VandenBroeck and Keller, Equation (23)

Percentage Accurate: 76.0% → 99.4%

Time: 19.3s

Alternatives: 32

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: 76.0% 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: 99.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -50000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 0.0062:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\frac{\sin B}{F}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -50000000000.0)
     (- (/ -1.0 (sin B)) (/ (* x (cos B)) (sin B)))
     (if (<= F 0.0062)
       (- (/ (pow (fma 2.0 x (fma F F 2.0)) -0.5) (/ (sin B) F)) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -50000000000.0) {
		tmp = (-1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	} else if (F <= 0.0062) {
		tmp = (pow(fma(2.0, x, fma(F, F, 2.0)), -0.5) / (sin(B) / F)) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -50000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(Float64(x * cos(B)) / sin(B)));
	elseif (F <= 0.0062)
		tmp = Float64(Float64((fma(2.0, x, fma(F, F, 2.0)) ^ -0.5) / Float64(sin(B) / F)) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -50000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0062], N[(N[(N[Power[N[(2.0 * x + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -5e10

   1. Initial program 59.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in F around -inf 99.8%

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

   if -5e10 < F < 0.00619999999999999978

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

      1. associate-*r/ 99.7%

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

      2. associate-*l/ 99.7%

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

      3. *-commutative 99.7%

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

      4. clear-num 99.7%

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

      5. un-div-inv 99.7%

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

      6. fma-define 99.7%

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

      7. fma-undefine 99.7%

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

      8. *-commutative 99.7%

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

      9. fma-define 99.7%

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

      10. fma-define 99.7%

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

   6. Applied egg-rr 99.7%

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

   if 0.00619999999999999978 < F

   1. Initial program 59.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -50000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 0.0062:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\frac{\sin B}{F}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -42000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 10^{+37}:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -42000000000.0)
     (- (/ -1.0 (sin B)) (/ (* x (cos B)) (sin B)))
     (if (<= F 1e+37)
       (- (* F (/ (pow (fma x 2.0 (fma F F 2.0)) -0.5) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -42000000000.0) {
		tmp = (-1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	} else if (F <= 1e+37) {
		tmp = (F * (pow(fma(x, 2.0, fma(F, F, 2.0)), -0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -42000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(Float64(x * cos(B)) / sin(B)));
	elseif (F <= 1e+37)
		tmp = Float64(Float64(F * Float64((fma(x, 2.0, fma(F, F, 2.0)) ^ -0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -42000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1e+37], N[(N[(F * N[(N[Power[N[(x * 2.0 + N[(F * F + 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.2e10

   1. Initial program 59.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in F around -inf 99.8%

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

   if -4.2e10 < F < 9.99999999999999954e36

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   if 9.99999999999999954e36 < 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 67.3%

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

   5. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -42000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 10^{+37}:\\ \;\;\;\;F \cdot \frac{{\left(\mathsf{fma}\left(x, 2, \mathsf{fma}\left(F, F, 2\right)\right)\right)}^{-0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -36000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{\sin B}{F}} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -36000000000.0)
   (- (/ -1.0 (sin B)) (/ (* x (cos B)) (sin B)))
   (if (<= F 7.8e+36)
     (+
      (* x (/ -1.0 (tan B)))
      (* (/ 1.0 (/ (sin B) F)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5)))
     (- (/ 1.0 (sin B)) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	} else if (F <= 7.8e+36) {
		tmp = (x * (-1.0 / tan(B))) + ((1.0 / (sin(B) / F)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / sin(B)) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-36000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - ((x * cos(b)) / sin(b))
    else if (f <= 7.8d+36) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((1.0d0 / (sin(b) / f)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)))
    else
        tmp = (1.0d0 / sin(b)) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - ((x * Math.cos(B)) / Math.sin(B));
	} else if (F <= 7.8e+36) {
		tmp = (x * (-1.0 / Math.tan(B))) + ((1.0 / (Math.sin(B) / F)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -36000000000.0:
		tmp = (-1.0 / math.sin(B)) - ((x * math.cos(B)) / math.sin(B))
	elif F <= 7.8e+36:
		tmp = (x * (-1.0 / math.tan(B))) + ((1.0 / (math.sin(B) / F)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5))
	else:
		tmp = (1.0 / math.sin(B)) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -36000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(Float64(x * cos(B)) / sin(B)));
	elseif (F <= 7.8e+36)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64(Float64(1.0 / Float64(sin(B) / F)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -36000000000.0)
		tmp = (-1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	elseif (F <= 7.8e+36)
		tmp = (x * (-1.0 / tan(B))) + ((1.0 / (sin(B) / F)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5));
	else
		tmp = (1.0 / sin(B)) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -36000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7.8e+36], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / N[(N[Sin[B], $MachinePrecision] / F), $MachinePrecision]), $MachinePrecision] * N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.6e10

   1. Initial program 59.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in F around -inf 99.8%

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

   if -3.6e10 < F < 7.80000000000000042e36

   1. Initial program 99.5%

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

      1. clear-num 99.5%

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

      2. inv-pow 99.5%

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

   4. Applied egg-rr 99.5%

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

      1. unpow-1 99.5%

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

   6. Simplified 99.5%

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

   if 7.80000000000000042e36 < 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 67.3%

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

   5. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -36000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 7.8 \cdot 10^{+36}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + \frac{1}{\frac{\sin B}{F}} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -50000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{+33}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -50000000000.0)
   (- (/ -1.0 (sin B)) (/ (* x (cos B)) (sin B)))
   (if (<= F 2.2e+33)
     (+
      (* x (/ -1.0 (tan B)))
      (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F (sin B))))
     (- (/ 1.0 (sin B)) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -50000000000.0) {
		tmp = (-1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	} else if (F <= 2.2e+33) {
		tmp = (x * (-1.0 / tan(B))) + (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / sin(B)));
	} else {
		tmp = (1.0 / sin(B)) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-50000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - ((x * cos(b)) / sin(b))
    else if (f <= 2.2d+33) then
        tmp = (x * ((-1.0d0) / tan(b))) + ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / sin(b)))
    else
        tmp = (1.0d0 / sin(b)) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -50000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - ((x * Math.cos(B)) / Math.sin(B));
	} else if (F <= 2.2e+33) {
		tmp = (x * (-1.0 / Math.tan(B))) + (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / Math.sin(B)));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -50000000000.0:
		tmp = (-1.0 / math.sin(B)) - ((x * math.cos(B)) / math.sin(B))
	elif F <= 2.2e+33:
		tmp = (x * (-1.0 / math.tan(B))) + (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / math.sin(B)))
	else:
		tmp = (1.0 / math.sin(B)) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -50000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(Float64(x * cos(B)) / sin(B)));
	elseif (F <= 2.2e+33)
		tmp = Float64(Float64(x * Float64(-1.0 / tan(B))) + Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / sin(B))));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -50000000000.0)
		tmp = (-1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	elseif (F <= 2.2e+33)
		tmp = (x * (-1.0 / tan(B))) + ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / sin(B)));
	else
		tmp = (1.0 / sin(B)) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -50000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.2e+33], N[(N[(x * N[(-1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[Power[N[(N[(2.0 + N[(F * F), $MachinePrecision]), $MachinePrecision] + N[(x * 2.0), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision] * N[(F / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5e10

   1. Initial program 59.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in F around -inf 99.8%

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

   if -5e10 < F < 2.19999999999999994e33

   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

   if 2.19999999999999994e33 < F

   1. Initial program 57.2%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -50000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{+33}:\\ \;\;\;\;x \cdot \frac{-1}{\tan B} + {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -60000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{+33}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -60000000000.0)
   (- (/ -1.0 (sin B)) (/ (* x (cos B)) (sin B)))
   (if (<= F 2.2e+33)
     (-
      (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F (sin B)))
      (/ 1.0 (/ (tan B) x)))
     (- (/ 1.0 (sin B)) (/ x (tan B))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -60000000000.0) {
		tmp = (-1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	} else if (F <= 2.2e+33) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / sin(B))) - (1.0 / (tan(B) / x));
	} else {
		tmp = (1.0 / sin(B)) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-60000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - ((x * cos(b)) / sin(b))
    else if (f <= 2.2d+33) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / sin(b))) - (1.0d0 / (tan(b) / x))
    else
        tmp = (1.0d0 / sin(b)) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -60000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - ((x * Math.cos(B)) / Math.sin(B));
	} else if (F <= 2.2e+33) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / Math.sin(B))) - (1.0 / (Math.tan(B) / x));
	} else {
		tmp = (1.0 / Math.sin(B)) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -60000000000.0:
		tmp = (-1.0 / math.sin(B)) - ((x * math.cos(B)) / math.sin(B))
	elif F <= 2.2e+33:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / math.sin(B))) - (1.0 / (math.tan(B) / x))
	else:
		tmp = (1.0 / math.sin(B)) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -60000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(Float64(x * cos(B)) / sin(B)));
	elseif (F <= 2.2e+33)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / sin(B))) - Float64(1.0 / Float64(tan(B) / x)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -60000000000.0)
		tmp = (-1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	elseif (F <= 2.2e+33)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / sin(B))) - (1.0 / (tan(B) / x));
	else
		tmp = (1.0 / sin(B)) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -60000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.2e+33], 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[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -6e10

   1. Initial program 59.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in F around -inf 99.8%

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

   if -6e10 < F < 2.19999999999999994e33

   1. Initial program 99.5%

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

      1. div-inv 99.7%

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

      2. neg-mul-1 99.7%

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

      3. clear-num 99.5%

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

      4. un-div-inv 99.5%

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

   4. Applied egg-rr 99.5%

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

   if 2.19999999999999994e33 < F

   1. Initial program 57.2%

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

   3. Simplified 68.8%

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

   5. Taylor expanded in F around inf 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -60000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{+33}:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{1}{\frac{\tan B}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -36000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 0.0062:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -36000000000.0)
     (- (/ -1.0 (sin B)) (/ (* x (cos B)) (sin B)))
     (if (<= F 0.0062)
       (- (/ F (/ (sin B) (sqrt 0.5))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	} else if (F <= 0.0062) {
		tmp = (F / (sin(B) / sqrt(0.5))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-36000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - ((x * cos(b)) / sin(b))
    else if (f <= 0.0062d0) then
        tmp = (f / (sin(b) / sqrt(0.5d0))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - ((x * Math.cos(B)) / Math.sin(B));
	} else if (F <= 0.0062) {
		tmp = (F / (Math.sin(B) / Math.sqrt(0.5))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -36000000000.0:
		tmp = (-1.0 / math.sin(B)) - ((x * math.cos(B)) / math.sin(B))
	elif F <= 0.0062:
		tmp = (F / (math.sin(B) / math.sqrt(0.5))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -36000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(Float64(x * cos(B)) / sin(B)));
	elseif (F <= 0.0062)
		tmp = Float64(Float64(F / Float64(sin(B) / sqrt(0.5))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -36000000000.0)
		tmp = (-1.0 / sin(B)) - ((x * cos(B)) / sin(B));
	elseif (F <= 0.0062)
		tmp = (F / (sin(B) / sqrt(0.5))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -36000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0062], N[(N[(F / N[(N[Sin[B], $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.6e10

