = 0
	if F <= -8.8e-32:
		tmp = t_1 - (x / math.tan(B))
	elif F <= -1.15e-186:
		tmp = t_0
	elif F <= 2.4e-228:
		tmp = (math.cos(B) * -x) / math.sin(B)
	elif F <= 0.029:
		tmp = t_0
	else:
		tmp = (-x / math.tan(B)) - t_1
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(F * Float64(sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0)))) / sin(B))) - Float64(x / B))
	t_1 = Float64(-1.0 / sin(B))
	tmp = 0.0
	if (F <= -8.8e-32)
		tmp = Float64(t_1 - Float64(x / tan(B)));
	elseif (F <= -1.15e-186)
		tmp = t_0;
	elseif (F <= 2.4e-228)
		tmp = Float64(Float64(cos(B) * Float64(-x)) / sin(B));
	elseif (F <= 0.029)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(-x) / tan(B)) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (F * (sqrt((1.0 / (2.0 + (x * 2.0)))) / sin(B))) - (x / B);
	t_1 = -1.0 / sin(B);
	tmp = 0.0;
	if (F <= -8.8e-32)
		tmp = t_1 - (x / tan(B));
	elseif (F <= -1.15e-186)
		tmp = t_0;
	elseif (F <= 2.4e-228)
		tmp = (cos(B) * -x) / sin(B);
	elseif (F <= 0.029)
		tmp = t_0;
	else
		tmp = (-x / tan(B)) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(F * N[(N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -8.8e-32], N[(t$95$1 - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1.15e-186], t$95$0, If[LessEqual[F, 2.4e-228], N[(N[(N[Cos[B], $MachinePrecision] * (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.029], t$95$0, N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -8.7999999999999999e-32

   1. Initial program 55.5%

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

   3. Taylor expanded in F around -inf 99.7%

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

      1. div-inv 99.8%

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

   5. Applied egg-rr 99.8%

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

   if -8.7999999999999999e-32 < F < -1.15e-186 or 2.40000000000000002e-228 < F < 0.0290000000000000015

   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. Taylor expanded in B around 0 87.7%