   1. Initial program 59.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in F around -inf 99.8%

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

   if -3.6e10 < F < 0.00619999999999999978

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.2%

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

      1. *-commutative 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in x around 0 99.2%

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

      1. clear-num 99.2%

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

      2. inv-pow 99.2%

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

   10. Applied egg-rr 99.2%

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

       1. unpow-1 99.2%

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

   12. Simplified 99.2%

       \[\leadsto F \cdot \color{blue}{\frac{1}{\frac{\sin B}{\sqrt{0.5}}}} - \frac{x}{\tan B} \]
   13. Step-by-step derivation

       1. clear-num 99.2%

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

       2. sub-neg 99.2%

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

       3. distribute-neg-frac2 99.2%

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

   14. Applied egg-rr 99.2%

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

       1. distribute-frac-neg2 99.2%

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

       2. sub-neg 99.2%

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

       3. associate-*r/ 99.2%

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

       4. *-rgt-identity 99.2%

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

       5. times-frac 99.2%

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

       6. /-rgt-identity 99.2%

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

       7. associate-/r/ 99.3%

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

   16. Simplified 99.3%

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

   if 0.00619999999999999978 < F

   1. Initial program 59.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -36000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x \cdot \cos B}{\sin B}\\ \mathbf{elif}\;F \leq 0.0062:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -36000000000:\\ \;\;\;\;\frac{-1}{\sin B} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;F \leq 0.0062:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -36000000000.0)
     (+ (/ -1.0 (sin B)) (/ -1.0 (/ (tan B) x)))
     (if (<= F 0.0062)
       (- (/ F (/ (sin B) (sqrt 0.5))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / sin(B)) + (-1.0 / (tan(B) / x));
	} else if (F <= 0.0062) {
		tmp = (F / (sin(B) / sqrt(0.5))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-36000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) + ((-1.0d0) / (tan(b) / x))
    else if (f <= 0.0062d0) then
        tmp = (f / (sin(b) / sqrt(0.5d0))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / Math.sin(B)) + (-1.0 / (Math.tan(B) / x));
	} else if (F <= 0.0062) {
		tmp = (F / (Math.sin(B) / Math.sqrt(0.5))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -36000000000.0:
		tmp = (-1.0 / math.sin(B)) + (-1.0 / (math.tan(B) / x))
	elif F <= 0.0062:
		tmp = (F / (math.sin(B) / math.sqrt(0.5))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -36000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) + Float64(-1.0 / Float64(tan(B) / x)));
	elseif (F <= 0.0062)
		tmp = Float64(Float64(F / Float64(sin(B) / sqrt(0.5))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -36000000000.0)
		tmp = (-1.0 / sin(B)) + (-1.0 / (tan(B) / x));
	elseif (F <= 0.0062)
		tmp = (F / (sin(B) / sqrt(0.5))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -36000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0062], N[(N[(F / N[(N[Sin[B], $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.6e10

   1. Initial program 59.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in F around -inf 99.8%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

   9. Simplified 99.8%

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

   if -3.6e10 < F < 0.00619999999999999978

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.2%

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

      1. *-commutative 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in x around 0 99.2%

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

      1. clear-num 99.2%

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

      2. inv-pow 99.2%

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

   10. Applied egg-rr 99.2%

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

       1. unpow-1 99.2%

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

   12. Simplified 99.2%

       \[\leadsto F \cdot \color{blue}{\frac{1}{\frac{\sin B}{\sqrt{0.5}}}} - \frac{x}{\tan B} \]
   13. Step-by-step derivation

       1. clear-num 99.2%

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

       2. sub-neg 99.2%

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

       3. distribute-neg-frac2 99.2%

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

   14. Applied egg-rr 99.2%

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

       1. distribute-frac-neg2 99.2%

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

       2. sub-neg 99.2%

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

       3. associate-*r/ 99.2%

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

       4. *-rgt-identity 99.2%

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

       5. times-frac 99.2%

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

       6. /-rgt-identity 99.2%

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

       7. associate-/r/ 99.3%

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

   16. Simplified 99.3%

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

   if 0.00619999999999999978 < F

   1. Initial program 59.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -36000000000:\\ \;\;\;\;\frac{-1}{\sin B} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;F \leq 0.0062:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -36000000000:\\ \;\;\;\;\frac{-1}{\sin B} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;F \leq 0.0062:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -36000000000.0)
     (+ (/ -1.0 (sin B)) (/ -1.0 (/ (tan B) x)))
     (if (<= F 0.0062)
       (- (* F (/ (sqrt 0.5) (sin B))) t_0)
       (- (/ 1.0 (sin B)) t_0)))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / sin(B)) + (-1.0 / (tan(B) / x));
	} else if (F <= 0.0062) {
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-36000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) + ((-1.0d0) / (tan(b) / x))
    else if (f <= 0.0062d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - t_0
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / Math.sin(B)) + (-1.0 / (Math.tan(B) / x));
	} else if (F <= 0.0062) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - t_0;
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -36000000000.0:
		tmp = (-1.0 / math.sin(B)) + (-1.0 / (math.tan(B) / x))
	elif F <= 0.0062:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - t_0
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -36000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) + Float64(-1.0 / Float64(tan(B) / x)));
	elseif (F <= 0.0062)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - t_0);
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -36000000000.0)
		tmp = (-1.0 / sin(B)) + (-1.0 / (tan(B) / x));
	elseif (F <= 0.0062)
		tmp = (F * (sqrt(0.5) / sin(B))) - t_0;
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -36000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 0.0062], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.6e10

   1. Initial program 59.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in F around -inf 99.8%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

   9. Simplified 99.8%

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

   if -3.6e10 < F < 0.00619999999999999978

   1. Initial program 99.5%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in F around 0 99.2%

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

      1. *-commutative 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in x around 0 99.2%

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

   if 0.00619999999999999978 < F

   1. Initial program 59.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -36000000000:\\ \;\;\;\;\frac{-1}{\sin B} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;F \leq 0.0062:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 92.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -36000000000:\\ \;\;\;\;\frac{-1}{\sin B} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-93}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - t\_0\\ \mathbf{elif}\;F \leq 0.0062:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F}} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -36000000000.0)
     (+ (/ -1.0 (sin B)) (/ -1.0 (/ (tan B) x)))
     (if (<= F 2e-93)
       (- (* (/ F B) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) t_0)
       (if (<= F 0.0062)
         (-
          (* (/ 1.0 (/ (sin B) F)) (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5))
          (/ x B))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / sin(B)) + (-1.0 / (tan(B) / x));
	} else if (F <= 2e-93) {
		tmp = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0;
	} else if (F <= 0.0062) {
		tmp = ((1.0 / (sin(B) / F)) * pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-36000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) + ((-1.0d0) / (tan(b) / x))
    else if (f <= 2d-93) then
        tmp = ((f / b) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - t_0
    else if (f <= 0.0062d0) then
        tmp = ((1.0d0 / (sin(b) / f)) * (((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0))) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / Math.sin(B)) + (-1.0 / (Math.tan(B) / x));
	} else if (F <= 2e-93) {
		tmp = ((F / B) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0;
	} else if (F <= 0.0062) {
		tmp = ((1.0 / (Math.sin(B) / F)) * Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -36000000000.0:
		tmp = (-1.0 / math.sin(B)) + (-1.0 / (math.tan(B) / x))
	elif F <= 2e-93:
		tmp = ((F / B) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0
	elif F <= 0.0062:
		tmp = ((1.0 / (math.sin(B) / F)) * math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5)) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -36000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) + Float64(-1.0 / Float64(tan(B) / x)));
	elseif (F <= 2e-93)
		tmp = Float64(Float64(Float64(F / B) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - t_0);
	elseif (F <= 0.0062)
		tmp = Float64(Float64(Float64(1.0 / Float64(sin(B) / F)) * (Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -36000000000.0)
		tmp = (-1.0 / sin(B)) + (-1.0 / (tan(B) / x));
	elseif (F <= 2e-93)
		tmp = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0;
	elseif (F <= 0.0062)
		tmp = ((1.0 / (sin(B) / F)) * (((2.0 + (F * F)) + (x * 2.0)) ^ -0.5)) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -36000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2e-93], N[(N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.0062], N[(N[(N[(1.0 / N[(N[Sin[B], $MachinePrecision] / F), $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[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.6e10

   1. Initial program 59.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in F around -inf 99.8%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

   9. Simplified 99.8%

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

   if -3.6e10 < F < 1.9999999999999998e-93

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around 0 99.7%

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

      1. *-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in B around 0 86.3%

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

   if 1.9999999999999998e-93 < F < 0.00619999999999999978

   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. Step-by-step derivation

      1. clear-num 99.4%

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

      2. inv-pow 99.4%

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

   4. Applied egg-rr 99.4%

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

      1. unpow-1 99.4%

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

   6. Simplified 99.4%

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

   7. Taylor expanded in B around 0 99.4%

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

      1. associate-*r/ 99.4%

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

      2. neg-mul-1 99.4%

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

   9. Simplified 99.4%

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

   if 0.00619999999999999978 < F

   1. Initial program 59.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -36000000000:\\ \;\;\;\;\frac{-1}{\sin B} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;F \leq 2 \cdot 10^{-93}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.0062:\\ \;\;\;\;\frac{1}{\frac{\sin B}{F}} \cdot {\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 92.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -36000000000:\\ \;\;\;\;\frac{-1}{\sin B} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - t\_0\\ \mathbf{elif}\;F \leq 0.0062:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -36000000000.0)
     (+ (/ -1.0 (sin B)) (/ -1.0 (/ (tan B) x)))
     (if (<= F 2.8e-93)
       (- (* (/ F B) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) t_0)
       (if (<= F 0.0062)
         (- (* (pow (+ (+ 2.0 (* F F)) (* x 2.0)) -0.5) (/ F (sin B))) (/ x B))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / sin(B)) + (-1.0 / (tan(B) / x));
	} else if (F <= 2.8e-93) {
		tmp = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0;
	} else if (F <= 0.0062) {
		tmp = (pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / sin(B))) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-36000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) + ((-1.0d0) / (tan(b) / x))
    else if (f <= 2.8d-93) then
        tmp = ((f / b) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - t_0
    else if (f <= 0.0062d0) then
        tmp = ((((2.0d0 + (f * f)) + (x * 2.0d0)) ** (-0.5d0)) * (f / sin(b))) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / Math.sin(B)) + (-1.0 / (Math.tan(B) / x));
	} else if (F <= 2.8e-93) {
		tmp = ((F / B) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0;
	} else if (F <= 0.0062) {
		tmp = (Math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / Math.sin(B))) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -36000000000.0:
		tmp = (-1.0 / math.sin(B)) + (-1.0 / (math.tan(B) / x))
	elif F <= 2.8e-93:
		tmp = ((F / B) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0
	elif F <= 0.0062:
		tmp = (math.pow(((2.0 + (F * F)) + (x * 2.0)), -0.5) * (F / math.sin(B))) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -36000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) + Float64(-1.0 / Float64(tan(B) / x)));
	elseif (F <= 2.8e-93)
		tmp = Float64(Float64(Float64(F / B) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - t_0);
	elseif (F <= 0.0062)
		tmp = Float64(Float64((Float64(Float64(2.0 + Float64(F * F)) + Float64(x * 2.0)) ^ -0.5) * Float64(F / sin(B))) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -36000000000.0)
		tmp = (-1.0 / sin(B)) + (-1.0 / (tan(B) / x));
	elseif (F <= 2.8e-93)
		tmp = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0;
	elseif (F <= 0.0062)
		tmp = ((((2.0 + (F * F)) + (x * 2.0)) ^ -0.5) * (F / sin(B))) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -36000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.8e-93], N[(N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.0062], 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[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.6e10