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

   4. Taylor expanded in F around 0 87.6%

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

      1. associate-*l/ 87.6%

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

      2. associate-*r/ 87.7%

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

   6. Simplified 87.7%

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

   if -1.15e-186 < F < 2.40000000000000002e-228

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 47.9%

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

   4. Taylor expanded in x around inf 92.5%

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

      1. associate-*r/ 92.5%

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

      2. neg-mul-1 92.5%

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

      3. distribute-rgt-neg-in 92.5%

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

   6. Simplified 92.5%

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

   if 0.0290000000000000015 < F

   1. Initial program 58.7%

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

   3. Simplified 74.8%

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

   5. Taylor expanded in F around inf 98.6%

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

      1. associate-/r* 98.7%

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

      2. associate-*r/ 98.8%

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

      3. rgt-mult-inverse 98.9%

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

      4. frac-2neg 98.9%

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

      5. metadata-eval 98.9%

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

   7. Applied egg-rr 98.9%

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

4. Final simplification 94.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.8 \cdot 10^{-32}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -1.15 \cdot 10^{-186}:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{-228}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.029:\\ \;\;\;\;F \cdot \frac{\sqrt{\frac{1}{2 + x \cdot 2}}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B} - \frac{-1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1}{\sin B}\\ \mathbf{if}\;F \leq -7.5 \cdot 10^{-134}:\\ \;\;\;\;t\_0 - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-228}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.0055:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ -1.0 (sin B))))
   (if (<= F -7.5e-134)
     (- t_0 (/ x (tan B)))
     (if (<= F 2.5e-228)
       (/ (* (cos B) (- x)) (sin B))
       (if (<= F 0.0055)
         (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)) (/ x B))
         (- (/ (- x) (tan B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = -1.0 / sin(B);
	double tmp;
	if (F <= -7.5e-134) {
		tmp = t_0 - (x / tan(B));
	} else if (F <= 2.5e-228) {
		tmp = (cos(B) * -x) / sin(B);
	} else if (F <= 0.0055) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (-x / tan(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 = (-1.0d0) / sin(b)
    if (f <= (-7.5d-134)) then
        tmp = t_0 - (x / tan(b))
    else if (f <= 2.5d-228) then
        tmp = (cos(b) * -x) / sin(b)
    else if (f <= 0.0055d0) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else
        tmp = (-x / tan(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = -1.0 / Math.sin(B);
	double tmp;
	if (F <= -7.5e-134) {
		tmp = t_0 - (x / Math.tan(B));
	} else if (F <= 2.5e-228) {
		tmp = (Math.cos(B) * -x) / Math.sin(B);
	} else if (F <= 0.0055) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (-x / Math.tan(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = -1.0 / math.sin(B)
	tmp = 0
	if F <= -7.5e-134:
		tmp = t_0 - (x / math.tan(B))
	elif F <= 2.5e-228:
		tmp = (math.cos(B) * -x) / math.sin(B)
	elif F <= 0.0055:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	else:
		tmp = (-x / math.tan(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(-1.0 / sin(B))
	tmp = 0.0
	if (F <= -7.5e-134)
		tmp = Float64(t_0 - Float64(x / tan(B)));
	elseif (F <= 2.5e-228)
		tmp = Float64(Float64(cos(B) * Float64(-x)) / sin(B));
	elseif (F <= 0.0055)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	else
		tmp = Float64(Float64(Float64(-x) / tan(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = -1.0 / sin(B);
	tmp = 0.0;
	if (F <= -7.5e-134)
		tmp = t_0 - (x / tan(B));
	elseif (F <= 2.5e-228)
		tmp = (cos(B) * -x) / sin(B);
	elseif (F <= 0.0055)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	else
		tmp = (-x / tan(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -7.5e-134], N[(t$95$0 - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.5e-228], N[(N[(N[Cos[B], $MachinePrecision] * (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0055], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / N[Tan[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7.50000000000000048e-134

   1. Initial program 62.6%

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

   3. Taylor expanded in F around -inf 88.7%

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

      1. div-inv 88.8%

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

   5. Applied egg-rr 88.8%

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

   if -7.50000000000000048e-134 < F < 2.49999999999999986e-228

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 43.0%

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

   4. Taylor expanded in x around inf 85.8%

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

      1. associate-*r/ 85.8%

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

      2. neg-mul-1 85.8%

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

      3. distribute-rgt-neg-in 85.8%

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

   6. Simplified 85.8%

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

   if 2.49999999999999986e-228 < F < 0.0054999999999999997

   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. Taylor expanded in B around 0 86.5%

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

   4. Taylor expanded in B around 0 66.7%

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

   if 0.0054999999999999997 < F

   1. Initial program 58.7%

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

   3. Simplified 74.8%

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

   5. Taylor expanded in F around inf 98.6%

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

      1. associate-/r* 98.7%

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

      2. associate-*r/ 98.8%

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

      3. rgt-mult-inverse 98.9%

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

      4. frac-2neg 98.9%

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

      5. metadata-eval 98.9%

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

   7. Applied egg-rr 98.9%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.5 \cdot 10^{-134}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.5 \cdot 10^{-228}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 0.0055:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{\tan B} - \frac{-1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 76.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{-133}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-228}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 850:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.7e-133)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F 3.2e-228)
     (/ (* (cos B) (- x)) (sin B))
     (if (<= F 850.0)
       (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)) (/ x B))
       (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.7e-133) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= 3.2e-228) {
		tmp = (cos(B) * -x) / sin(B);
	} else if (F <= 850.0) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.7d-133)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= 3.2d-228) then
        tmp = (cos(b) * -x) / sin(b)
    else if (f <= 850.0d0) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.7e-133) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= 3.2e-228) {
		tmp = (Math.cos(B) * -x) / Math.sin(B);
	} else if (F <= 850.0) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.7e-133:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= 3.2e-228:
		tmp = (math.cos(B) * -x) / math.sin(B)
	elif F <= 850.0:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.7e-133)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= 3.2e-228)
		tmp = Float64(Float64(cos(B) * Float64(-x)) / sin(B));
	elseif (F <= 850.0)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.7e-133)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= 3.2e-228)
		tmp = (cos(B) * -x) / sin(B);
	elseif (F <= 850.0)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.7e-133], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.2e-228], N[(N[(N[Cos[B], $MachinePrecision] * (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 850.0], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.70000000000000003e-133

   1. Initial program 62.6%

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

   3. Taylor expanded in F around -inf 88.7%

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

      1. div-inv 88.8%

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

   5. Applied egg-rr 88.8%

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

   if -1.70000000000000003e-133 < F < 3.20000000000000022e-228

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 43.0%

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

   4. Taylor expanded in x around inf 85.8%

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

      1. associate-*r/ 85.8%

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

      2. neg-mul-1 85.8%

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

      3. distribute-rgt-neg-in 85.8%

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

   6. Simplified 85.8%

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

   if 3.20000000000000022e-228 < F < 850

   1. Initial program 99.3%

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

   3. Taylor expanded in B around 0 85.3%

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

   4. Taylor expanded in B around 0 66.2%

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

   if 850 < F

   1. Initial program 57.5%

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

   3. Taylor expanded in B around 0 34.8%

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

   4. Taylor expanded in F around inf 76.9%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{-133}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-228}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 850:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 69.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{-133}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-228}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 160000:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.7e-133)
   (- (/ -1.0 B) (* x (/ 1.0 (tan B))))
   (if (<= F 3.2e-228)
     (/ (* (cos B) (- x)) (sin B))
     (if (<= F 160000.0)
       (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)) (/ x B))
       (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.7e-133) {
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	} else if (F <= 3.2e-228) {
		tmp = (cos(B) * -x) / sin(B);
	} else if (F <= 160000.0) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.7d-133)) then
        tmp = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    else if (f <= 3.2d-228) then
        tmp = (cos(b) * -x) / sin(b)
    else if (f <= 160000.0d0) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.7e-133) {
		tmp = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else if (F <= 3.2e-228) {
		tmp = (Math.cos(B) * -x) / Math.sin(B);
	} else if (F <= 160000.0) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.7e-133:
		tmp = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	elif F <= 3.2e-228:
		tmp = (math.cos(B) * -x) / math.sin(B)
	elif F <= 160000.0:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.7e-133)
		tmp = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= 3.2e-228)
		tmp = Float64(Float64(cos(B) * Float64(-x)) / sin(B));
	elseif (F <= 160000.0)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.7e-133)
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	elseif (F <= 3.2e-228)
		tmp = (cos(B) * -x) / sin(B);
	elseif (F <= 160000.0)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.7e-133], N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.2e-228], N[(N[(N[Cos[B], $MachinePrecision] * (-x)), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 160000.0], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.70000000000000003e-133

   1. Initial program 62.6%

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

   3. Taylor expanded in F around -inf 88.7%

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

   4. Taylor expanded in B around 0 73.0%

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

   if -1.70000000000000003e-133 < F < 3.20000000000000022e-228

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 43.0%

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

   4. Taylor expanded in x around inf 85.8%

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

      1. associate-*r/ 85.8%

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

      2. neg-mul-1 85.8%

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

      3. distribute-rgt-neg-in 85.8%

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

   6. Simplified 85.8%

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

   if 3.20000000000000022e-228 < F < 1.6e5

   1. Initial program 99.3%

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

   3. Taylor expanded in B around 0 85.3%

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

   4. Taylor expanded in B around 0 66.2%

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

   if 1.6e5 < F

   1. Initial program 57.5%

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

   3. Taylor expanded in B around 0 34.8%

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

   4. Taylor expanded in F around inf 76.9%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{-133}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-228}:\\ \;\;\;\;\frac{\cos B \cdot \left(-x\right)}{\sin B}\\ \mathbf{elif}\;F \leq 160000:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 69.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.5 \cdot 10^{-134}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-228}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \mathbf{elif}\;F \leq 1800:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.5e-134)
   (- (/ -1.0 B) (* x (/ 1.0 (tan B))))
   (if (<= F 3.2e-228)
     (* x (/ -1.0 (tan B)))
     (if (<= F 1800.0)
       (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F B)) (/ x B))
       (- (/ 1.0 (sin B)) (/ x B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.5e-134) {
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	} else if (F <= 3.2e-228) {
		tmp = x * (-1.0 / tan(B));
	} else if (F <= 1800.0) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-7.5d-134)) then
        tmp = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    else if (f <= 3.2d-228) then
        tmp = x * ((-1.0d0) / tan(b))
    else if (f <= 1800.0d0) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / b)) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.5e-134) {
		tmp = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else if (F <= 3.2e-228) {
		tmp = x * (-1.0 / Math.tan(B));
	} else if (F <= 1800.0) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -7.5e-134:
		tmp = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	elif F <= 3.2e-228:
		tmp = x * (-1.0 / math.tan(B))
	elif F <= 1800.0:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / B)) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.5e-134)
		tmp = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= 3.2e-228)
		tmp = Float64(x * Float64(-1.0 / tan(B)));
	elseif (F <= 1800.0)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -7.5e-134)
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	elseif (F <= 3.2e-228)
		tmp = x * (-1.0 / tan(B));
	elseif (F <= 1800.0)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / B)) - (x / B);
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.5e-134], N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.2e-228], N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1800.0], N[(N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -7.50000000000000048e-134