   1. Initial program 59.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in F around -inf 99.8%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

   9. Simplified 99.8%

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

   if -3.6e10 < F < 2.79999999999999998e-93

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around 0 99.7%

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

      1. *-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in B around 0 86.3%

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

   if 2.79999999999999998e-93 < F < 0.00619999999999999978

   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 99.3%

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

      1. associate-*r/ 99.4%

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

      2. neg-mul-1 99.4%

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

   5. Simplified 99.3%

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

   if 0.00619999999999999978 < F

   1. Initial program 59.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -36000000000:\\ \;\;\;\;\frac{-1}{\sin B} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.0062:\\ \;\;\;\;{\left(\left(2 + F \cdot F\right) + x \cdot 2\right)}^{-0.5} \cdot \frac{F}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 86.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq -2.7 \cdot 10^{-66}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-93}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 0.0046:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -1e-5)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F -2.7e-66)
       (* F (/ (sqrt 0.5) (sin B)))
       (if (<= F 2.3e-93)
         (/ (* x (cos B)) (- (sin B)))
         (if (<= F 0.0046)
           (/ F (/ (sin B) (sqrt 0.5)))
           (- (/ 1.0 (sin B)) t_0)))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -1e-5) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= -2.7e-66) {
		tmp = F * (sqrt(0.5) / sin(B));
	} else if (F <= 2.3e-93) {
		tmp = (x * cos(B)) / -sin(B);
	} else if (F <= 0.0046) {
		tmp = F / (sin(B) / sqrt(0.5));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-1d-5)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= (-2.7d-66)) then
        tmp = f * (sqrt(0.5d0) / sin(b))
    else if (f <= 2.3d-93) then
        tmp = (x * cos(b)) / -sin(b)
    else if (f <= 0.0046d0) then
        tmp = f / (sin(b) / sqrt(0.5d0))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -1e-5) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= -2.7e-66) {
		tmp = F * (Math.sqrt(0.5) / Math.sin(B));
	} else if (F <= 2.3e-93) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else if (F <= 0.0046) {
		tmp = F / (Math.sin(B) / Math.sqrt(0.5));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -1e-5:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= -2.7e-66:
		tmp = F * (math.sqrt(0.5) / math.sin(B))
	elif F <= 2.3e-93:
		tmp = (x * math.cos(B)) / -math.sin(B)
	elif F <= 0.0046:
		tmp = F / (math.sin(B) / math.sqrt(0.5))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -1e-5)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= -2.7e-66)
		tmp = Float64(F * Float64(sqrt(0.5) / sin(B)));
	elseif (F <= 2.3e-93)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	elseif (F <= 0.0046)
		tmp = Float64(F / Float64(sin(B) / sqrt(0.5)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -1e-5)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= -2.7e-66)
		tmp = F * (sqrt(0.5) / sin(B));
	elseif (F <= 2.3e-93)
		tmp = (x * cos(B)) / -sin(B);
	elseif (F <= 0.0046)
		tmp = F / (sin(B) / sqrt(0.5));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e-5], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, -2.7e-66], N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.3e-93], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 0.0046], N[(F / N[(N[Sin[B], $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -1.00000000000000008e-5

   1. Initial program 61.2%

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

   3. Simplified 75.5%

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

   5. Taylor expanded in F around -inf 99.7%

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

   if -1.00000000000000008e-5 < F < -2.69999999999999996e-66

   1. Initial program 99.3%

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

   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 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 90.6%

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

      1. associate-/l* 90.6%

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

   10. Simplified 90.6%

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

   if -2.69999999999999996e-66 < F < 2.2999999999999998e-93

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around -inf 33.0%

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

      1. clear-num 32.8%

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

      2. inv-pow 32.8%

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

   7. Applied egg-rr 32.8%

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

      1. unpow-1 32.8%

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

   9. Simplified 32.8%

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

   10. Taylor expanded in x around inf 83.4%

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

       1. mul-1-neg 83.4%

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

   12. Simplified 83.4%

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

   if 2.2999999999999998e-93 < F < 0.0045999999999999999

   1. Initial program 99.3%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in F around 0 96.4%

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

      1. *-commutative 96.4%

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

   7. Simplified 96.4%

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

   8. Taylor expanded in x around 0 96.4%

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

      1. clear-num 96.6%

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

      2. inv-pow 96.6%

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

   10. Applied egg-rr 96.6%

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

       1. unpow-1 96.6%

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

   12. Simplified 96.6%

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

   13. Taylor expanded in F around inf 77.4%

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

       1. *-rgt-identity 77.4%

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

       2. times-frac 77.5%

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

       3. /-rgt-identity 77.5%

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

       4. associate-/r/ 77.9%

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

   15. Simplified 77.9%

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

   if 0.0045999999999999999 < F

   1. Initial program 59.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.7 \cdot 10^{-66}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-93}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 0.0046:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 91.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -36000000000:\\ \;\;\;\;\frac{-1}{\sin B} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-84}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - t\_0\\ \mathbf{elif}\;F \leq 0.0062:\\ \;\;\;\;F \cdot \frac{1}{\frac{\sin B}{\sqrt{0.5}}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -36000000000.0)
     (+ (/ -1.0 (sin B)) (/ -1.0 (/ (tan B) x)))
     (if (<= F 7e-84)
       (- (* (/ F B) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) t_0)
       (if (<= F 0.0062)
         (- (* F (/ 1.0 (/ (sin B) (sqrt 0.5)))) (/ x B))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / sin(B)) + (-1.0 / (tan(B) / x));
	} else if (F <= 7e-84) {
		tmp = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0;
	} else if (F <= 0.0062) {
		tmp = (F * (1.0 / (sin(B) / sqrt(0.5)))) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-36000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) + ((-1.0d0) / (tan(b) / x))
    else if (f <= 7d-84) then
        tmp = ((f / b) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - t_0
    else if (f <= 0.0062d0) then
        tmp = (f * (1.0d0 / (sin(b) / sqrt(0.5d0)))) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / Math.sin(B)) + (-1.0 / (Math.tan(B) / x));
	} else if (F <= 7e-84) {
		tmp = ((F / B) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0;
	} else if (F <= 0.0062) {
		tmp = (F * (1.0 / (Math.sin(B) / Math.sqrt(0.5)))) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -36000000000.0:
		tmp = (-1.0 / math.sin(B)) + (-1.0 / (math.tan(B) / x))
	elif F <= 7e-84:
		tmp = ((F / B) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0
	elif F <= 0.0062:
		tmp = (F * (1.0 / (math.sin(B) / math.sqrt(0.5)))) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -36000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) + Float64(-1.0 / Float64(tan(B) / x)));
	elseif (F <= 7e-84)
		tmp = Float64(Float64(Float64(F / B) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - t_0);
	elseif (F <= 0.0062)
		tmp = Float64(Float64(F * Float64(1.0 / Float64(sin(B) / sqrt(0.5)))) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -36000000000.0)
		tmp = (-1.0 / sin(B)) + (-1.0 / (tan(B) / x));
	elseif (F <= 7e-84)
		tmp = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - t_0;
	elseif (F <= 0.0062)
		tmp = (F * (1.0 / (sin(B) / sqrt(0.5)))) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -36000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 7e-84], N[(N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.0062], N[(N[(F * N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.6e10

   1. Initial program 59.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in F around -inf 99.8%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

   9. Simplified 99.8%

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

   if -3.6e10 < F < 7.0000000000000002e-84

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around 0 99.7%

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

      1. *-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in B around 0 86.4%

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

   if 7.0000000000000002e-84 < F < 0.00619999999999999978

   1. Initial program 99.2%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in F around 0 96.2%

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

      1. *-commutative 96.2%

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

   7. Simplified 96.2%

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

   8. Taylor expanded in x around 0 96.2%

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

   9. Taylor expanded in B around 0 96.2%

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

       1. clear-num 96.5%

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

       2. inv-pow 96.5%

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

   11. Applied egg-rr 96.5%

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

       1. unpow-1 96.5%

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

   13. Simplified 96.5%

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

   if 0.00619999999999999978 < F

   1. Initial program 59.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -36000000000:\\ \;\;\;\;\frac{-1}{\sin B} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;F \leq 7 \cdot 10^{-84}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.0062:\\ \;\;\;\;F \cdot \frac{1}{\frac{\sin B}{\sqrt{0.5}}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 91.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -36000000000:\\ \;\;\;\;\frac{-1}{\sin B} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;F \leq 9.6 \cdot 10^{-81}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B} - t\_0\\ \mathbf{elif}\;F \leq 0.0048:\\ \;\;\;\;F \cdot \frac{1}{\frac{\sin B}{\sqrt{0.5}}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -36000000000.0)
     (+ (/ -1.0 (sin B)) (/ -1.0 (/ (tan B) x)))
     (if (<= F 9.6e-81)
       (- (/ (* F (sqrt 0.5)) B) t_0)
       (if (<= F 0.0048)
         (- (* F (/ 1.0 (/ (sin B) (sqrt 0.5)))) (/ x B))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / sin(B)) + (-1.0 / (tan(B) / x));
	} else if (F <= 9.6e-81) {
		tmp = ((F * sqrt(0.5)) / B) - t_0;
	} else if (F <= 0.0048) {
		tmp = (F * (1.0 / (sin(B) / sqrt(0.5)))) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-36000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) + ((-1.0d0) / (tan(b) / x))
    else if (f <= 9.6d-81) then
        tmp = ((f * sqrt(0.5d0)) / b) - t_0
    else if (f <= 0.0048d0) then
        tmp = (f * (1.0d0 / (sin(b) / sqrt(0.5d0)))) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / Math.sin(B)) + (-1.0 / (Math.tan(B) / x));
	} else if (F <= 9.6e-81) {
		tmp = ((F * Math.sqrt(0.5)) / B) - t_0;
	} else if (F <= 0.0048) {
		tmp = (F * (1.0 / (Math.sin(B) / Math.sqrt(0.5)))) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -36000000000.0:
		tmp = (-1.0 / math.sin(B)) + (-1.0 / (math.tan(B) / x))
	elif F <= 9.6e-81:
		tmp = ((F * math.sqrt(0.5)) / B) - t_0
	elif F <= 0.0048:
		tmp = (F * (1.0 / (math.sin(B) / math.sqrt(0.5)))) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -36000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) + Float64(-1.0 / Float64(tan(B) / x)));
	elseif (F <= 9.6e-81)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) / B) - t_0);
	elseif (F <= 0.0048)
		tmp = Float64(Float64(F * Float64(1.0 / Float64(sin(B) / sqrt(0.5)))) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -36000000000.0)
		tmp = (-1.0 / sin(B)) + (-1.0 / (tan(B) / x));
	elseif (F <= 9.6e-81)
		tmp = ((F * sqrt(0.5)) / B) - t_0;
	elseif (F <= 0.0048)
		tmp = (F * (1.0 / (sin(B) / sqrt(0.5)))) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -36000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 9.6e-81], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.0048], N[(N[(F * N[(1.0 / N[(N[Sin[B], $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.6e10