   1. Initial program 62.6%

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

   3. Taylor expanded in F around -inf 88.7%

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

   4. Taylor expanded in B around 0 73.0%

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

   if -7.50000000000000048e-134 < F < 3.20000000000000022e-228

   1. Initial program 99.5%

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

   3. Taylor expanded in F around -inf 43.0%

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

   4. Taylor expanded in x around inf 85.8%

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

      1. mul-1-neg 85.8%

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

      2. associate-/l* 85.7%

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

      3. distribute-rgt-neg-in 85.7%

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

   6. Simplified 85.7%

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

      1. clear-num 85.7%

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

      2. tan-quot 85.7%

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

      3. frac-2neg 85.7%

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

      4. metadata-eval 85.7%

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

      5. div-inv 85.7%

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

   8. Applied egg-rr 85.7%

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

      1. mul-1-neg 85.7%

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

      2. distribute-frac-neg2 85.7%

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

      3. remove-double-neg 85.7%

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

   10. Simplified 85.7%

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

   if 3.20000000000000022e-228 < F < 1800

   1. Initial program 99.3%

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

   3. Taylor expanded in B around 0 85.3%

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

   4. Taylor expanded in B around 0 66.2%

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

   if 1800 < F

   1. Initial program 57.5%

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

   3. Taylor expanded in B around 0 34.8%

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

   4. Taylor expanded in F around inf 76.9%

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

4. Final simplification 75.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.5 \cdot 10^{-134}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-228}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \mathbf{elif}\;F \leq 1800:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 59.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-183} \lor \neg \left(x \leq 2.6 \cdot 10^{-124}\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -9.6e-183) (not (<= x 2.6e-124)))
   (* x (/ -1.0 (tan B)))
   (- (/ -1.0 (sin B)) (/ x B))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -9.6e-183) || !(x <= 2.6e-124)) {
		tmp = x * (-1.0 / tan(B));
	} else {
		tmp = (-1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-9.6d-183)) .or. (.not. (x <= 2.6d-124))) then
        tmp = x * ((-1.0d0) / tan(b))
    else
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -9.6e-183) || !(x <= 2.6e-124)) {
		tmp = x * (-1.0 / Math.tan(B));
	} else {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -9.6e-183) or not (x <= 2.6e-124):
		tmp = x * (-1.0 / math.tan(B))
	else:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -9.6e-183) || !(x <= 2.6e-124))
		tmp = Float64(x * Float64(-1.0 / tan(B)));
	else
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -9.6e-183) || ~((x <= 2.6e-124)))
		tmp = x * (-1.0 / tan(B));
	else
		tmp = (-1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -9.6e-183], N[Not[LessEqual[x, 2.6e-124]], $MachinePrecision]], N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.59999999999999972e-183 or 2.6e-124 < x

   1. Initial program 81.3%

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

   3. Taylor expanded in F around -inf 67.2%

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

   4. Taylor expanded in x around inf 81.3%

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

      1. mul-1-neg 81.3%

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

      2. associate-/l* 81.1%

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

      3. distribute-rgt-neg-in 81.1%

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

   6. Simplified 81.1%

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

      1. clear-num 81.1%

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

      2. tan-quot 81.2%

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

      3. frac-2neg 81.2%

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

      4. metadata-eval 81.2%

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

      5. div-inv 81.2%

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

   8. Applied egg-rr 81.2%

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

      1. mul-1-neg 81.2%

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

      2. distribute-frac-neg2 81.2%

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

      3. remove-double-neg 81.2%

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

   10. Simplified 81.2%

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

   if -9.59999999999999972e-183 < x < 2.6e-124

   1. Initial program 70.5%

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

   3. Taylor expanded in B around 0 64.0%

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

   4. Taylor expanded in F around -inf 23.7%

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

      1. distribute-lft-in 23.7%

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

      2. associate-*r/ 23.7%

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

      3. metadata-eval 23.7%

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

      4. mul-1-neg 23.7%

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

      5. unsub-neg 23.7%

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

   6. Simplified 23.7%

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

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.6 \cdot 10^{-183} \lor \neg \left(x \leq 2.6 \cdot 10^{-124}\right):\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 70.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{-133}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 240:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.7e-133)
   (- (/ -1.0 B) (* x (/ 1.0 (tan B))))
   (if (<= F 240.0) (* x (/ -1.0 (tan B))) (- (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.7e-133) {
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	} else if (F <= 240.0) {
		tmp = x * (-1.0 / tan(B));
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.7d-133)) then
        tmp = ((-1.0d0) / b) - (x * (1.0d0 / tan(b)))
    else if (f <= 240.0d0) then
        tmp = x * ((-1.0d0) / tan(b))
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.7e-133) {
		tmp = (-1.0 / B) - (x * (1.0 / Math.tan(B)));
	} else if (F <= 240.0) {
		tmp = x * (-1.0 / Math.tan(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.7e-133:
		tmp = (-1.0 / B) - (x * (1.0 / math.tan(B)))
	elif F <= 240.0:
		tmp = x * (-1.0 / math.tan(B))
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.7e-133)
		tmp = Float64(Float64(-1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	elseif (F <= 240.0)
		tmp = Float64(x * Float64(-1.0 / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.7e-133)
		tmp = (-1.0 / B) - (x * (1.0 / tan(B)));
	elseif (F <= 240.0)
		tmp = x * (-1.0 / tan(B));
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.7e-133], N[(N[(-1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 240.0], N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.70000000000000003e-133

   1. Initial program 62.6%

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

   3. Taylor expanded in F around -inf 88.7%

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

   4. Taylor expanded in B around 0 73.0%

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

   if -1.70000000000000003e-133 < F < 240

   1. Initial program 99.4%

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

   3. Taylor expanded in F around -inf 38.6%

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

   4. Taylor expanded in x around inf 69.0%

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

      1. mul-1-neg 69.0%

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

      2. associate-/l* 68.9%

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

      3. distribute-rgt-neg-in 68.9%

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

   6. Simplified 68.9%

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

      1. clear-num 69.0%

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

      2. tan-quot 69.0%

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

      3. frac-2neg 69.0%

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

      4. metadata-eval 69.0%

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

      5. div-inv 69.0%

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

   8. Applied egg-rr 69.0%

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

      1. mul-1-neg 69.0%

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

      2. distribute-frac-neg2 69.0%

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

      3. remove-double-neg 69.0%

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

   10. Simplified 69.0%

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

   if 240 < F

   1. Initial program 57.5%

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

   3. Taylor expanded in B around 0 34.8%

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

   4. Taylor expanded in F around inf 76.9%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{-133}:\\ \;\;\;\;\frac{-1}{B} - x \cdot \frac{1}{\tan B}\\ \mathbf{elif}\;F \leq 240:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 67.9% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.4 \cdot 10^{+132}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 88:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.4e+132)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 88.0) (* x (/ -1.0 (tan B))) (- (/ 1.0 (sin B)) (/ x B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.4e+132) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 88.0) {
		tmp = x * (-1.0 / tan(B));
	} else {
		tmp = (1.0 / sin(B)) - (x / B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-5.4d+132)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 88.0d0) then
        tmp = x * ((-1.0d0) / tan(b))
    else
        tmp = (1.0d0 / sin(b)) - (x / b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.4e+132) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 88.0) {
		tmp = x * (-1.0 / Math.tan(B));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5.4e+132:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 88.0:
		tmp = x * (-1.0 / math.tan(B))
	else:
		tmp = (1.0 / math.sin(B)) - (x / B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -5.4e+132)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 88.0)
		tmp = Float64(x * Float64(-1.0 / tan(B)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -5.4e+132)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 88.0)
		tmp = x * (-1.0 / tan(B));
	else
		tmp = (1.0 / sin(B)) - (x / B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.4e+132], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 88.0], N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.3999999999999999e132