   1. Initial program 59.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in F around -inf 99.8%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

   9. Simplified 99.8%

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

   if -3.6e10 < F < 9.5999999999999996e-81

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around 0 99.7%

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

      1. *-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 99.7%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. unpow-1 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in B around 0 86.5%

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

   if 9.5999999999999996e-81 < F < 0.00479999999999999958

   1. Initial program 99.3%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in F around 0 96.1%

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

      1. *-commutative 96.1%

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

   7. Simplified 96.1%

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

   8. Taylor expanded in x around 0 96.1%

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

   9. Taylor expanded in B around 0 96.1%

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

       1. clear-num 96.4%

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

       2. inv-pow 96.4%

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

   11. Applied egg-rr 96.4%

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

       1. unpow-1 96.4%

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

   13. Simplified 96.4%

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

   if 0.00479999999999999958 < F

   1. Initial program 59.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -36000000000:\\ \;\;\;\;\frac{-1}{\sin B} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;F \leq 9.6 \cdot 10^{-81}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.0048:\\ \;\;\;\;F \cdot \frac{1}{\frac{\sin B}{\sqrt{0.5}}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 91.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ t_1 := \frac{1}{\sin B}\\ \mathbf{if}\;F \leq -36000000000:\\ \;\;\;\;\frac{-1}{\sin B} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B} - t\_0\\ \mathbf{elif}\;F \leq 0.0062:\\ \;\;\;\;F \cdot \left(t\_1 \cdot \sqrt{0.5}\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;t\_1 - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))) (t_1 (/ 1.0 (sin B))))
   (if (<= F -36000000000.0)
     (+ (/ -1.0 (sin B)) (/ -1.0 (/ (tan B) x)))
     (if (<= F 2.8e-93)
       (- (/ (* F (sqrt 0.5)) B) t_0)
       (if (<= F 0.0062) (- (* F (* t_1 (sqrt 0.5))) (/ x B)) (- t_1 t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double t_1 = 1.0 / sin(B);
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / sin(B)) + (-1.0 / (tan(B) / x));
	} else if (F <= 2.8e-93) {
		tmp = ((F * sqrt(0.5)) / B) - t_0;
	} else if (F <= 0.0062) {
		tmp = (F * (t_1 * sqrt(0.5))) - (x / B);
	} else {
		tmp = t_1 - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x / tan(b)
    t_1 = 1.0d0 / sin(b)
    if (f <= (-36000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) + ((-1.0d0) / (tan(b) / x))
    else if (f <= 2.8d-93) then
        tmp = ((f * sqrt(0.5d0)) / b) - t_0
    else if (f <= 0.0062d0) then
        tmp = (f * (t_1 * sqrt(0.5d0))) - (x / b)
    else
        tmp = t_1 - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double t_1 = 1.0 / Math.sin(B);
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / Math.sin(B)) + (-1.0 / (Math.tan(B) / x));
	} else if (F <= 2.8e-93) {
		tmp = ((F * Math.sqrt(0.5)) / B) - t_0;
	} else if (F <= 0.0062) {
		tmp = (F * (t_1 * Math.sqrt(0.5))) - (x / B);
	} else {
		tmp = t_1 - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	t_1 = 1.0 / math.sin(B)
	tmp = 0
	if F <= -36000000000.0:
		tmp = (-1.0 / math.sin(B)) + (-1.0 / (math.tan(B) / x))
	elif F <= 2.8e-93:
		tmp = ((F * math.sqrt(0.5)) / B) - t_0
	elif F <= 0.0062:
		tmp = (F * (t_1 * math.sqrt(0.5))) - (x / B)
	else:
		tmp = t_1 - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	t_1 = Float64(1.0 / sin(B))
	tmp = 0.0
	if (F <= -36000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) + Float64(-1.0 / Float64(tan(B) / x)));
	elseif (F <= 2.8e-93)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) / B) - t_0);
	elseif (F <= 0.0062)
		tmp = Float64(Float64(F * Float64(t_1 * sqrt(0.5))) - Float64(x / B));
	else
		tmp = Float64(t_1 - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	t_1 = 1.0 / sin(B);
	tmp = 0.0;
	if (F <= -36000000000.0)
		tmp = (-1.0 / sin(B)) + (-1.0 / (tan(B) / x));
	elseif (F <= 2.8e-93)
		tmp = ((F * sqrt(0.5)) / B) - t_0;
	elseif (F <= 0.0062)
		tmp = (F * (t_1 * sqrt(0.5))) - (x / B);
	else
		tmp = t_1 - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -36000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.8e-93], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.0062], N[(N[(F * N[(t$95$1 * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(t$95$1 - t$95$0), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.6e10

   1. Initial program 59.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in F around -inf 99.8%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

   9. Simplified 99.8%

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

   if -3.6e10 < F < 2.79999999999999998e-93

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around 0 99.7%

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

      1. *-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 99.7%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

   10. Applied egg-rr 99.8%

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

       1. unpow-1 99.8%

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

   12. Simplified 99.8%

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

   13. Taylor expanded in B around 0 86.3%

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

   if 2.79999999999999998e-93 < F < 0.00619999999999999978

   1. Initial program 99.3%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in F around 0 96.4%

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

      1. *-commutative 96.4%

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

   7. Simplified 96.4%

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

   8. Taylor expanded in x around 0 96.4%

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

   9. Taylor expanded in B around 0 96.4%

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

       1. div-inv 96.5%

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

   11. Applied egg-rr 96.5%

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

   if 0.00619999999999999978 < F

   1. Initial program 59.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -36000000000:\\ \;\;\;\;\frac{-1}{\sin B} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.0062:\\ \;\;\;\;F \cdot \left(\frac{1}{\sin B} \cdot \sqrt{0.5}\right) - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 64.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.085:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.8 \cdot 10^{-156} \lor \neg \left(F \leq 2.4 \cdot 10^{-93}\right) \land F \leq 0.0057:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.085)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (or (<= F -3.8e-156) (and (not (<= F 2.4e-93)) (<= F 0.0057)))
     (/ F (/ (sin B) (sqrt 0.5)))
     (- (/ 1.0 B) (* x (/ 1.0 (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.085) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if ((F <= -3.8e-156) || (!(F <= 2.4e-93) && (F <= 0.0057))) {
		tmp = F / (sin(B) / sqrt(0.5));
	} else {
		tmp = (1.0 / B) - (x * (1.0 / 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 <= (-0.085d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if ((f <= (-3.8d-156)) .or. (.not. (f <= 2.4d-93)) .and. (f <= 0.0057d0)) then
        tmp = f / (sin(b) / sqrt(0.5d0))
    else
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.085) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if ((F <= -3.8e-156) || (!(F <= 2.4e-93) && (F <= 0.0057))) {
		tmp = F / (Math.sin(B) / Math.sqrt(0.5));
	} else {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.085:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif (F <= -3.8e-156) or (not (F <= 2.4e-93) and (F <= 0.0057)):
		tmp = F / (math.sin(B) / math.sqrt(0.5))
	else:
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.085)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif ((F <= -3.8e-156) || (!(F <= 2.4e-93) && (F <= 0.0057)))
		tmp = Float64(F / Float64(sin(B) / sqrt(0.5)));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.085)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif ((F <= -3.8e-156) || (~((F <= 2.4e-93)) && (F <= 0.0057)))
		tmp = F / (sin(B) / sqrt(0.5));
	else
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.085], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, -3.8e-156], And[N[Not[LessEqual[F, 2.4e-93]], $MachinePrecision], LessEqual[F, 0.0057]]], N[(F / N[(N[Sin[B], $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.0850000000000000061

   1. Initial program 61.2%

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

   3. Simplified 75.5%

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

   5. Taylor expanded in F around -inf 99.7%

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

   6. Taylor expanded in B around 0 79.5%

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

   if -0.0850000000000000061 < F < -3.80000000000000008e-156 or 2.4000000000000001e-93 < F < 0.0057000000000000002

   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 99.3%

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

   5. Taylor expanded in F around 0 98.0%

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

      1. *-commutative 98.0%

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

   7. Simplified 98.0%

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

   8. Taylor expanded in x around 0 98.0%

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

      1. clear-num 98.2%

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

      2. inv-pow 98.2%

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

   10. Applied egg-rr 98.2%

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

       1. unpow-1 98.2%

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

   12. Simplified 98.2%

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

   13. Taylor expanded in F around inf 72.4%

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

       1. *-rgt-identity 72.4%

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

       2. times-frac 72.4%

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

       3. /-rgt-identity 72.4%

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

       4. associate-/r/ 72.7%

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

   15. Simplified 72.7%

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

   if -3.80000000000000008e-156 < F < 2.4000000000000001e-93 or 0.0057000000000000002 < F

   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 B around 0 69.7%

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

   4. Taylor expanded in F around inf 61.3%

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

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.085:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.8 \cdot 10^{-156} \lor \neg \left(F \leq 2.4 \cdot 10^{-93}\right) \land F \leq 0.0057:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 64.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.0021:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.9 \cdot 10^{-156} \lor \neg \left(F \leq 1.95 \cdot 10^{-93}\right) \land F \leq 0.0048:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.0021)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (or (<= F -3.9e-156) (and (not (<= F 1.95e-93)) (<= F 0.0048)))
     (* F (/ (sqrt 0.5) (sin B)))
     (- (/ 1.0 B) (* x (/ 1.0 (tan B)))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0021) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if ((F <= -3.9e-156) || (!(F <= 1.95e-93) && (F <= 0.0048))) {
		tmp = F * (sqrt(0.5) / sin(B));
	} else {
		tmp = (1.0 / B) - (x * (1.0 / 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 <= (-0.0021d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if ((f <= (-3.9d-156)) .or. (.not. (f <= 1.95d-93)) .and. (f <= 0.0048d0)) then
        tmp = f * (sqrt(0.5d0) / sin(b))
    else
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0021) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if ((F <= -3.9e-156) || (!(F <= 1.95e-93) && (F <= 0.0048))) {
		tmp = F * (Math.sqrt(0.5) / Math.sin(B));
	} else {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.0021:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif (F <= -3.9e-156) or (not (F <= 1.95e-93) and (F <= 0.0048)):
		tmp = F * (math.sqrt(0.5) / math.sin(B))
	else:
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.0021)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif ((F <= -3.9e-156) || (!(F <= 1.95e-93) && (F <= 0.0048)))
		tmp = Float64(F * Float64(sqrt(0.5) / sin(B)));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.0021)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif ((F <= -3.9e-156) || (~((F <= 1.95e-93)) && (F <= 0.0048)))
		tmp = F * (sqrt(0.5) / sin(B));
	else
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.0021], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[F, -3.9e-156], And[N[Not[LessEqual[F, 1.95e-93]], $MachinePrecision], LessEqual[F, 0.0048]]], N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -0.00209999999999999987