   1. Initial program 40.5%

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

   3. Taylor expanded in B around 0 19.3%

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

   4. Taylor expanded in F around -inf 78.6%

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

      1. distribute-lft-in 78.6%

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

      2. associate-*r/ 78.6%

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

      3. metadata-eval 78.6%

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

      4. mul-1-neg 78.6%

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

      5. unsub-neg 78.6%

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

   6. Simplified 78.6%

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

   if -5.3999999999999999e132 < F < 88

   1. Initial program 98.8%

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

   3. Taylor expanded in F around -inf 44.9%

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

   4. Taylor expanded in x around inf 66.7%

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

      1. mul-1-neg 66.7%

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

      2. associate-/l* 66.6%

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

      3. distribute-rgt-neg-in 66.6%

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

   6. Simplified 66.6%

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

      1. clear-num 66.6%

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

      2. tan-quot 66.6%

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

      3. frac-2neg 66.6%

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

      4. metadata-eval 66.6%

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

      5. div-inv 66.6%

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

   8. Applied egg-rr 66.6%

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

      1. mul-1-neg 66.6%

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

      2. distribute-frac-neg2 66.6%

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

      3. remove-double-neg 66.6%

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

   10. Simplified 66.6%

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

   if 88 < F

   1. Initial program 57.5%

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

   3. Taylor expanded in B around 0 34.8%

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

   4. Taylor expanded in F around inf 76.9%

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

4. Final simplification 71.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.4 \cdot 10^{+132}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 88:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 66.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -5.4 \cdot 10^{+132}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{-117}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -5.4e+132)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 1.6e-117) (* x (/ -1.0 (tan B))) (- (/ 1.0 B) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -5.4e+132) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 1.6e-117) {
		tmp = x * (-1.0 / 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 <= (-5.4d+132)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 1.6d-117) then
        tmp = x * ((-1.0d0) / 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 <= -5.4e+132) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 1.6e-117) {
		tmp = x * (-1.0 / Math.tan(B));
	} else {
		tmp = (1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -5.4e+132:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 1.6e-117:
		tmp = x * (-1.0 / 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 <= -5.4e+132)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 1.6e-117)
		tmp = Float64(x * Float64(-1.0 / 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 <= -5.4e+132)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 1.6e-117)
		tmp = x * (-1.0 / tan(B));
	else
		tmp = (1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -5.4e+132], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.6e-117], N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $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 < -5.3999999999999999e132