   1. Initial program 61.2%

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

   3. Simplified 75.5%

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

   5. Taylor expanded in F around -inf 99.7%

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

   6. Taylor expanded in B around 0 79.5%

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

   if -0.00209999999999999987 < F < -3.9000000000000001e-156 or 1.95000000000000009e-93 < F < 0.00479999999999999958

   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 99.3%

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

   5. Taylor expanded in F around 0 98.0%

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

      1. *-commutative 98.0%

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

   7. Simplified 98.0%

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

   8. Taylor expanded in x around 0 72.4%

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

      1. associate-/l* 72.3%

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

   10. Simplified 72.3%

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

   if -3.9000000000000001e-156 < F < 1.95000000000000009e-93 or 0.00479999999999999958 < F

   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 B around 0 69.7%

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

   4. Taylor expanded in F around inf 61.3%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0021:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -3.9 \cdot 10^{-156} \lor \neg \left(F \leq 1.95 \cdot 10^{-93}\right) \land F \leq 0.0048:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 79.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -7.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.35 \cdot 10^{-66}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-93}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 0.0052:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -7.2e-5)
   (- (/ -1.0 (sin B)) (/ x (tan B)))
   (if (<= F -2.35e-66)
     (* F (/ (sqrt 0.5) (sin B)))
     (if (<= F 3.1e-93)
       (/ (* x (cos B)) (- (sin B)))
       (if (<= F 0.0052)
         (/ F (/ (sin B) (sqrt 0.5)))
         (- (/ 1.0 B) (* x (/ 1.0 (tan B)))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.2e-5) {
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	} else if (F <= -2.35e-66) {
		tmp = F * (sqrt(0.5) / sin(B));
	} else if (F <= 3.1e-93) {
		tmp = (x * cos(B)) / -sin(B);
	} else if (F <= 0.0052) {
		tmp = F / (sin(B) / sqrt(0.5));
	} else {
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-7.2d-5)) then
        tmp = ((-1.0d0) / sin(b)) - (x / tan(b))
    else if (f <= (-2.35d-66)) then
        tmp = f * (sqrt(0.5d0) / sin(b))
    else if (f <= 3.1d-93) then
        tmp = (x * cos(b)) / -sin(b)
    else if (f <= 0.0052d0) then
        tmp = f / (sin(b) / sqrt(0.5d0))
    else
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -7.2e-5) {
		tmp = (-1.0 / Math.sin(B)) - (x / Math.tan(B));
	} else if (F <= -2.35e-66) {
		tmp = F * (Math.sqrt(0.5) / Math.sin(B));
	} else if (F <= 3.1e-93) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else if (F <= 0.0052) {
		tmp = F / (Math.sin(B) / Math.sqrt(0.5));
	} else {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -7.2e-5:
		tmp = (-1.0 / math.sin(B)) - (x / math.tan(B))
	elif F <= -2.35e-66:
		tmp = F * (math.sqrt(0.5) / math.sin(B))
	elif F <= 3.1e-93:
		tmp = (x * math.cos(B)) / -math.sin(B)
	elif F <= 0.0052:
		tmp = F / (math.sin(B) / math.sqrt(0.5))
	else:
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -7.2e-5)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / tan(B)));
	elseif (F <= -2.35e-66)
		tmp = Float64(F * Float64(sqrt(0.5) / sin(B)));
	elseif (F <= 3.1e-93)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	elseif (F <= 0.0052)
		tmp = Float64(F / Float64(sin(B) / sqrt(0.5)));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -7.2e-5)
		tmp = (-1.0 / sin(B)) - (x / tan(B));
	elseif (F <= -2.35e-66)
		tmp = F * (sqrt(0.5) / sin(B));
	elseif (F <= 3.1e-93)
		tmp = (x * cos(B)) / -sin(B);
	elseif (F <= 0.0052)
		tmp = F / (sin(B) / sqrt(0.5));
	else
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -7.2e-5], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.35e-66], N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.1e-93], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 0.0052], N[(F / N[(N[Sin[B], $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -7.20000000000000018e-5

   1. Initial program 61.2%

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

   3. Simplified 75.5%

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

   5. Taylor expanded in F around -inf 99.7%

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

   if -7.20000000000000018e-5 < F < -2.35e-66

   1. Initial program 99.3%

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

   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 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 90.6%

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

      1. associate-/l* 90.6%

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

   10. Simplified 90.6%

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

   if -2.35e-66 < F < 3.1e-93

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around -inf 33.0%

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

      1. clear-num 32.8%

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

      2. inv-pow 32.8%

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

   7. Applied egg-rr 32.8%

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

      1. unpow-1 32.8%

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

   9. Simplified 32.8%

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

   10. Taylor expanded in x around inf 83.4%

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

       1. mul-1-neg 83.4%

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

   12. Simplified 83.4%

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

   if 3.1e-93 < F < 0.0051999999999999998

   1. Initial program 99.3%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in F around 0 96.4%

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

      1. *-commutative 96.4%

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

   7. Simplified 96.4%

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

   8. Taylor expanded in x around 0 96.4%

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

      1. clear-num 96.6%

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

      2. inv-pow 96.6%

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

   10. Applied egg-rr 96.6%

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

       1. unpow-1 96.6%

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

   12. Simplified 96.6%

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

   13. Taylor expanded in F around inf 77.4%

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

       1. *-rgt-identity 77.4%

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

       2. times-frac 77.5%

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

       3. /-rgt-identity 77.5%

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

       4. associate-/r/ 77.9%

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

   15. Simplified 77.9%

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

   if 0.0051999999999999998 < F

   1. Initial program 59.2%

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

   3. Taylor expanded in B around 0 43.5%

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

   4. Taylor expanded in F around inf 70.0%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -7.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq -2.35 \cdot 10^{-66}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 3.1 \cdot 10^{-93}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 0.0052:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 73.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -0.0017:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.4 \cdot 10^{-66}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 0.0047:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -0.0017)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F -2.4e-66)
     (* F (/ (sqrt 0.5) (sin B)))
     (if (<= F 3.2e-93)
       (/ (* x (cos B)) (- (sin B)))
       (if (<= F 0.0047)
         (/ F (/ (sin B) (sqrt 0.5)))
         (- (/ 1.0 B) (* x (/ 1.0 (tan B)))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0017) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= -2.4e-66) {
		tmp = F * (sqrt(0.5) / sin(B));
	} else if (F <= 3.2e-93) {
		tmp = (x * cos(B)) / -sin(B);
	} else if (F <= 0.0047) {
		tmp = F / (sin(B) / sqrt(0.5));
	} else {
		tmp = (1.0 / B) - (x * (1.0 / 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 <= (-0.0017d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= (-2.4d-66)) then
        tmp = f * (sqrt(0.5d0) / sin(b))
    else if (f <= 3.2d-93) then
        tmp = (x * cos(b)) / -sin(b)
    else if (f <= 0.0047d0) then
        tmp = f / (sin(b) / sqrt(0.5d0))
    else
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -0.0017) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= -2.4e-66) {
		tmp = F * (Math.sqrt(0.5) / Math.sin(B));
	} else if (F <= 3.2e-93) {
		tmp = (x * Math.cos(B)) / -Math.sin(B);
	} else if (F <= 0.0047) {
		tmp = F / (Math.sin(B) / Math.sqrt(0.5));
	} else {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -0.0017:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= -2.4e-66:
		tmp = F * (math.sqrt(0.5) / math.sin(B))
	elif F <= 3.2e-93:
		tmp = (x * math.cos(B)) / -math.sin(B)
	elif F <= 0.0047:
		tmp = F / (math.sin(B) / math.sqrt(0.5))
	else:
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -0.0017)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= -2.4e-66)
		tmp = Float64(F * Float64(sqrt(0.5) / sin(B)));
	elseif (F <= 3.2e-93)
		tmp = Float64(Float64(x * cos(B)) / Float64(-sin(B)));
	elseif (F <= 0.0047)
		tmp = Float64(F / Float64(sin(B) / sqrt(0.5)));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -0.0017)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= -2.4e-66)
		tmp = F * (sqrt(0.5) / sin(B));
	elseif (F <= 3.2e-93)
		tmp = (x * cos(B)) / -sin(B);
	elseif (F <= 0.0047)
		tmp = F / (sin(B) / sqrt(0.5));
	else
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -0.0017], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -2.4e-66], N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.2e-93], N[(N[(x * N[Cos[B], $MachinePrecision]), $MachinePrecision] / (-N[Sin[B], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 0.0047], N[(F / N[(N[Sin[B], $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if F < -0.00169999999999999991