   1. Initial program 40.5%

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

   3. Taylor expanded in B around 0 19.3%

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

   4. Taylor expanded in F around -inf 78.6%

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

      1. distribute-lft-in 78.6%

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

      2. associate-*r/ 78.6%

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

      3. metadata-eval 78.6%

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

      4. mul-1-neg 78.6%

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

      5. unsub-neg 78.6%

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

   6. Simplified 78.6%

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

   if -5.3999999999999999e132 < F < 1.59999999999999998e-117

   1. Initial program 98.7%

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

   3. Taylor expanded in F around -inf 46.9%

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

   4. Taylor expanded in x around inf 74.2%

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

      1. mul-1-neg 74.2%

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

      2. associate-/l* 74.1%

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

      3. distribute-rgt-neg-in 74.1%

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

   6. Simplified 74.1%

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

      1. clear-num 74.1%

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

      2. tan-quot 74.1%

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

      3. frac-2neg 74.1%

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

      4. metadata-eval 74.1%

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

      5. div-inv 74.1%

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

   8. Applied egg-rr 74.1%

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

      1. mul-1-neg 74.1%

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

      2. distribute-frac-neg2 74.1%

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

      3. remove-double-neg 74.1%

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

   10. Simplified 74.1%

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

   if 1.59999999999999998e-117 < F

   1. Initial program 70.5%

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

   3. Simplified 81.9%

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

   5. Taylor expanded in F around inf 81.4%

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

   6. Taylor expanded in B around 0 64.7%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.4 \cdot 10^{+132}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 1.6 \cdot 10^{-117}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 36.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{-1 - x}{B}\\ \mathbf{if}\;F \leq -1.7 \cdot 10^{-133}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;F \leq 9 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\left|t\_0\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ (- -1.0 x) B)))
   (if (<= F -1.7e-133) t_0 (if (<= F 9e+29) (/ x (- B)) (fabs t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -1.7e-133) {
		tmp = t_0;
	} else if (F <= 9e+29) {
		tmp = x / -B;
	} else {
		tmp = fabs(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 = ((-1.0d0) - x) / b
    if (f <= (-1.7d-133)) then
        tmp = t_0
    else if (f <= 9d+29) then
        tmp = x / -b
    else
        tmp = abs(t_0)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = (-1.0 - x) / B;
	double tmp;
	if (F <= -1.7e-133) {
		tmp = t_0;
	} else if (F <= 9e+29) {
		tmp = x / -B;
	} else {
		tmp = Math.abs(t_0);
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = (-1.0 - x) / B
	tmp = 0
	if F <= -1.7e-133:
		tmp = t_0
	elif F <= 9e+29:
		tmp = x / -B
	else:
		tmp = math.fabs(t_0)
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(Float64(-1.0 - x) / B)
	tmp = 0.0
	if (F <= -1.7e-133)
		tmp = t_0;
	elseif (F <= 9e+29)
		tmp = Float64(x / Float64(-B));
	else
		tmp = abs(t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = (-1.0 - x) / B;
	tmp = 0.0;
	if (F <= -1.7e-133)
		tmp = t_0;
	elseif (F <= 9e+29)
		tmp = x / -B;
	else
		tmp = abs(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision]}, If[LessEqual[F, -1.7e-133], t$95$0, If[LessEqual[F, 9e+29], N[(x / (-B)), $MachinePrecision], N[Abs[t$95$0], $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.70000000000000003e-133

   1. Initial program 62.6%

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

   3. Taylor expanded in F around -inf 88.7%

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

   4. Taylor expanded in B around 0 44.5%

      \[\leadsto \color{blue}{\frac{{B}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.16666666666666666\right) - \left(1 + x\right)}{B}} \]

   5. Taylor expanded in B around 0 44.7%

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

      1. associate-*r/ 44.7%

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

      2. distribute-lft-in 44.7%

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

      3. metadata-eval 44.7%

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

      4. neg-mul-1 44.7%

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

      5. unsub-neg 44.7%

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

   7. Simplified 44.7%

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

   if -1.70000000000000003e-133 < F < 9.0000000000000005e29

   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. Taylor expanded in B around 0 75.5%

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

   4. Taylor expanded in x around inf 47.1%

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

      1. associate-*r/ 47.1%

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

      2. neg-mul-1 47.1%

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

   6. Simplified 47.1%

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

   if 9.0000000000000005e29 < F

   1. Initial program 56.9%

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

   3. Taylor expanded in F around -inf 45.8%

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

   4. Taylor expanded in B around 0 24.4%

      \[\leadsto \color{blue}{\frac{{B}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.16666666666666666\right) - \left(1 + x\right)}{B}} \]

   5. Taylor expanded in B around 0 23.8%

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

      1. associate-*r/ 23.8%

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

      2. distribute-lft-in 23.8%

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

      3. metadata-eval 23.8%

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

      4. neg-mul-1 23.8%

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

      5. unsub-neg 23.8%

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

   7. Simplified 23.8%

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

      1. add-sqr-sqrt 11.1%

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

      2. sqrt-unprod 15.5%

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

      3. pow2 15.5%

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

   9. Applied egg-rr 15.5%

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

       1. unpow2 15.5%

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

       2. rem-sqrt-square 28.9%

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

   11. Simplified 28.9%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{-133}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{elif}\;F \leq 9 \cdot 10^{+29}:\\ \;\;\;\;\frac{x}{-B}\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{-1 - x}{B}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 55.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ x \cdot \frac{-1}{\tan B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (* x (/ -1.0 (tan B))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 78.0%

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

3. Taylor expanded in F around -inf 54.1%

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

4. Taylor expanded in x around inf 60.2%

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

   1. mul-1-neg 60.2%

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

   2. associate-/l* 60.1%

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

   3. distribute-rgt-neg-in 60.1%

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

6. Simplified 60.1%

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

   1. clear-num 60.1%

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

   2. tan-quot 60.1%

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

   3. frac-2neg 60.1%

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

   4. metadata-eval 60.1%

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

   5. div-inv 60.1%

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

8. Applied egg-rr 60.1%

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

   1. mul-1-neg 60.1%

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

   2. distribute-frac-neg2 60.1%

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

   3. remove-double-neg 60.1%

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

10. Simplified 60.1%

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

11. Final simplification 60.1%

    \[\leadsto x \cdot \frac{-1}{\tan B} \]
12. Add Preprocessing

Alternative 20: 36.7% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{-133}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1.7e-133) (/ (- -1.0 x) B) (/ x (- B))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.7e-133) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = x / -B;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-1.7d-133)) then
        tmp = ((-1.0d0) - x) / b
    else
        tmp = x / -b
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1.7e-133) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = x / -B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1.7e-133:
		tmp = (-1.0 - x) / B
	else:
		tmp = x / -B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1.7e-133)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(x / Float64(-B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1.7e-133)
		tmp = (-1.0 - x) / B;
	else
		tmp = x / -B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1.7e-133], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[(x / (-B)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.70000000000000003e-133

   1. Initial program 62.6%

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

   3. Taylor expanded in F around -inf 88.7%

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

   4. Taylor expanded in B around 0 44.5%

      \[\leadsto \color{blue}{\frac{{B}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.16666666666666666\right) - \left(1 + x\right)}{B}} \]

   5. Taylor expanded in B around 0 44.7%

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

      1. associate-*r/ 44.7%

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

      2. distribute-lft-in 44.7%

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

      3. metadata-eval 44.7%

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

      4. neg-mul-1 44.7%

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

      5. unsub-neg 44.7%

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

   7. Simplified 44.7%

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

   if -1.70000000000000003e-133 < F

   1. Initial program 83.6%

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

   3. Taylor expanded in B around 0 60.0%

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

   4. Taylor expanded in x around inf 38.8%

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

      1. associate-*r/ 38.8%

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

      2. neg-mul-1 38.8%

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

   6. Simplified 38.8%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.7 \cdot 10^{-133}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 29.7% accurate, 81.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{-B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ x (- B)))

C

double code(double F, double B, double x) {
	return x / -B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = x / -b
end function

Java

public static double code(double F, double B, double x) {
	return x / -B;
}

Python

def code(F, B, x):
	return x / -B

Julia

function code(F, B, x)
	return Float64(x / Float64(-B))
end

MATLAB

function tmp = code(F, B, x)
	tmp = x / -B;
end

Wolfram

code[F_, B_, x_] := N[(x / (-B)), $MachinePrecision]

Derivation

1. Initial program 78.0%

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

3. Taylor expanded in B around 0 53.1%

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

4. Taylor expanded in x around inf 36.2%

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

   1. associate-*r/ 36.2%

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

   2. neg-mul-1 36.2%

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

6. Simplified 36.2%

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

7. Final simplification 36.2%

   \[\leadsto \frac{x}{-B} \]
8. Add Preprocessing

Alternative 22: 10.7% accurate, 108.0× speedup

Math

\[\begin{array}{l} \\ \frac{-1}{B} \end{array} \]

FPCore

(FPCore (F B x) :precision binary64 (/ -1.0 B))

C

double code(double F, double B, double x) {
	return -1.0 / B;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    code = (-1.0d0) / b
end function

Java

public static double code(double F, double B, double x) {
	return -1.0 / B;
}

Python

def code(F, B, x):
	return -1.0 / B

Julia

function code(F, B, x)
	return Float64(-1.0 / B)
end

MATLAB

function tmp = code(F, B, x)
	tmp = -1.0 / B;
end

Wolfram

code[F_, B_, x_] := N[(-1.0 / B), $MachinePrecision]

Derivation

1. Initial program 78.0%

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

3. Taylor expanded in F around -inf 54.1%

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

4. Taylor expanded in B around 0 31.1%

   \[\leadsto \color{blue}{\frac{{B}^{2} \cdot \left(0.3333333333333333 \cdot x - 0.16666666666666666\right) - \left(1 + x\right)}{B}} \]

5. Taylor expanded in B around 0 31.2%

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

   1. associate-*r/ 31.2%

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

   2. distribute-lft-in 31.2%

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

   3. metadata-eval 31.2%

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

   4. neg-mul-1 31.2%

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

   5. unsub-neg 31.2%

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

7. Simplified 31.2%

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

8. Taylor expanded in x around 0 7.5%

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

Reproduce

herbie shell --seed 2024086 
(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))))))