   1. Initial program 61.2%

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

   3. Simplified 75.5%

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

   5. Taylor expanded in F around -inf 99.7%

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

   6. Taylor expanded in B around 0 79.5%

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

   if -0.00169999999999999991 < F < -2.40000000000000026e-66

   1. Initial program 99.3%

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

   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 99.6%

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

      1. *-commutative 99.6%

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

   7. Simplified 99.6%

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

   8. Taylor expanded in x around 0 90.6%

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

      1. associate-/l* 90.6%

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

   10. Simplified 90.6%

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

   if -2.40000000000000026e-66 < F < 3.1999999999999999e-93

   1. Initial program 99.6%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around -inf 33.0%

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

      1. clear-num 32.8%

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

      2. inv-pow 32.8%

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

   7. Applied egg-rr 32.8%

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

      1. unpow-1 32.8%

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

   9. Simplified 32.8%

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

   10. Taylor expanded in x around inf 83.4%

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

       1. mul-1-neg 83.4%

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

   12. Simplified 83.4%

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

   if 3.1999999999999999e-93 < F < 0.00470000000000000018

   1. Initial program 99.3%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in F around 0 96.4%

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

      1. *-commutative 96.4%

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

   7. Simplified 96.4%

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

   8. Taylor expanded in x around 0 96.4%

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

      1. clear-num 96.6%

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

      2. inv-pow 96.6%

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

   10. Applied egg-rr 96.6%

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

       1. unpow-1 96.6%

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

   12. Simplified 96.6%

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

   13. Taylor expanded in F around inf 77.4%

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

       1. *-rgt-identity 77.4%

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

       2. times-frac 77.5%

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

       3. /-rgt-identity 77.5%

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

       4. associate-/r/ 77.9%

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

   15. Simplified 77.9%

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

   if 0.00470000000000000018 < F

   1. Initial program 59.2%

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

   3. Taylor expanded in B around 0 43.5%

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

   4. Taylor expanded in F around inf 70.0%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -0.0017:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq -2.4 \cdot 10^{-66}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B}\\ \mathbf{elif}\;F \leq 3.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{x \cdot \cos B}{-\sin B}\\ \mathbf{elif}\;F \leq 0.0047:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 91.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -36000000000:\\ \;\;\;\;\frac{-1}{\sin B} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-77}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B} - t\_0\\ \mathbf{elif}\;F \leq 0.006:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -36000000000.0)
     (+ (/ -1.0 (sin B)) (/ -1.0 (/ (tan B) x)))
     (if (<= F 2.7e-77)
       (- (/ (* F (sqrt 0.5)) B) t_0)
       (if (<= F 0.006)
         (- (* F (/ (sqrt 0.5) (sin B))) (/ x B))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / sin(B)) + (-1.0 / (tan(B) / x));
	} else if (F <= 2.7e-77) {
		tmp = ((F * sqrt(0.5)) / B) - t_0;
	} else if (F <= 0.006) {
		tmp = (F * (sqrt(0.5) / sin(B))) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-36000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) + ((-1.0d0) / (tan(b) / x))
    else if (f <= 2.7d-77) then
        tmp = ((f * sqrt(0.5d0)) / b) - t_0
    else if (f <= 0.006d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / Math.sin(B)) + (-1.0 / (Math.tan(B) / x));
	} else if (F <= 2.7e-77) {
		tmp = ((F * Math.sqrt(0.5)) / B) - t_0;
	} else if (F <= 0.006) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -36000000000.0:
		tmp = (-1.0 / math.sin(B)) + (-1.0 / (math.tan(B) / x))
	elif F <= 2.7e-77:
		tmp = ((F * math.sqrt(0.5)) / B) - t_0
	elif F <= 0.006:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -36000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) + Float64(-1.0 / Float64(tan(B) / x)));
	elseif (F <= 2.7e-77)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) / B) - t_0);
	elseif (F <= 0.006)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -36000000000.0)
		tmp = (-1.0 / sin(B)) + (-1.0 / (tan(B) / x));
	elseif (F <= 2.7e-77)
		tmp = ((F * sqrt(0.5)) / B) - t_0;
	elseif (F <= 0.006)
		tmp = (F * (sqrt(0.5) / sin(B))) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -36000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.7e-77], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.006], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.6e10

   1. Initial program 59.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in F around -inf 99.8%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

   9. Simplified 99.8%

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

   if -3.6e10 < F < 2.7e-77

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around 0 99.7%

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

      1. *-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 99.7%

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

      1. clear-num 99.7%

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

      2. inv-pow 99.7%

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

   10. Applied egg-rr 99.7%

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

       1. unpow-1 99.7%

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

   12. Simplified 99.7%

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

   13. Taylor expanded in B around 0 86.5%

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

   if 2.7e-77 < F < 0.0060000000000000001

   1. Initial program 99.3%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in F around 0 96.1%

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

      1. *-commutative 96.1%

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

   7. Simplified 96.1%

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

   8. Taylor expanded in x around 0 96.1%

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

   9. Taylor expanded in B around 0 96.1%

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

   if 0.0060000000000000001 < F

   1. Initial program 59.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -36000000000:\\ \;\;\;\;\frac{-1}{\sin B} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{-77}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.006:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 91.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -36000000000:\\ \;\;\;\;\frac{-1}{\sin B} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-93}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{elif}\;F \leq 0.0058:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -36000000000.0)
     (+ (/ -1.0 (sin B)) (/ -1.0 (/ (tan B) x)))
     (if (<= F 2.8e-93)
       (- (* F (/ (sqrt 0.5) B)) t_0)
       (if (<= F 0.0058)
         (- (* F (/ (sqrt 0.5) (sin B))) (/ x B))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / sin(B)) + (-1.0 / (tan(B) / x));
	} else if (F <= 2.8e-93) {
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	} else if (F <= 0.0058) {
		tmp = (F * (sqrt(0.5) / sin(B))) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-36000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) + ((-1.0d0) / (tan(b) / x))
    else if (f <= 2.8d-93) then
        tmp = (f * (sqrt(0.5d0) / b)) - t_0
    else if (f <= 0.0058d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / Math.sin(B)) + (-1.0 / (Math.tan(B) / x));
	} else if (F <= 2.8e-93) {
		tmp = (F * (Math.sqrt(0.5) / B)) - t_0;
	} else if (F <= 0.0058) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -36000000000.0:
		tmp = (-1.0 / math.sin(B)) + (-1.0 / (math.tan(B) / x))
	elif F <= 2.8e-93:
		tmp = (F * (math.sqrt(0.5) / B)) - t_0
	elif F <= 0.0058:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -36000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) + Float64(-1.0 / Float64(tan(B) / x)));
	elseif (F <= 2.8e-93)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - t_0);
	elseif (F <= 0.0058)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -36000000000.0)
		tmp = (-1.0 / sin(B)) + (-1.0 / (tan(B) / x));
	elseif (F <= 2.8e-93)
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	elseif (F <= 0.0058)
		tmp = (F * (sqrt(0.5) / sin(B))) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -36000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] + N[(-1.0 / N[(N[Tan[B], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.8e-93], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.0058], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.6e10

   1. Initial program 59.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in F around -inf 99.8%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

   9. Simplified 99.8%

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

   if -3.6e10 < F < 2.79999999999999998e-93

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around 0 99.7%

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

      1. *-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 99.7%

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

   9. Taylor expanded in B around 0 86.3%

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

   if 2.79999999999999998e-93 < F < 0.0058

   1. Initial program 99.3%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in F around 0 96.4%

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

      1. *-commutative 96.4%

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

   7. Simplified 96.4%

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

   8. Taylor expanded in x around 0 96.4%

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

   9. Taylor expanded in B around 0 96.4%

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

   if 0.0058 < F

   1. Initial program 59.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in F around inf 99.8%

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

4. Final simplification 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -36000000000:\\ \;\;\;\;\frac{-1}{\sin B} + \frac{-1}{\frac{\tan B}{x}}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-93}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 0.0058:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - \frac{x}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 91.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -36000000000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{-93}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{elif}\;F \leq 0.0062:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -36000000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 2.1e-93)
       (- (* F (/ (sqrt 0.5) B)) t_0)
       (if (<= F 0.0062)
         (- (* F (/ (sqrt 0.5) (sin B))) (/ x B))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 2.1e-93) {
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	} else if (F <= 0.0062) {
		tmp = (F * (sqrt(0.5) / sin(B))) - (x / B);
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-36000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 2.1d-93) then
        tmp = (f * (sqrt(0.5d0) / b)) - t_0
    else if (f <= 0.0062d0) then
        tmp = (f * (sqrt(0.5d0) / sin(b))) - (x / b)
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 2.1e-93) {
		tmp = (F * (Math.sqrt(0.5) / B)) - t_0;
	} else if (F <= 0.0062) {
		tmp = (F * (Math.sqrt(0.5) / Math.sin(B))) - (x / B);
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -36000000000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 2.1e-93:
		tmp = (F * (math.sqrt(0.5) / B)) - t_0
	elif F <= 0.0062:
		tmp = (F * (math.sqrt(0.5) / math.sin(B))) - (x / B)
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -36000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 2.1e-93)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - t_0);
	elseif (F <= 0.0062)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / sin(B))) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -36000000000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 2.1e-93)
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	elseif (F <= 0.0062)
		tmp = (F * (sqrt(0.5) / sin(B))) - (x / B);
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -36000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2.1e-93], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.0062], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / N[Sin[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.6e10

   1. Initial program 59.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in F around -inf 99.8%

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

   if -3.6e10 < F < 2.1000000000000001e-93

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around 0 99.7%

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

      1. *-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 99.7%

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

   9. Taylor expanded in B around 0 86.3%

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

   if 2.1000000000000001e-93 < F < 0.00619999999999999978

   1. Initial program 99.3%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in F around 0 96.4%

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

      1. *-commutative 96.4%

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

   7. Simplified 96.4%

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

   8. Taylor expanded in x around 0 96.4%

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

   9. Taylor expanded in B around 0 96.4%

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

   if 0.00619999999999999978 < F

   1. Initial program 59.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in F around inf 99.8%

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

Alternative 22: 91.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\tan B}\\ \mathbf{if}\;F \leq -36000000000:\\ \;\;\;\;\frac{-1}{\sin B} - t\_0\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{-42}:\\ \;\;\;\;F \cdot \frac{\sqrt{0.5}}{B} - t\_0\\ \mathbf{elif}\;F \leq 0.0057:\\ \;\;\;\;\frac{F}{\frac{\sin B}{\sqrt{0.5}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\sin B} - t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (let* ((t_0 (/ x (tan B))))
   (if (<= F -36000000000.0)
     (- (/ -1.0 (sin B)) t_0)
     (if (<= F 2.4e-42)
       (- (* F (/ (sqrt 0.5) B)) t_0)
       (if (<= F 0.0057)
         (/ F (/ (sin B) (sqrt 0.5)))
         (- (/ 1.0 (sin B)) t_0))))))

C

double code(double F, double B, double x) {
	double t_0 = x / tan(B);
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / sin(B)) - t_0;
	} else if (F <= 2.4e-42) {
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	} else if (F <= 0.0057) {
		tmp = F / (sin(B) / sqrt(0.5));
	} else {
		tmp = (1.0 / sin(B)) - t_0;
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x / tan(b)
    if (f <= (-36000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - t_0
    else if (f <= 2.4d-42) then
        tmp = (f * (sqrt(0.5d0) / b)) - t_0
    else if (f <= 0.0057d0) then
        tmp = f / (sin(b) / sqrt(0.5d0))
    else
        tmp = (1.0d0 / sin(b)) - t_0
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double t_0 = x / Math.tan(B);
	double tmp;
	if (F <= -36000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - t_0;
	} else if (F <= 2.4e-42) {
		tmp = (F * (Math.sqrt(0.5) / B)) - t_0;
	} else if (F <= 0.0057) {
		tmp = F / (Math.sin(B) / Math.sqrt(0.5));
	} else {
		tmp = (1.0 / Math.sin(B)) - t_0;
	}
	return tmp;
}

Python

def code(F, B, x):
	t_0 = x / math.tan(B)
	tmp = 0
	if F <= -36000000000.0:
		tmp = (-1.0 / math.sin(B)) - t_0
	elif F <= 2.4e-42:
		tmp = (F * (math.sqrt(0.5) / B)) - t_0
	elif F <= 0.0057:
		tmp = F / (math.sin(B) / math.sqrt(0.5))
	else:
		tmp = (1.0 / math.sin(B)) - t_0
	return tmp

Julia

function code(F, B, x)
	t_0 = Float64(x / tan(B))
	tmp = 0.0
	if (F <= -36000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - t_0);
	elseif (F <= 2.4e-42)
		tmp = Float64(Float64(F * Float64(sqrt(0.5) / B)) - t_0);
	elseif (F <= 0.0057)
		tmp = Float64(F / Float64(sin(B) / sqrt(0.5)));
	else
		tmp = Float64(Float64(1.0 / sin(B)) - t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	t_0 = x / tan(B);
	tmp = 0.0;
	if (F <= -36000000000.0)
		tmp = (-1.0 / sin(B)) - t_0;
	elseif (F <= 2.4e-42)
		tmp = (F * (sqrt(0.5) / B)) - t_0;
	elseif (F <= 0.0057)
		tmp = F / (sin(B) / sqrt(0.5));
	else
		tmp = (1.0 / sin(B)) - t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := Block[{t$95$0 = N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -36000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 2.4e-42], N[(N[(F * N[(N[Sqrt[0.5], $MachinePrecision] / B), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision], If[LessEqual[F, 0.0057], N[(F / N[(N[Sin[B], $MachinePrecision] / N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.6e10

   1. Initial program 59.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in F around -inf 99.8%

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

   if -3.6e10 < F < 2.40000000000000003e-42

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around 0 99.7%

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

      1. *-commutative 99.7%

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

   7. Simplified 99.7%

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

   8. Taylor expanded in x around 0 99.7%

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

   9. Taylor expanded in B around 0 85.1%

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

   if 2.40000000000000003e-42 < F < 0.0057000000000000002

   1. Initial program 99.0%

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

   3. Simplified 98.8%

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

   5. Taylor expanded in F around 0 92.9%

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

      1. *-commutative 92.9%

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

   7. Simplified 92.9%

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

   8. Taylor expanded in x around 0 92.9%

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

      1. clear-num 93.3%

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

      2. inv-pow 93.3%

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

   10. Applied egg-rr 93.3%

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

       1. unpow-1 93.3%

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

   12. Simplified 93.3%

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

   13. Taylor expanded in F around inf 92.8%

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

       1. *-rgt-identity 92.8%

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

       2. times-frac 93.0%

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

       3. /-rgt-identity 93.0%

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

       4. associate-/r/ 93.6%

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

   15. Simplified 93.6%

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

   if 0.0057000000000000002 < F

   1. Initial program 59.2%

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

   3. Simplified 70.2%

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

   5. Taylor expanded in F around inf 99.8%

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

Alternative 23: 63.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1500000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1500000000000.0)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 2.2e-93)
     (- (/ -1.0 B) (/ x (tan B)))
     (if (<= F 2.3e-9)
       (- (* (/ F B) (sqrt (/ 1.0 (+ 2.0 (* x 2.0))))) (/ x B))
       (- (/ 1.0 B) (* x (/ 1.0 (tan B))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1500000000000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 2.2e-93) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= 2.3e-9) {
		tmp = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / 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 <= (-1500000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 2.2d-93) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (f <= 2.3d-9) then
        tmp = ((f / b) * sqrt((1.0d0 / (2.0d0 + (x * 2.0d0))))) - (x / b)
    else
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1500000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 2.2e-93) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (F <= 2.3e-9) {
		tmp = ((F / B) * Math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1500000000000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 2.2e-93:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif F <= 2.3e-9:
		tmp = ((F / B) * math.sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B)
	else:
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1500000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 2.2e-93)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= 2.3e-9)
		tmp = Float64(Float64(Float64(F / B) * sqrt(Float64(1.0 / Float64(2.0 + Float64(x * 2.0))))) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1500000000000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 2.2e-93)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (F <= 2.3e-9)
		tmp = ((F / B) * sqrt((1.0 / (2.0 + (x * 2.0))))) - (x / B);
	else
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1500000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.2e-93], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.3e-9], N[(N[(N[(F / B), $MachinePrecision] * N[Sqrt[N[(1.0 / N[(2.0 + N[(x * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.5e12

   1. Initial program 58.8%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in F around -inf 99.8%

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

   6. Taylor expanded in B around 0 79.7%

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

   if -1.5e12 < F < 2.19999999999999996e-93

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around -inf 33.0%

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

   6. Taylor expanded in B around 0 50.1%

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

   if 2.19999999999999996e-93 < F < 2.2999999999999999e-9

   1. Initial program 99.2%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in F around 0 99.0%

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

      1. *-commutative 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in B around 0 67.4%

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

   9. Taylor expanded in F around 0 67.6%

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

   if 2.2999999999999999e-9 < F

   1. Initial program 60.4%

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

   3. Taylor expanded in B around 0 42.3%

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

   4. Taylor expanded in F around inf 68.1%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1500000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.2 \cdot 10^{-93}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.3 \cdot 10^{-9}:\\ \;\;\;\;\frac{F}{B} \cdot \sqrt{\frac{1}{2 + x \cdot 2}} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 24: 63.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -1460000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{-93}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -1460000000000.0)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 2.4e-93)
     (- (/ -1.0 B) (/ x (tan B)))
     (if (<= F 3.9e-10)
       (- (* (sqrt 0.5) (/ F B)) (/ x B))
       (- (/ 1.0 B) (* x (/ 1.0 (tan B))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -1460000000000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 2.4e-93) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= 3.9e-10) {
		tmp = (sqrt(0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / 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 <= (-1460000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 2.4d-93) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (f <= 3.9d-10) then
        tmp = (sqrt(0.5d0) * (f / b)) - (x / b)
    else
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -1460000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 2.4e-93) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (F <= 3.9e-10) {
		tmp = (Math.sqrt(0.5) * (F / B)) - (x / B);
	} else {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -1460000000000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 2.4e-93:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif F <= 3.9e-10:
		tmp = (math.sqrt(0.5) * (F / B)) - (x / B)
	else:
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -1460000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 2.4e-93)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= 3.9e-10)
		tmp = Float64(Float64(sqrt(0.5) * Float64(F / B)) - Float64(x / B));
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -1460000000000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 2.4e-93)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (F <= 3.9e-10)
		tmp = (sqrt(0.5) * (F / B)) - (x / B);
	else
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -1460000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.4e-93], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.9e-10], N[(N[(N[Sqrt[0.5], $MachinePrecision] * N[(F / B), $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -1.46e12

   1. Initial program 58.8%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in F around -inf 99.8%

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

   6. Taylor expanded in B around 0 79.7%

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

   if -1.46e12 < F < 2.4000000000000001e-93

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around -inf 33.0%

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

   6. Taylor expanded in B around 0 50.1%

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

   if 2.4000000000000001e-93 < F < 3.9e-10

   1. Initial program 99.2%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in F around 0 99.0%

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

      1. *-commutative 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in B around 0 67.4%

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

   9. Taylor expanded in F around 0 67.6%

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

   10. Taylor expanded in x around 0 67.6%

       \[\leadsto -1 \cdot \frac{x}{B} + \frac{F}{B} \cdot \color{blue}{\sqrt{0.5}} \]

   if 3.9e-10 < F

   1. Initial program 60.4%

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

   3. Taylor expanded in B around 0 42.3%

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

   4. Taylor expanded in F around inf 68.1%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1460000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.4 \cdot 10^{-93}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 3.9 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{0.5} \cdot \frac{F}{B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 25: 63.2% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -29500000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -29500000000000.0)
   (- (/ -1.0 (sin B)) (/ x B))
   (if (<= F 2.8e-93)
     (- (/ -1.0 B) (/ x (tan B)))
     (if (<= F 2.1e-9)
       (/ (- (* F (sqrt 0.5)) x) B)
       (- (/ 1.0 B) (* x (/ 1.0 (tan B))))))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -29500000000000.0) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else if (F <= 2.8e-93) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else if (F <= 2.1e-9) {
		tmp = ((F * sqrt(0.5)) - x) / B;
	} else {
		tmp = (1.0 / B) - (x * (1.0 / 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 <= (-29500000000000.0d0)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else if (f <= 2.8d-93) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else if (f <= 2.1d-9) then
        tmp = ((f * sqrt(0.5d0)) - x) / b
    else
        tmp = (1.0d0 / b) - (x * (1.0d0 / tan(b)))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -29500000000000.0) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else if (F <= 2.8e-93) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else if (F <= 2.1e-9) {
		tmp = ((F * Math.sqrt(0.5)) - x) / B;
	} else {
		tmp = (1.0 / B) - (x * (1.0 / Math.tan(B)));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -29500000000000.0:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	elif F <= 2.8e-93:
		tmp = (-1.0 / B) - (x / math.tan(B))
	elif F <= 2.1e-9:
		tmp = ((F * math.sqrt(0.5)) - x) / B
	else:
		tmp = (1.0 / B) - (x * (1.0 / math.tan(B)))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -29500000000000.0)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	elseif (F <= 2.8e-93)
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	elseif (F <= 2.1e-9)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B);
	else
		tmp = Float64(Float64(1.0 / B) - Float64(x * Float64(1.0 / tan(B))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -29500000000000.0)
		tmp = (-1.0 / sin(B)) - (x / B);
	elseif (F <= 2.8e-93)
		tmp = (-1.0 / B) - (x / tan(B));
	elseif (F <= 2.1e-9)
		tmp = ((F * sqrt(0.5)) - x) / B;
	else
		tmp = (1.0 / B) - (x * (1.0 / tan(B)));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -29500000000000.0], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.8e-93], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.1e-9], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(1.0 / B), $MachinePrecision] - N[(x * N[(1.0 / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.95e13

   1. Initial program 58.8%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in F around -inf 99.8%

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

   6. Taylor expanded in B around 0 79.7%

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

   if -2.95e13 < F < 2.79999999999999998e-93

   1. Initial program 99.5%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in F around -inf 33.0%

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

   6. Taylor expanded in B around 0 50.1%

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

   if 2.79999999999999998e-93 < F < 2.10000000000000019e-9

   1. Initial program 99.2%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in F around 0 99.0%

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

      1. *-commutative 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in B around 0 67.4%

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

   9. Taylor expanded in x around 0 67.4%

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

   if 2.10000000000000019e-9 < F

   1. Initial program 60.4%

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

   3. Taylor expanded in B around 0 42.3%

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

   4. Taylor expanded in F around inf 68.1%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -29500000000000:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{elif}\;F \leq 2.8 \cdot 10^{-93}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{elif}\;F \leq 2.1 \cdot 10^{-9}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{B} - x \cdot \frac{1}{\tan B}\\ \end{array} \]
5. Add Preprocessing

Alternative 26: 56.1% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-129} \lor \neg \left(x \leq 1.6 \cdot 10^{-116}\right):\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (or (<= x -2.4e-129) (not (<= x 1.6e-116)))
   (- (/ -1.0 B) (/ x (tan B)))
   (/ -1.0 (sin B))))

C

double code(double F, double B, double x) {
	double tmp;
	if ((x <= -2.4e-129) || !(x <= 1.6e-116)) {
		tmp = (-1.0 / B) - (x / tan(B));
	} else {
		tmp = -1.0 / sin(B);
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-2.4d-129)) .or. (.not. (x <= 1.6d-116))) then
        tmp = ((-1.0d0) / b) - (x / tan(b))
    else
        tmp = (-1.0d0) / sin(b)
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if ((x <= -2.4e-129) || !(x <= 1.6e-116)) {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	} else {
		tmp = -1.0 / Math.sin(B);
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if (x <= -2.4e-129) or not (x <= 1.6e-116):
		tmp = (-1.0 / B) - (x / math.tan(B))
	else:
		tmp = -1.0 / math.sin(B)
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if ((x <= -2.4e-129) || !(x <= 1.6e-116))
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	else
		tmp = Float64(-1.0 / sin(B));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if ((x <= -2.4e-129) || ~((x <= 1.6e-116)))
		tmp = (-1.0 / B) - (x / tan(B));
	else
		tmp = -1.0 / sin(B);
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[Or[LessEqual[x, -2.4e-129], N[Not[LessEqual[x, 1.6e-116]], $MachinePrecision]], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.39999999999999989e-129 or 1.60000000000000005e-116 < x

   1. Initial program 82.8%

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

   3. Simplified 91.4%

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

   5. Taylor expanded in F around -inf 62.1%

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

   6. Taylor expanded in B around 0 67.8%

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

   if -2.39999999999999989e-129 < x < 1.60000000000000005e-116

   1. Initial program 71.9%

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

   3. Simplified 75.1%

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

   5. Taylor expanded in F around -inf 30.5%

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

      1. clear-num 30.5%

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

      2. inv-pow 30.5%

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

   7. Applied egg-rr 30.5%

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

      1. unpow-1 30.5%

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

   9. Simplified 30.5%

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

   10. Taylor expanded in x around 0 30.5%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{-129} \lor \neg \left(x \leq 1.6 \cdot 10^{-116}\right):\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\sin B}\\ \end{array} \]
5. Add Preprocessing

Alternative 27: 42.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;B \leq 6 \cdot 10^{-45}:\\ \;\;\;\;\frac{F \cdot \sqrt{0.5} - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= B 6e-45) (/ (- (* F (sqrt 0.5)) x) B) (- (/ -1.0 B) (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (B <= 6e-45) {
		tmp = ((F * sqrt(0.5)) - x) / B;
	} else {
		tmp = (-1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (b <= 6d-45) then
        tmp = ((f * sqrt(0.5d0)) - x) / b
    else
        tmp = ((-1.0d0) / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (B <= 6e-45) {
		tmp = ((F * Math.sqrt(0.5)) - x) / B;
	} else {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if B <= 6e-45:
		tmp = ((F * math.sqrt(0.5)) - x) / B
	else:
		tmp = (-1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (B <= 6e-45)
		tmp = Float64(Float64(Float64(F * sqrt(0.5)) - x) / B);
	else
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (B <= 6e-45)
		tmp = ((F * sqrt(0.5)) - x) / B;
	else
		tmp = (-1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[B, 6e-45], N[(N[(N[(F * N[Sqrt[0.5], $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / B), $MachinePrecision], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 6.00000000000000022e-45

   1. Initial program 77.4%

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

   3. Simplified 85.8%

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

   5. Taylor expanded in F around 0 59.1%

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

      1. *-commutative 59.1%

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

   7. Simplified 59.1%

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

   8. Taylor expanded in B around 0 38.5%

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

   9. Taylor expanded in x around 0 39.9%

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

   if 6.00000000000000022e-45 < B

   1. Initial program 83.3%

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

   3. Simplified 85.1%

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

   5. Taylor expanded in F around -inf 47.1%

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

   6. Taylor expanded in B around 0 48.6%

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

Alternative 28: 54.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.72 \cdot 10^{+14}:\\ \;\;\;\;\frac{-1}{\sin B} - \frac{x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{B} - \frac{x}{\tan B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.72e+14)
   (- (/ -1.0 (sin B)) (/ x B))
   (- (/ -1.0 B) (/ x (tan B)))))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.72e+14) {
		tmp = (-1.0 / sin(B)) - (x / B);
	} else {
		tmp = (-1.0 / B) - (x / tan(B));
	}
	return tmp;
}

Fortran

real(8) function code(f, b, x)
    real(8), intent (in) :: f
    real(8), intent (in) :: b
    real(8), intent (in) :: x
    real(8) :: tmp
    if (f <= (-2.72d+14)) then
        tmp = ((-1.0d0) / sin(b)) - (x / b)
    else
        tmp = ((-1.0d0) / b) - (x / tan(b))
    end if
    code = tmp
end function

Java

public static double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.72e+14) {
		tmp = (-1.0 / Math.sin(B)) - (x / B);
	} else {
		tmp = (-1.0 / B) - (x / Math.tan(B));
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.72e+14:
		tmp = (-1.0 / math.sin(B)) - (x / B)
	else:
		tmp = (-1.0 / B) - (x / math.tan(B))
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.72e+14)
		tmp = Float64(Float64(-1.0 / sin(B)) - Float64(x / B));
	else
		tmp = Float64(Float64(-1.0 / B) - Float64(x / tan(B)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.72e+14)
		tmp = (-1.0 / sin(B)) - (x / B);
	else
		tmp = (-1.0 / B) - (x / tan(B));
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.72e+14], N[(N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision] - N[(x / B), $MachinePrecision]), $MachinePrecision], N[(N[(-1.0 / B), $MachinePrecision] - N[(x / N[Tan[B], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -2.72e14

   1. Initial program 58.8%

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

   3. Simplified 74.0%

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

   5. Taylor expanded in F around -inf 99.8%

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

   6. Taylor expanded in B around 0 79.7%

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

   if -2.72e14 < F

   1. Initial program 85.6%

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

   3. Simplified 89.5%

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

   5. Taylor expanded in F around -inf 34.6%

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

   6. Taylor expanded in B around 0 44.6%

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

Alternative 29: 37.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -36000000000:\\ \;\;\;\;\frac{-1}{\sin B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -36000000000.0) (/ -1.0 (sin B)) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -36000000000.0) {
		tmp = -1.0 / sin(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 <= (-36000000000.0d0)) then
        tmp = (-1.0d0) / sin(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 <= -36000000000.0) {
		tmp = -1.0 / Math.sin(B);
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -36000000000.0:
		tmp = -1.0 / math.sin(B)
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -36000000000.0)
		tmp = Float64(-1.0 / sin(B));
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -36000000000.0)
		tmp = -1.0 / sin(B);
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -36000000000.0], N[(-1.0 / N[Sin[B], $MachinePrecision]), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -3.6e10

   1. Initial program 59.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in F around -inf 99.8%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

   7. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

   9. Simplified 99.8%

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

   10. Taylor expanded in x around 0 64.2%

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

   if -3.6e10 < F

   1. Initial program 85.6%

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

   3. Simplified 89.5%

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

   5. Taylor expanded in F around -inf 34.2%

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

   6. Taylor expanded in B around 0 18.8%

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

      1. mul-1-neg 18.8%

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

      2. distribute-neg-frac2 18.8%

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

      3. +-commutative 18.8%

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

   8. Simplified 18.8%

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

   9. Taylor expanded in x around inf 29.6%

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

       1. associate-*r/ 29.6%

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

       2. neg-mul-1 29.6%

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

   11. Simplified 29.6%

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

Alternative 30: 36.6% accurate, 32.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{-87}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -2.3e-87) (/ (- -1.0 x) B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -2.3e-87) {
		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 <= (-2.3d-87)) 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 <= -2.3e-87) {
		tmp = (-1.0 - x) / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -2.3e-87:
		tmp = (-1.0 - x) / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -2.3e-87)
		tmp = Float64(Float64(-1.0 - x) / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -2.3e-87)
		tmp = (-1.0 - x) / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -2.3e-87], N[(N[(-1.0 - x), $MachinePrecision] / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -2.3000000000000001e-87

   1. Initial program 68.4%

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

   3. Simplified 80.0%

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

   5. Taylor expanded in F around -inf 85.7%

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

   6. Taylor expanded in B around 0 40.3%

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

      1. mul-1-neg 40.3%

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

      2. distribute-neg-frac2 40.3%

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

      3. +-commutative 40.3%

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

   8. Simplified 40.3%

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

   if -2.3000000000000001e-87 < F

   1. Initial program 84.0%

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

   3. Simplified 88.3%

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

   5. Taylor expanded in F around -inf 33.8%

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

   6. Taylor expanded in B around 0 18.6%

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

      1. mul-1-neg 18.6%

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

      2. distribute-neg-frac2 18.6%

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

      3. +-commutative 18.6%

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

   8. Simplified 18.6%

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

   9. Taylor expanded in x around inf 30.9%

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

       1. associate-*r/ 30.9%

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

       2. neg-mul-1 30.9%

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

   11. Simplified 30.9%

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

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.3 \cdot 10^{-87}:\\ \;\;\;\;\frac{-1 - x}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 31: 30.1% accurate, 35.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;F \leq -95000000000:\\ \;\;\;\;\frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{B}\\ \end{array} \end{array} \]

FPCore

(FPCore (F B x)
 :precision binary64
 (if (<= F -95000000000.0) (/ -1.0 B) (/ (- x) B)))

C

double code(double F, double B, double x) {
	double tmp;
	if (F <= -95000000000.0) {
		tmp = -1.0 / 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 <= (-95000000000.0d0)) then
        tmp = (-1.0d0) / 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 <= -95000000000.0) {
		tmp = -1.0 / B;
	} else {
		tmp = -x / B;
	}
	return tmp;
}

Python

def code(F, B, x):
	tmp = 0
	if F <= -95000000000.0:
		tmp = -1.0 / B
	else:
		tmp = -x / B
	return tmp

Julia

function code(F, B, x)
	tmp = 0.0
	if (F <= -95000000000.0)
		tmp = Float64(-1.0 / B);
	else
		tmp = Float64(Float64(-x) / B);
	end
	return tmp
end

MATLAB

function tmp_2 = code(F, B, x)
	tmp = 0.0;
	if (F <= -95000000000.0)
		tmp = -1.0 / B;
	else
		tmp = -x / B;
	end
	tmp_2 = tmp;
end

Wolfram

code[F_, B_, x_] := If[LessEqual[F, -95000000000.0], N[(-1.0 / B), $MachinePrecision], N[((-x) / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -9.5e10

   1. Initial program 59.4%

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

   3. Simplified 74.4%

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

   5. Taylor expanded in F around -inf 99.8%

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

   6. Taylor expanded in B around 0 46.0%

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

      1. mul-1-neg 46.0%

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

      2. distribute-neg-frac2 46.0%

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

      3. +-commutative 46.0%

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

   8. Simplified 46.0%

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

   9. Taylor expanded in x around 0 30.2%

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

   if -9.5e10 < F

   1. Initial program 85.6%

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

   3. Simplified 89.5%

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

   5. Taylor expanded in F around -inf 34.2%

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

   6. Taylor expanded in B around 0 18.8%

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

      1. mul-1-neg 18.8%

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

      2. distribute-neg-frac2 18.8%

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

      3. +-commutative 18.8%

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

   8. Simplified 18.8%

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

   9. Taylor expanded in x around inf 29.6%

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

       1. associate-*r/ 29.6%

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

       2. neg-mul-1 29.6%

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

   11. Simplified 29.6%

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

Alternative 32: 10.6% 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 78.9%

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

3. Simplified 85.6%

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

5. Taylor expanded in F around -inf 50.9%

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

6. Taylor expanded in B around 0 25.7%

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

   1. mul-1-neg 25.7%

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

   2. distribute-neg-frac2 25.7%

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

   3. +-commutative 25.7%

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

8. Simplified 25.7%

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

9. Taylor expanded in x around 0 10.4%

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

Reproduce

herbie shell --seed 2024150 
(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))))))